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href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_hopf_groups'>Relation to Hopf groups</a></li> <li><a href='#the_theorem_of_hopf_modules'>The theorem of Hopf modules</a></li> <li><a href='#relation_to_lie_algebras'>Relation to Lie algebras</a></li> <li><a href='#TannakaDuality'>Tannaka duality</a></li> <li><a href='#as_3vector_spaces'>As 3-vector spaces</a></li> <li><a href='#relation_to_frobenius_algebras'>Relation to Frobenius algebras</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#the_kacpaljutkin__hopf_algebra'>The Kac-Paljutkin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">H_8</annotation></semantics></math> Hopf algebra</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>Hopf algebra</em> is an abstraction of the properties of</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> on a finite group, and more generally, the algebra of regular functions on an affine algebraic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-group;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebra">universal enveloping algebra</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>,</p> </li> </ul> <p>where not only the <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> structure is remembered, but also the natural <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> structure, making it a <a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, as well as the algebraic structure induced by the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a>-operation in the group, called the <em><a class="existingWikiWord" href="/nlab/show/antipode">antipode</a></em>.</p> <p>More intrinsically, a Hopf algebra structure on an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> is precisely the structure such as to make its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> into a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> equipped with a <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> – this is the statement of <em><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></em> for Hopf algebras.</p> <p>Hopf algebras and their generalization to <a class="existingWikiWord" href="/nlab/show/Hopf+algebroids">Hopf algebroids</a> arise notably as <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebras">groupoid convolution algebras</a>. Another important source of Hopf algebras is <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a>, see at <em><a class="existingWikiWord" href="/nlab/show/combinatorial+Hopf+algebras">combinatorial Hopf algebras</a></em>.</p> <p>There is a wide variety of variations of the notion of Hopf algebra, relaxing <a class="existingWikiWord" href="/nlab/show/properties">properties</a> or adding <a class="existingWikiWord" href="/nlab/show/structure">structure</a>. Examples are <em><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebras">weak Hopf algebras</a></em>, <em><a class="existingWikiWord" href="/nlab/show/quasi-Hopf+algebras">quasi-Hopf algebras</a></em>, <em>(<a class="existingWikiWord" href="/nlab/show/quasi-triangular+Hopf+algebra">quasi</a>-)<a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebras">triangular Hopf algebras</a></em>, <em><a class="existingWikiWord" href="/nlab/show/quantum+groups">quantum groups</a></em>, <em><a class="existingWikiWord" href="/nlab/show/hopfish+algebras">hopfish algebras</a></em> etc. Most of these notions are systematized via <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> by the properties and structures on the coresponding <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a>, see at <em><a href="#TannakaDuality">Tannaka duality</a></em> below.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="Antipode"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/antipode">antipode</a>)</strong></p> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>Δ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,m,\eta,\Delta,\epsilon)</annotation></semantics></math> with multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>, comultiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>, unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>k</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\eta \colon k\to A</annotation></semantics></math> and counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\epsilon \colon A\to k</annotation></semantics></math> is called a <strong>Hopf algebra</strong> if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+function">linear function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S \colon A \longrightarrow A \,, </annotation></semantics></math></div> <p>then called the <strong><a class="existingWikiWord" href="/nlab/show/antipode">antipode</a></strong> or <strong>coinverse</strong>, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∘</mo><mo stretchy="false">(</mo><mi mathvariant="normal">id</mi><mo>⊗</mo><mi>S</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>Δ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>m</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>S</mi><mo>⊗</mo><mi mathvariant="normal">id</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>Δ</mi><mo>=</mo><mi>η</mi><mo>∘</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> m\circ(\mathrm{id}\otimes S)\circ \Delta \;=\; m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon </annotation></semantics></math></div> <p>(as a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A\to A</annotation></semantics></math>).