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Leibniz algebra in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><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="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_lie_algebras_in_lodaypirashvili_category'>Relation to Lie algebras in Loday-Pirashvili category</a></li> <li><a href='#corepresentation_representation_crossed_module'>Corepresentation, representation, crossed module</a></li> <li><a href='#abelian_extensions'>Abelian extensions</a></li> <li><a href='#homology_and_cohomology'>Homology and cohomology</a></li> <li><a href='#relation_to_zinbiel_algebras'>Relation to Zinbiel algebras</a></li> <li><a href='#LieThirdTheorem'>Lie’s third theorem for Leibniz algebras</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#basic_examples'>Basic examples</a></li> <li><a href='#FromLieModulesAndEmbeddingTensors'>From Lie modules and embedding tensors</a></li> <li><a href='#LeibnizAlgebrasFromDgLieAlgebras'>From dg-Lie algebras</a></li> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_dglie_algebras_and_tensor_hierarchies'>Relation to dg-Lie algebras and tensor hierarchies</a></li> <li><a href='#lie_integration'>Lie integration</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Leibniz algebra</em> is like a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, but without the condition that the <a class="existingWikiWord" href="/nlab/show/magma">product</a>, often still written as a bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math>, is skew-symmetric. The <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> however is retained as a condition in its form as the <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>-property of the product over itself. In view of the analogous <a class="existingWikiWord" href="/nlab/show/product+law">product law</a> of <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> (also a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>-property) attributed to <a class="existingWikiWord" href="/nlab/show/Gottfried+Leibniz">Gottfried Leibniz</a>, this is then called the <em>Leibniz identity</em> which gives Leibniz algebras their modern name (<a href="#Loday93">Loday 93</a>, <a href="#LodayPirashvili93">Loday-Pirashvili 93</a>) even though the concept itself is older (<a href="#Blokh65">Blokh 65</a>).</p> <p>Leibniz algebras were motivated in <a href="#Cuvier91">Cuvier 91</a>, <a href="#LodayPirashvili93">Loday-Pirashvili 93</a> as generalizing the relation between <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> and <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> (<a href="cyclic+homology#LodayQuillen84">Loday-Quillen 84</a>) to one between <em>Leibniz cohomology</em> and <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>: Where the nilpotency of the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> in the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> that compute <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> is equivalent to the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> in the corresponding Lie algebra, Leibniz cohomology is defined on non-skew symmetric <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> where now it is the generalization of the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> in form of the <em>Leibniz rule</em> <a class="maruku-eqref" href="#eq:LeibnizRule">(1)</a> which still guarantees the nilpotency of the differential.</p> <p>More recently, Leibniz algebras have been argued to clarify the nature of the <a class="existingWikiWord" href="/nlab/show/embedding+tensor">embedding tensor</a> and the resulting <a class="existingWikiWord" href="/nlab/show/tensor+hierarchies">tensor hierarchies</a> in <a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a> (<a href="#Lavau17">Lavau 17</a>).</p> <h2 id="definition">Definition</h2> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> (typically a <a class="existingWikiWord" href="/nlab/show/field">field</a>).</p> <div class="num_defn" id="LeftLeibnizAlgebra"> <h6 id="definition_2">Definition</h6> <p><strong>(left Leibniz algebra)</strong></p> <p>A <em>left Leibniz <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</em> (or: <em>left Loday algebra</em>) is</p> <ul> <li>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/module">module</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> (hence 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/vector+space">vector space</a> if <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 a <a class="existingWikiWord" href="/nlab/show/field">field</a> and then often required to be <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a>)</li> </ul> <p>equipped with</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>, the <em><a class="existingWikiWord" href="/nlab/show/magma">product</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>v</mi><mo>⋅</mo><mi>w</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A &\longrightarrow& A \\ (v,w) &\mapsto& v \cdot w } </annotation></semantics></math></div></li> </ul> <p>such that</p> <ul> <li> <p>the product satisfies the <em>left Leibniz identity</em>, saying that</p> <div class="maruku-equation" id="eq:LeibnizRule"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><msub><mi>v</mi> <mn>2</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> v_1 \cdot (v_ 2 \cdot v_3) \;=\; (v_1 \cdot v_2) \cdot v_3 \;+\; v_2 \cdot (v_1 \cdot v_3) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v_i \in A</annotation></semantics></math>.