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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> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#nomenclature'>Nomenclature</a></li> <li><a href='#motivation_in_tannakian_formalism'>Motivation in Tannakian formalism</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#via_monoidal_categories'>Via monoidal categories</a></li> <li><a href='#an_explicit_definition'>An explicit definition</a></li> <li><a href='#via_rings'>Via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A\otimes A^{op}</annotation></semantics></math>-rings</a></li> </ul> <li><a href='#literature'>Literature</a></li> <ul> <li><a href='#commutative_case'>Commutative case</a></li> <li><a href='#noncommutative_case'>Noncommutative case</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A bialgebroid may be viewed as a multiobject generalization of a concept of a <a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, or a possibly noncommutative generalization of a space-algebra dual version of the concept of an internal category in spaces.</p> <h4 id="nomenclature">Nomenclature</h4> <p>This entry is about “associative” bialgebroid, see also the different concept of a <a class="existingWikiWord" href="/nlab/show/Lie+bialgebroid">Lie bialgebroid</a>.</p> <h4 id="motivation_in_tannakian_formalism">Motivation in Tannakian formalism</h4> <p>When a monoidal category has a <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> to a category of vector spaces over a field, one tries to “reconstruct” the category as the category of representations of the endomorphism object of a fiber functor. One often does not have a fiber functor to vector spaces but only to bimodules over some base 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>. Sometimes in such cases, the object of endomorphisms of the fiber functor form a bialgebroid over <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> and the category is equivalent to the category of representations of that bialgebroid.</p> <h2 id="definition">Definition</h2> <h3 id="via_monoidal_categories">Via monoidal categories</h3> <p>Given a unital (possibly noncommutative) ring <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> an <strong><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>-bialgebroid</strong> is an <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>-<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>-bimodule <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> (object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>R</mi></msub><msub><mi>ℳ</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">{}_R \mathcal{M}_R</annotation></semantics></math>) equipped with a structure of a comonoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>R</mi></msub><msub><mi>ℳ</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">{}_R \mathcal{M}_R</annotation></semantics></math> (i.e. an <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>-coring) and of a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mrow><msup><mi>R</mi> <mi>e</mi></msup></mrow></msub><msub><mi>ℳ</mi> <mrow><msup><mi>R</mi> <mi>e</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">{}_{R^e}\mathcal{M}_{R^e}</annotation></semantics></math> (i.e. an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>e</mi></msup></mrow><annotation encoding="application/x-tex">R^e</annotation></semantics></math>-ring), where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>e</mi></msup><mo>=</mo><msup><mi>R</mi> <mi>op</mi></msup><mo>⊗</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R^e = R^{op}\otimes R</annotation></semantics></math> is the enveloping ring of <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>; and the structures of a monoid and a comonoid satisfy certain compatibility conditions. These compatibility conditions are equivalent to the fact that the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mrow><msub><mo>⊗</mo> <mrow><msup><mi>R</mi> <mi>e</mi></msup></mrow></msub></mrow></msub><mi>H</mi><mo>:</mo><msub><mi>ℳ</mi> <mrow><msup><mi>R</mi> <mi>e</mi></msup></mrow></msub><mo>→</mo><msub><mi>ℳ</mi> <mrow><msup><mi>R</mi> <mi>e</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">{}_{\otimes_{R^e}} H : \mathcal{M}_{R^e}\to \mathcal{M}_{R^e}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/opmonoidal+monad">opmonoidal</a>. The category of <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>-comodules is by definition the category of comodules over the underlying <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>-coring.</p> <h3 id="an_explicit_definition">An explicit definition</h3> <p>If <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> is an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over some <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</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>, then a left associative <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>-bialgebroid is another associative <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>-algebra <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> together with the following additional maps: an algebra map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\alpha:A\to H</annotation></semantics></math> called the source map, an algebra map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\beta:A^{op}\to H</annotation></semantics></math> called the target map, so that the elements of the images of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> commute in <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>, therefore inducing an <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>-bimodule structure on <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> via the rule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>.</mo><mi>h</mi><mo>.