</p> </div> <div class="num_defn" id="InvolutiveHopfAlgebra"> <h6 id="definition_3">Definition</h6> <p>If the antipode of a Hopf algebra (Def. <a class="maruku-ref" href="#Antipode"></a>) is an <a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involution</a>, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∘</mo><mi>S</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">S \circ S = id</annotation></semantics></math>, then one speaks of an <em><a class="existingWikiWord" href="/nlab/show/involutive+Hopf+algebra">involutive Hopf algebra</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If an antipode exists then it is unique, just the way that if inverses exist in a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> then they are unique. One sometimes prefers to have a <strong>skew-antipode</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>S</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{S}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mover><mi>S</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mover><mi>S</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>η</mi><mo>∘</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h_{(2)}\tilde{S}(h_{(1)}) = \tilde{S}(h_{(2)}) h_{(1)} = (\eta\circ\epsilon) (h)</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is an invertible antipode then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>S</mi><mo stretchy="false">˜</mo></mover><mo>−</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\tilde{S} - S^{-1}</annotation></semantics></math> is a skew-antipode. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is a bialgebra with a skew-antipode iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">H^{op}</annotation></semantics></math> (the same vector space, opposite product, the same coproduct) is a Hopf algebra.</p> <p>The unit of a Hopf algebra is a <a class="existingWikiWord" href="/nlab/show/grouplike+element">grouplike element</a>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">S(1)1=1</annotation></semantics></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">S(1)=1</annotation></semantics></math>. By linearity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∘</mo><mi>η</mi><mo>∘</mo><mi>ϵ</mi><mo>=</mo><mi>η</mi><mo>∘</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">S\circ\eta\circ\epsilon = \eta\circ\epsilon</annotation></semantics></math>.</p> </div> <div class="num_prop" id="AntipodeIsAnAntihomomorphism"> <h6 id="proposition">Proposition</h6> <p>The antipode is an <a class="existingWikiWord" href="/nlab/show/anti-homomorphism">anti-homomorphism</a> both of algebras and coalgebras (i.e. a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><msubsup><mi>A</mi> <mi>op</mi> <mi>cop</mi></msubsup></mrow><annotation encoding="application/x-tex">S \colon A\to A^{cop}_{op}</annotation></semantics></math>).</p> <p>In particular, an <a class="existingWikiWord" href="/nlab/show/involutive+Hopf+algebra">involutive Hopf algebra</a> (Def. <a class="maruku-ref" href="#InvolutiveHopfAlgebra"></a>) is a <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a>.</p> </div> <div class="proof"> <h6 id="proof_algebra_part">Proof (algebra part)</h6> <p>In <a class="existingWikiWord" href="/nlab/show/Sweedler+notation">Sweedler notation</a>, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">g,h\in A</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S((h g)_{(1)}) (h g)_{(2)} = \epsilon(h g)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S((h g)_{(1)}) h_{(2)}g_{(2)} = \epsilon(h)\epsilon(g)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mi>S</mi><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>S</mi><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">S((h g)_{(1)}) h_{(2)}g_{(2)} S g_{(3)} S h_{(3)} = \epsilon(h_{(1)})\epsilon(g_{(1)}) S g_{(2)} S h_{(2)}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(h_{(1)}g_{(1)}) \epsilon(h_{(2)})\epsilon(g_{(2)}) = (S g) (S h)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(h_{(1)}\epsilon(h_{(2)})g_{(1)}\epsilon(g_{(2)})) = (S g)(S h)</annotation></semantics></math></div> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>h</mi><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(h g) = (S g) (S h)</annotation></semantics></math>.</p> <p>For the coalgebra part, notice first that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mn>1</mn><mo>⊗</mo><mn>1</mn><mo>=</mo><mi>τ</mi><mo>∘</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi><mo>∘</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon(h)1\otimes 1 = \tau\circ\Delta(\epsilon(h)1)=\tau\circ\Delta(S h_{(1)}h_{(2)})</annotation></semantics></math>. Expand this as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(4)}\otimes h_{(3)})</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) (S h_{(5)}\otimes S h_{(6)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(3)}\otimes h_{(2)})(S h_{(4)}\otimes S h_{(5)})</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(1\otimes\epsilon(h_{(2)}))</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>τ</mi><mo>∘</mo><mi>Δ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>τ</mi><mo>∘</mo><mi>Δ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>ϵ</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = (\tau\circ\Delta)(S h_{(1)})(1\otimes\epsilon(h_{(2)})1) = (\tau\circ\Delta)(S h_{(1)}\epsilon(h_{(2)}))</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>S</mi><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>τ</mi><mo>∘</mo><mi>Δ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mi>h</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mo stretchy="false">(</mo><mi>S</mi><mi>h</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> (S h_{(1)}\otimes S h_{(2)})=(\tau\circ\Delta)(S h) = (S h)_{(2)}\otimes (S h)_{(1)}.