</p> </li> </ul> <p>This says equivalently that the operations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v \cdot (-) \colon A \to A</annotation></semantics></math> of left-multiplication by elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v \in A</annotation></semantics></math> via the given <a class="existingWikiWord" href="/nlab/show/magma">product</a> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of the product itself, whence the name (paying tribute to <a class="existingWikiWord" href="/nlab/show/Gottfried+Leibniz">Gottfried Leibniz</a>‘s <a class="existingWikiWord" href="/nlab/show/product+rule">product rule</a> of <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>).</p> </div> <p>Analogously there is the concept of <em>right Leibniz algebras</em> in the evident way.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_lie_algebras_in_lodaypirashvili_category">Relation to Lie algebras in Loday-Pirashvili category</h3> <p>There is a remarkable observation of Loday and Pirashvili that in the <a class="existingWikiWord" href="/nlab/show/Loday%E2%80%93Pirashvili+tensor+category">Loday–Pirashvili tensor category</a> of linear maps with (exotic) “infinitesimal tensor product”, the category of internal Lie algebras has the category of, say left, Leibniz <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 as a full subcategory.</p> <h3 id="corepresentation_representation_crossed_module">Corepresentation, representation, crossed module</h3> <p>Both a representation and a corepresentation of a right Leibniz <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>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> involve 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>-module <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> and two <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 maps “actions” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>⊗</mo><mi>𝔤</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">M\otimes\mathfrak{g}\to M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>⊗</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}\otimes M\to M</annotation></semantics></math> with 3 axioms.</p> <p>For representations:</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>m</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[m, [x, y]] = [[m, x], y] - [[m, y], x]</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>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[x, [a, y]] = [[x, m], y] - [[x, y], m]</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>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[x, [y, m]] = [[x, y], m] - [[x, m], y]</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x,y\in\mathfrak{g}</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">m\in M</annotation></semantics></math>.</p> <p>For corepresentatons:</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><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[[x, y], m] = [x, [y, m]] - [y, [x, m]] </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>y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[y, [a, x]] = [[y, m], x] - [m, [x, 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><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">[[m, x], y] = [m, [x, y]] - [[y, m], x].</annotation></semantics></math></div> <p>If the two “actions” are symmetric, i.e. <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>m</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[x,m] + [m,x] = 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">m\in M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x\in\mathfrak{g}</annotation></semantics></math> then all the 6 axioms of representation or corepresentation are equivalent. If <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> is underlying a Leibniz algebra then an action 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">\mathfrak{g}</annotation></semantics></math> on <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> is by definition symmetric, hence all the 6 equivalent conditions hold.</p> <p>A map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">t : \mathfrak{g}\to\mathfrak{b}</annotation></semantics></math> together with an action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">\mathfrak{b}</annotation></semantics></math> on <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">\mathfrak{g}</annotation></semantics></math> is a Leibniz crossed module if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>b</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>b</mi><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>t</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>for</mi><mi>all</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>b</mi><mo>∈</mo><mi>𝔟</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> t([b,g])= [b,t(g)],\,\,\,t([g,b])=[t(g),b],\,\,\,\, for all\,\,\, b\in\mathfrak{b}, g' \in\mathfrak{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><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo>′</mo><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>for</mi><mi>all</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> [g, t(g')] = [g, g'] = [t(g), g'],\,\,\,\, for all\,\,\, g, g' \in\mathfrak{g} </annotation></semantics></math></div> <h3 id="abelian_extensions">Abelian extensions</h3> <p>Abelian extension of right Leibniz algebras is a split short exact sequence 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>-modules</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>M</mi><mo>→</mo><mi>𝔥</mi><mo>→</mo><mi>𝔤</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0\to M \to \mathfrak{h}\to \mathfrak{g}\to 0 </annotation></semantics></math></div> <p>where the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔥</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{h}\to\mathfrak{g}</annotation></semantics></math> is a morphism of Leibniz algebras, and <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> is equipped with induced action 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">\mathfrak{g}</annotation></semantics></math>. The isomorphisms of extensions 