</mo><mi>b</mi><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>β</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">a.h.b = \alpha(a)\beta(b) h</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>A</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">a,b\in A, h\in H</annotation></semantics></math>; an <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>-bimodule morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta:H\to H\otimes_A H</annotation></semantics></math> which is required to be a counital coassociative comultiplication on <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> in the monoidal category of <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>-bimodules with monoidal product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_A</annotation></semantics></math>. The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊗</mo><mi>A</mi><mo>∋</mo><mi>h</mi><mo>⊗</mo><mi>a</mi><mo>↦</mo><mi>ϵ</mi><mo stretchy="false">(</mo><mi>h</mi><mi>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">H\otimes A\ni h\otimes a\mapsto \epsilon(h\alpha(a))\in </annotation></semantics></math> must be a left action extending the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A\otimes A\to A</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>⊗</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\alpha\otimes id_A</annotation></semantics></math>. Furthermore, a compatibility between the 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 multiplications on <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 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊗</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">H\otimes H</annotation></semantics></math> is required. For a noncommutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> the tensor square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">H\otimes_A</annotation></semantics></math> is not an algebra, hence asking for a bialgebra-like compatibility that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta:H\to H\otimes_A H</annotation></semantics></math> is a morphism of <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>-algebras does not make sense. Instead, one requires that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">H\otimes_A H</annotation></semantics></math> has 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>-subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> which contains the image of <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 has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊗</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">H\otimes H</annotation></semantics></math>. Then one requires that the <a class="existingWikiWord" href="/nlab/show/corestriction">corestriction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><msup><mo stretchy="false">|</mo> <mi>T</mi></msup><mo>:</mo><mi>H</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\Delta|^T :H\to T</annotation></semantics></math> is a homomorphism of unital algebras. Under these conditions, one can make a canonical choice for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, namely the so called Takeuchi’s product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>×</mo> <mi>A</mi></msub><mi>H</mi><mo>⊂</mo><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">H\times_A H\subset H\otimes_A H</annotation></semantics></math>, which always inherits an associative multiplication along the projection from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊗</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">H\otimes H</annotation></semantics></math>.</p> <h3 id="via_rings">Via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A\otimes A^{op}</annotation></semantics></math>-rings</h3> <p>All modules and morphisms will be over a fixed ground commutative ring <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>.</p> <p>A <strong>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>-bialgebroid</strong> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A\otimes_k A^{op}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><msub><mi>μ</mi> <mi>H</mi></msub><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H,\mu_H,\eta)</annotation></semantics></math>, together with the <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>-bimodule map “comultiplication” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta : H\to H\otimes_A H</annotation></semantics></math>, which is coassociative and counital with a 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>, such that</p> <p>(i) the <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>-bimodule structure used on <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>.</mo><mi>h</mi><mo>.</mo><mi>a</mi><mo>′</mo><mo>:</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">a.h.a':= s(a)t(a')h</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mo>=</mo><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊗</mo><msub><mn>1</mn> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">s := \eta(-\otimes 1_A):A\to H</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>:</mo><mo>=</mo><mi>η</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>A</mi></msub><mo>⊗</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">t:=\eta(1_A\otimes -):A^{op}\to H</annotation></semantics></math> are the algebra maps induced by the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A\otimes A^{op}</annotation></semantics></math>-ring <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></p> <p>(ii) the coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta : H\to H\otimes_A H</annotation></semantics></math> corestricts to the <a class="existingWikiWord" href="/nlab/show/Takeuchi+product">Takeuchi product</a> and the corestriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi><msub><mo>×</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">\Delta : H\to H\times_A H</annotation></semantics></math> is 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>-algebra map, where the Takeuchi product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>×</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">H\times_A H</annotation></semantics></math> has a multiplication induced factorwise</p> <p>(iii) <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> is a left <a class="existingWikiWord" href="/nlab/show/character">character</a> on the <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>-ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><msub><mi>μ</mi> <mi>H</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H,\mu_H,s)</annotation></semantics></math>.</p> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">H\otimes_A H</annotation></semantics></math> is in general not an algebra, just an <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>-bimodule. That is why (ii) is needed. An equivalent condition to (ii) is the following: the formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>.