</annotation></semantics></math></div></div> <p>The axiom that must be satisfied by the antipode looks like a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-linear version of the identity satisfied by the inverse map in a group bimonoid: taking a group element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, duplicating by the diagonal map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,g)</annotation></semantics></math>, taking the inverse of either component of this ordered pair, and then multiplying the two components, we obtain the identity element of our group.</p> <p>Just as an <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">algebra</a> is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> and a <a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a> is a <a class="existingWikiWord" href="/nlab/show/bimonoid">bimonoid</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>, a Hopf algebra is a <a class="existingWikiWord" href="/nlab/show/Hopf+monoid">Hopf monoid</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p><strong>Caution: convention in topology</strong></p> <p>In <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, it is common to define Hopf algebras without mentioning the antipode, since in many topological cases of interest it exists automatically. For example, this is the case if it is <a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> and “connected” in the sense that its degree-0 part is just the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> (a property possessed by the homology or cohomology of any connected space). In algebraic topology also the strict coassociativity is not always taken for granted.</p> </div> <h2 id="properties">Properties</h2> <h3 id="relation_to_hopf_groups">Relation to Hopf groups</h3> <p>Note that the definition of Hopf algebra (or, really, of <a class="existingWikiWord" href="/nlab/show/Hopf+monoid">Hopf monoid</a>) is <a class="existingWikiWord" href="/nlab/show/duality">self-dual</a>: a Hopf monoid in a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the same as a Hopf monoid in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">V^{op}</annotation></semantics></math> (i.e. a “Hopf comonoid”). Thus we can view a Hopf algebra as “like a group” in two different ways, depending on whether the group multiplication corresponds to the multiplication or the comultiplication of the Hopf algebra. The formal connections between Hopf monoids and group objects are:</p> <ol> <li> <p>A Hopf monoid in a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the same as a group object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. Such Hopf monoids are always <em>cocommutative</em> (that is, their underlying comonoid is cocommutative). This is because every object of a cartesian monoidal category is a cocommutative comonoid object in a unique way, and every morphism is a comonoid homomorphism.</p> </li> <li> <p>A <em>commutative</em> Hopf monoid in a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>V</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CMon(V)^{op}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(V)</annotation></semantics></math> is the category of commutative monoids in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, hence a <a class="existingWikiWord" href="/nlab/show/cogroup">cogroup</a> object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(V)</annotation></semantics></math> (a point highlighted by <a class="existingWikiWord" href="/nlab/show/Haynes+Miller">Haynes Miller</a>, see (<a href="#Ravenel86">Ravenel 86, appendix A1</a>)). This works because the tensor product of commutative algebras is the categorical <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, and hence the <a class="existingWikiWord" href="/nlab/show/product">product</a> in its <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a>. In particular, a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebra">commutative Hopf algebra</a> is the same as a group object in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Alg^{op}</annotation></semantics></math> of affine schemes.</p> </li> </ol> <p>Corresponding to these two, an ordinary group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> gives us two different Hopf algebras (here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/ground+ring">ground ring</a>):</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math> (the free vector space on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>), with multiplication given by the group operation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and comultiplication given by the diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>↦</mo><mi>g</mi><mo>⊗</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">g\mapsto g\otimes g</annotation></semantics></math>. This Hopf algebra is always cocommutative, and is commutative iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is abelian. It can be viewed as the result of applying the <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>Set</mi><mo>→</mo><mi>k</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">k[-]:Set \to k Mod</annotation></semantics></math> to the Hopf monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> </li> <li> <p>The function algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(G)</annotation></semantics></math> (the set of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">G\to k</annotation></semantics></math>), with comultiplication given by precomposition with the group operation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">k(G) \to k(G\times G) \cong k(G)\otimes k(G),</annotation></semantics></math></div> <p>and multiplication given by pointwise multiplication in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. In this case we need some finiteness or algebraicity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in order to guarantee <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(G\times G) \cong k(G)\otimes k(G)</annotation></semantics></math>. This Hopf algebra is always commutative, and is cocommutative iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is abelian.</p> </li> </ol> <p>Note that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is finite, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>≅</mo><mi>k</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k[G]\cong k(G)</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-modules, but the Hopf algebra structure is quite different.</p> <h3 id="the_theorem_of_hopf_modules">The theorem of Hopf modules</h3> <p>Hopf algebras can be characterized among bialgebras by the <a href="https://ncatlab.org/nlab/show/Hopf+module#fundamental_theorem_on_hopf_modules">fundamental theorem of Hopf modules</a>: the category of Hopf modules over a bialgebra is canonically equivalent to the category of vector spaces over the ground ring iff the bialgebra is a Hopf algebra. This categorical fact enables a definition of Hopf monoids in some setups that do not allow a sensible definition of antipode.</p> <h3 id="relation_to_lie_algebras">Relation to Lie algebras</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Milnor-Moore+theorem">Milnor-Moore theorem</a></li> </ul> <h3 id="TannakaDuality">Tannaka duality</h3> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> (finite dimensional) over the underlying <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> of a Hopf algebra canonically inherits the structure of an <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> such that the forgetful <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> to <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over the ground field is a <a class="existingWikiWord" href="/nlab/show/strict+monoidal+functor">strict monoidal functor</a>.</p> <p>The statement of <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for Hopf algebras is that this property characterizes Hopf algebras. (See for instance (<a href="#Bakke">Bakke</a>))</p> <p>For generalization of this characterization to <a class="existingWikiWord" href="/nlab/show/quasi-Hopf+algebras">quasi-Hopf algebras</a> and <a class="existingWikiWord" href="/nlab/show/hopfish+algebras">hopfish algebras</a> see (<a href="#Vercruysse">Vercruysse</a>).</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h3 id="as_3vector_spaces">As 3-vector spaces</h3> <p>A Hopf algebra structure on an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> canonically defines on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> the structure of an algebra object <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of algebras, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>s and bimodule homomorphisms.</p> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a> this allows to identify Hopf algebras with certain <em>3-vector spaces</em> .</p> <p>(For instance (<a href="http://ncatlab.org/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups">FHLT, p. 27</a>)).</p> <p>More general 3-vector spaces are given by <em><a class="existingWikiWord" href="/nlab/show/hopfish+algebras">hopfish algebras</a></em> and generally by <a class="existingWikiWord" href="/nlab/show/sesquiunital+sesquialgebras">sesquiunital sesquialgebras</a>.</p> <h3 id="relation_to_frobenius_algebras">Relation to Frobenius algebras</h3> <p>Both Hopf algebras and <a class="existingWikiWord" href="/nlab/show/Frobenius+algebras">Frobenius algebras</a> are at the same time an <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and a <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, albeit with different additional structures and properties. Nevertheless, one may ask whether there is any relation between these two. This leads to the following result due to <a href="#LS69">Larson & Sweedler (1969)</a>.