</mo><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>k</mi> <mi>i</mi></msub><mo>⊗</mo><msub><mi>l</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⋅</mo><msub><mi>k</mi> <mi>i</mi></msub><mo>⊗</mo><msub><mi>h</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>⋅</mo><msub><mi>l</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">h.(\sum_i k_i \otimes l_i) = \sum_i h_{(1)}\cdot k_i \otimes h_{(2)} \cdot l_i</annotation></semantics></math> defines a well-defined action 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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">H\otimes_A H</annotation></semantics></math>.</p> <p>The definition of a <strong>right <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>-bialgebroid</strong> differs by the <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>-bimodule structure on <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> given instead by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>.</mo><mi>h</mi><mo>.</mo><mi>a</mi><mo>′</mo><mo>:</mo><mo>=</mo><mi>h</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a.h.a':= h s(a')t(a)</annotation></semantics></math> and the 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> is a <em>right</em> character on the <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>-coring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><msub><mi>μ</mi> <mi>H</mi></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H,\mu_H,t)</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> and <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> can be interchanged in the last requirement).</p> <h2 id="literature">Literature</h2> <p>Related notions: <a class="existingWikiWord" href="/nlab/show/Hopf+algebroid">Hopf algebroid</a></p> <h4 id="commutative_case">Commutative case</h4> <p>The commutative case is rather classical. See for example the appendix to</p> <ul> <li>Douglas C. Ravenel, <em>Complex cobordism and stable homotopy groups of spheres</em>, Pure and Applied Mathematics <strong>121</strong>. Academic Press Inc., Orlando, FL, 1986.</li> </ul> <h4 id="noncommutative_case">Noncommutative case</h4> <p>The first version of a bialgebroid over a noncommutative base was more narrow:</p> <ul> <li>M. Sweedler, <em>Groups of simple algebras</em>, Publ. IHES 44:79–189, 1974, <a href="http://www.numdam.org/item?id=PMIHES_1974__44__79_0">numdam</a></li> </ul> <p>A modern generality, but in different early formalism, is due to Takeuchi (who was motivated to generalize the results from the Sweedler’s paper), under the name of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>×</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\times_A</annotation></semantics></math>-bialgebra (as it involves the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>×</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\times_A</annotation></semantics></math>-product, nowdays called <a class="existingWikiWord" href="/nlab/show/Takeuchi+product">Takeuchi product</a>):</p> <ul> <li>M. Takeuchi, <em>Groups of algebras over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">A \times \bar{A}</annotation></semantics></math>, J. Math. Soc. Japan <strong>29</strong>, 459–492, 1977, <a href="http://www.ams.org/mathscinet-getitem?mr=0506407">MR0506407</a>, <a href="http://projecteuclid.org/euclid.jmsj/1240432948">euclid</a></em></li> </ul> <p>Lu introduces the name bialgebroid for a structure which is equivalent to the Takeuchi’s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>×</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\times_A</annotation></semantics></math>-bialgebra (though differently axiomatized there):</p> <ul> <li>Jiang-Hua Lu, <em>Hopf algebroids and quantum groupoids</em>, Int. J. Math. <strong>7</strong>, 1 (1996) pp. 47-70, <a href="http://arxiv.org/abs/q-alg/9505024">q-alg/9505024</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=95e:16037">MR95e:16037</a>, <a href="http://dx.doi.org/10.1142/S0129167X96000050">doi</a></li> </ul> <p>Modern treatments are in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriella+B%C3%B6hm">Gabriella Böhm</a>, <em>Internal bialgebroids, entwining structures and corings</em>, <a href="http://arxiv.org/abs/math/0311244">math.QA/0311244</a>, in: Algebraic structures and their representations, 207–226, Contemp. Math. <strong>376</strong>, Amer. Math. Soc. 2005.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G.+B%C3%B6hm">G. Böhm</a>, <em>Hopf algebroids</em>, (a chapter of) Handbook of algebra, <a href="http://arxiv.org/abs/0805.3806">arxiv:math.RA/0805.3806</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Korn%C3%A9l+Szlach%C3%A1nyi">Kornél Szlachányi</a>, <em>The monoidal Eilenberg–Moore construction and bialgebroids</em>, J. Pure Appl. Algebra <strong>182</strong>, no. 2–3 (2003) 287–315; <em>Fiber functors, monoidal sites and Tannaka duality for bialgebroids</em>, <a href="http://arxiv.org/abs/0907.1578">arxiv/0907.1578</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, G. Militaru, <em>Bialgebroids, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>×</mo> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">\times_{R}</annotation></semantics></math>-bialgebras and duality</em>, J. Algebra 251: 279-294, 2002, <a href="http://arxiv.org/abs/math/0012164">math.QA/0012164</a></p> </li> <li> <p>J. Donin, <a class="existingWikiWord" href="/nlab/show/Andrey+Mudrov">Andrey Mudrov</a>, <em>Quantum groupoids and dynamical categories</em>, J. Algebra <strong>296</strong> (2006), no. 2, 348–384, <a href="https://arxiv.org/abs/math/0311316">math.QA/0311316</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=2201046">MR2007b:17022</a>, <a href="https://doi.org/10.1016/j.jalgebra.2006.01.001">doi</a></p> </li> </ul> <p>There is also a notion of quasibialgebroid, where the coassociativity is weakened by a <a class="existingWikiWord" href="/nlab/show/bialgebra+cocycle">bialgebroid 3-cocycle</a>. See also <a class="existingWikiWord" href="/nlab/show/Hopf+algebroid">Hopf algebroid</a>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on September 28, 2024 at 13:58:07. 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