</p> <div class="num_prop" id="HopfAlgebraIsFrobeniusAlgebra"> <h6 id="proposition_2">Proposition</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite-dimensional</a> Hopf algebra can be endowed with the structure of a Frobenius algebra.</p> </div> <p>The finite-dimensional condition is expected since all Frobenius algebras are finite-dimensional. A nontrivial part of this result is that, while a finite-dimensional Hopf algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> already has a counit, this is not be the same counit that realizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a Frobenius algebra. Rather, it is an <em>integral</em> (see <a href="#LS69">Larson & Sweedler (1969)</a> for details on this), which for finite-dimensional Hopf algebras always exist and are unique up to scaling.</p> <p>In fact, more is true. In <a href="#FSS11">Fuchs, Schweigert & Stigner (2011)</a> a <a href="https://ncatlab.org/nlab/show/Frobenius+algebra#special_frobenius_algebras">symmetric special Frobenius algebra</a> is constructed from a particular kind of Hopf algebra.</p> <div class="num_prop" id="DualHopfAlgebraIsSSFrobeniusAlgebra"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> be a finite-dimensional factorizable ribbon Hopf algebra, with multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>, comultiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>, and antipode <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> be a left-integral element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> a right-cointegral of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is finite-dimensional, this is equivalent to an integral of the dual Hopf algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">H^*</annotation></semantics></math>) such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∘</mo><mi>Λ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\lambda\circ\Lambda=1</annotation></semantics></math>. Then the dual vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">H^*</annotation></semantics></math> endowed with: a unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϵ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\epsilon^*</annotation></semantics></math>, a multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\Delta^*</annotation></semantics></math>, a counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\Lambda^*</annotation></semantics></math>, and comultiplication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mtext>id</mtext> <mi>H</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>∘</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mtext>id</mtext> <mi>H</mi></msub><mo>⊗</mo><mi>S</mi><mo>⊗</mo><msub><mtext>id</mtext> <mi>H</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo>⊗</mo><msub><mtext>id</mtext> <mi>H</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\Delta_F=((\text{id}_H\otimes (\lambda\circ \mu))\circ (\text{id}_H\otimes S\otimes\text{id}_H)\circ (\Delta\otimes\text{id}_H))^*</annotation></semantics></math></div> <p>is a symmetric Frobenius object in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Bimod</mtext><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Bimod}(H,H)</annotation></semantics></math> of bimodules of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, and it is furthermore special iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is is semisimple.</p> </div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of a <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> <a class="existingWikiWord" href="/nlab/show/H-space">H-space</a> (for instance a <a class="existingWikiWord" href="/nlab/show/based+loop+space">based loop space</a>) is a <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> via its <a class="existingWikiWord" href="/nlab/show/Pontrjagin+ring">Pontrjagin ring</a>-<a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a> – see at <em><a class="existingWikiWord" href="/nlab/show/homology+of+loop+spaces">homology of loop spaces</a></em> for more</p> </li> </ul> <h3 id="the_kacpaljutkin__hopf_algebra">The Kac-Paljutkin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">H_8</annotation></semantics></math> Hopf algebra</h3> <p>The Kac-Paljutkin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">H_8</annotation></semantics></math> Hopf algebra was first described in <a href="#KP66">Kac & Paljutkin (1966)</a> and it is the Hopf algebra with the smallest dimension that is <a class="existingWikiWord" href="/nlab/show/semisimple+object">semisimple</a>, <a class="existingWikiWord" href="/nlab/show/commutative+algebra">noncommutative</a>, and <a class="existingWikiWord" href="/nlab/show/cocommutative+coalgebra">noncocommutative</a>. It is also <a class="existingWikiWord" href="/nlab/show/self-dual+object">self-dual</a> (see e.g. <a href="#Burciu17">Burciu (2017)</a>). One presentation is in terms of generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x,y,z\}</annotation></semantics></math> satisfying the relations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>z</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2=y^2=z^2=1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>xz</mi><mo>=</mo><mi>zx</mi></mrow><annotation encoding="application/x-tex">xz=zx</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>zy</mi><mo>=</mo><mi>yz</mi></mrow><annotation encoding="application/x-tex">zy=yz</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>xyz</mi><mo>=</mo><mi>yx</mi></mrow><annotation encoding="application/x-tex">xyz=yx</annotation></semantics></math>, along with comultiplication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>xe</mi> <mn>0</mn></msub><mo>⊗</mo><mi>x</mi><mo>+</mo><msub><mi>xe</mi> <mn>1</mn></msub><mo>⊗</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\Delta(x)=xe_0\otimes x+xe_1\otimes y</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ye</mi> <mn>1</mn></msub><mo>⊗</mo><mi>x</mi><mo>+</mo><msub><mi>ye</mi> <mn>0</mn></msub><mo>⊗</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\Delta(y)=ye_1\otimes x+ye_0\otimes y</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>z</mi><mo>⊗</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">\Delta(z)=z\otimes z</annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>0</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_0=\frac{1}{2}(1+z)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_1=\frac{1}{2}(1-z)</annotation></semantics></math>) whose counit is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\epsilon(x)=\epsilon(y)=\epsilon(z)=1</annotation></semantics></math>. The antipode map is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>xe</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>ye</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S(x)=xe_0+ye_1</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>xe</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>ye</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">S(y)=xe_1+ye_0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">S(z)=z</annotation></semantics></math></div> <p>. Its <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> is the unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2\times\mathbb{Z}_2</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Tambara-Yamagami+category">Tambara-Yamagami category</a> that admits a <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and is not the representation category of some group.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-Hopf+algebra">quasi-Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a>/<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+algebroid">Hopf algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+C-star+algebra">Hopf C-star algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+monoid">Hopf monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+monad">Hopf monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/change+of+rings+theorem">change of rings theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+ring+spectrum">Hopf ring spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin+product">Pontrjagin product</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em>Hopf algebras</em>, lectures, <a href="http://www.maths.gla.ac.uk/~kab/Hopf%20lects%201-8.pdf">pdf</a></p> </li> <li> <p>Nicolas Andruskiewitsch, Walter Ferrer Santos, <em>The beginnings of the theory of Hopf algebras</em>, Acta Appl Math <strong>108</strong> (2009) 3-17 [<a href="https://arxiv.org/abs/0901.2460">arXiv:0901.2460</a>]</p> </li> </ul> <p>The diagrammatic definition of a Hopf algebra, is also in the <a href="http://en.wikipedia.org/wiki/Hopf_algebra#Formal_definition">Wikipedia entry</a>.</p> <ul> <li> <p>Eiichi Abe, <em>Hopf algebras</em>, Cambridge UP 1980.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Cartier">Pierre Cartier</a>, <em>A primer on Hopf algebras</em>, Frontiers in number theory, physics, and geometry II, 537–615, preprint IHÉS 2006-40, 81p (<a href="https://doi.org/10.1007/978-3-540-30308-4_12">doi:10.1007/978-3-540-30308-4_12</a>, <a href="http://preprints.ihes.fr/2006/M/M-06-40.pdf">pdf</a>, <a href="http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/CartierHopfAlgebras.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>V. G. <a class="existingWikiWord" href="/nlab/show/Drinfel%27d">Drinfel'd</a>, <em>Quantum groups</em>, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 2 798–820, AMS 1987, <a href="http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu">djvu:1.3 M</a>, <a href="http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf">pdf:2.5 M</a></p> </li> <li> <p>G. Hochschild, <em>Introduction to algebraic group schemes</em>, 1971</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shahn+Majid">Shahn Majid</a>, <em>Foundations of quantum group theory</em>, Cambridge University Press 1995, 2000.</p> </li> <li id="MilnorMoore65"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>On the structure of Hopf algebras</em>, Annals of Math. <strong>81</strong> (1965), 211-264 (<a href="https://doi.org/10.2307/1970615">doi:10.2307/1970615</a>, <a href="http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf">pdf</a>)</p> </li> <li> <p>Susan Montgomery, <em>Hopf algebras and their actions on rings</em>, AMS 1994, 240p.</p> </li> <li> <p>B. Parshall, J.Wang, <em>Quantum linear groups</em>, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Moss+Sweedler">Moss Sweedler</a>, <em>Hopf algebras</em>, Benjamin 1969.</p> </li> <li> <p>William C. Waterhouse, <em>Introduction to affine group schemes</em>, Graduate Texts in Mathematics <strong>66</strong>, Springer 1979. xi+164 pp.</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for Hopf algebras and their generalization is alluded to in</p> <ul> <li id="Vercruysse">Joost Vercruysse, <em>Hopf algebras—Variant notions and reconstruction theorems</em> (<a href="http://arxiv.org/abs/1202.3613">arXiv:1202.3613</a>)</li> </ul> <p>and discussed in detail in</p> <ul> <li id="Bakke">Tørris Koløen Bakke, <em>Hopf algebras and monoidal categories</em> (2007) (<a href="https://munin.uit.no/bitstream/handle/10037/1084/finalthesis.pdf">pdf</a>)</li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> with an eye towards <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a> and the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <a class="existingWikiWord" href="/nlab/show/W.+Stephen+Wilson">W. Stephen Wilson</a>, <em>The Hopf ring for complex cobordism</em>, Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (<a href="https://doi.org/10.1016/0022-4049(77)90070-6">doi:10.1016/0022-4049(77)90070-6</a>, <a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/The-Hopf-ring-for-complex-cobordism/bams/1183536024.full?tab=ArticleLink">euclid</a>, <a href="https://people.math.rochester.edu/faculty/doug/mypapers/hopfring.pdf">pdf</a>)</p> <blockquote> <p>(for <a class="existingWikiWord" href="/nlab/show/MU">MU</a>)</p> </blockquote> </li> <li id="Ravenel86"> <p><a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, appendix A1 of: <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em>, 1986 (<a href="http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Bott periodicity and Hopf rings</em>, 1992 (<a href="https://neil-strickland.staff.shef.ac.uk/research/thesis.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/StricklandHopfRings.pdf" title="pdf">pdf</a>)</p> <blockquote> <p>(for <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W.+Stephen+Wilson">W. Stephen Wilson</a>, <em>Hopf rings in algebraic topology</em>, Expositiones Mathematicae, 18:369–388, 2000 (<a href="https://math.jhu.edu/~wsw/papers2/math/39-hopf-rings-expo-2000.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <em>Hopf algebras, quantum groups and topological field theory</em> (2022) [<a href="https://www.math.uni-hamburg.de/home/schweigert/skripten/hskript.pdf">pdf</a>]</p> </li> </ul> <p>For Hopf algebras in generative linguistics, see:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, <a class="existingWikiWord" href="/nlab/show/Noam+Chomsky">Noam Chomsky</a>, Robert Berwick, <em>Mathematical Structure of Syntactic Merge</em> (<a href="https://arxiv.org/abs/2305.18278">arXiv:2305.18278</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, Robert Berwick, <a class="existingWikiWord" href="/nlab/show/Noam+Chomsky">Noam Chomsky</a>, <em>Old and New Minimalism: a Hopf algebra comparison</em> (<a href="https://arxiv.org/abs/2306.10270">arXiv:2306.10270</a>)</p> </li> </ul> <p>The construction of a Frobenius algebra structure on finite-dimensional Hopf algebras due to</p> <ul> <li id="LS69">Richard Larson, <a class="existingWikiWord" href="/nlab/show/Moss+Sweedler">Moss Sweedler</a>. <em>An Associative Orthogonal Bilinear Form for Hopf Algebras</em>. American Journal of Mathematics, Vol. 91, No. 1 (Jan., 1969), pp. 75-94 (20 pages). (<a href="https://doi.org/10.2307/2373270">doi</a>)</li> </ul> <p>See also</p> <ul> <li>Alfons Van Daele. <em>Reflections on the Larson-Sweedler theorem for (weak) multiplier Hopf algebras</em> (2024). (<a href="https://arxiv.org/abs/2404.15046">arXiv:2404.15046</a>).</li> </ul> <p>Lecture notes on Hopf algebras (and <a class="existingWikiWord" href="/nlab/show/quantum+groups">quantum groups</a>) in view of <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>: <em>Hopf algebras, quantum groups and topological field theory</em>, lecture notes (2022) [<a href="https://www.math.uni-hamburg.de/home/schweigert/skripten/hskript.pdf">pdf</a>]</li> </ul> <p>The proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">H^*</annotation></semantics></math> can be endowed with the structure of a symmetric special Frobenius object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Bimod</mtext><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Bimod}(H,H)</annotation></semantics></math>:</p> <ul> <li id="FSS11"><a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, Carl Stigner: <em>Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms</em>. Journal of Algebra <strong>363</strong>, 1 August 2012, Pages 29-72. (<a href="https://doi.org/10.1016/j.jalgebra.2012.04.008">doi</a>)</li> </ul> <p>On the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">H_8</annotation></semantics></math> Hopf algebra</p> <ul> <li id="KP66"> <p><a class="existingWikiWord" href="/nlab/show/G.+I.+Kac">G. I. Kac</a>, V. G. Paljutkin, <em>Finite ring groups_, Trans. Moscow Math. Soc. 15 (1966), 251–294.</em></p> </li> <li id="Burciu17"> <p>Sebastian Burciu. <em>Representations and conjugacy classes of semisimple quasitriangular Hopf algebras</em>. (2017) ((<a href="https://arxiv.org/abs/1709.02176">arXiv:1709.02176</a>).</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 7, 2025 at 19:21:50. 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