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">\mathfrak{g}</annotation></semantics></math> by <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> with fixed action are defined as usual. This way we obtain a set of equivalence classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext(\mathfrak{g},M)</annotation></semantics></math>. To classify the extensions one looks for compatible Leibniz brackets on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>⊕</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">M\oplus \mathfrak{g}</annotation></semantics></math>. The general form of a bracket is</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><mo stretchy="false">(</mo><msub><mi>m</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>m</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>m</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>m</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> [(m_1,x_1),(m_2,x_2)] = ([m_1, x_2] + [x_1, m_2] + f(x_1, x_2), [x_1, x_2]),</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1,x_2)</annotation></semantics></math> satisfy the following 2-cocycle identity:</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>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [x, f(y, z)] + [f(x, z), y] - [f(x, y), z] = f([x, y], z) - f([x, z], y) - f(x, [y, z]) </annotation></semantics></math></div> <p>The extension is <strong>split</strong> in the category of Leibniz algebras if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <em>boundary</em> i.e. 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>-module map <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>M</mi></mrow><annotation encoding="application/x-tex">g:\mathfrak{g}\to M</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> f(x, y) = [x, g(y)] + [g(x), y] - g([x, y]), \,\,\,x,y,\in\mathfrak{g} </annotation></semantics></math></div> <p>As for the Lie algebras, the group of abelian extensions agrees with the 2-cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>HL</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HL^2(\mathfrak{g},M)</annotation></semantics></math>.</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>-linear <strong>derivation</strong> of a right Leibniz algebra <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">\mathfrak{g}</annotation></semantics></math> with values in its representation <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> 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>-linear map satisfying the Leibniz property with respect to the bracket:</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><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \delta([x,y]) = [\delta(x),y]+[x,\delta(y)] </annotation></semantics></math></div> <p>Such derivations form 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>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Der(\mathfrak{g},M)</annotation></semantics></math>.</p> <h3 id="homology_and_cohomology">Homology and cohomology</h3> <p>The homology and cohomology of Leibniz algebra <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">\mathfrak{g}</annotation></semantics></math> with abelian <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>-module of coefficients, which is a corepresentation <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> in the case of homology and a representation <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> in the case of cohomology:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>HL</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Tor</mi> <mo>*</mo> <mrow><mi>U</mi><mi>𝔤</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mi>Lie</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> HL_*(\mathfrak{g},A) = Tor^{U\mathfrak{g}}_*(U(\mathfrak{g}_{Lie}),A) ,</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>HL</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Ext</mi> <mrow><mi>U</mi><mi>𝔤</mi></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mi>Lie</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> HL^*(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(U(\mathfrak{g}_{Lie}),A) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mi>Lie</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathfrak{g}_{Lie})</annotation></semantics></math> is the universal enveloping of the maximal Lie algebra quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>Lie</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{Lie}</annotation></semantics></math> 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">\mathfrak{g}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">U\mathfrak{g}</annotation></semantics></math> is the universal enveloping of a Leibniz algebra <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">\mathfrak{g}</annotation></semantics></math>.</p> <p>Fopr <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n\geq 0</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cocycles are elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^n(\mathfrak{g}, M) = Hom_k(\mathfrak{g}^{\otimes n}, M)</annotation></semantics></math>, satisfying the corresponding abelian cocycle condition determined by the differential</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mi>n</mi></msup><mo>:</mo><msup><mi>C</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d^n : C^n(\mathfrak{g}, M)\to C^{n+1}(\mathfrak{g}, M) </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><msup><mi>d</mi> <mi>n</mi></msup><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mover><mi>x</mi><mo stretchy="false">^</mo></mover> <mi>i</mi></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (d^n f) (x_1, . . . , x_{n+1}) = [x_1,f(x_2,\ldots,x_{n+1})] +\sum_{n+1}^{i=2} (-1)^i [f(x_1,\ldots, \hat{x}_i, \ldots, x_{n+1}), x_i] </annotation></semantics></math></div> <p>Notice a difference from the Lie algebra cocycles where instead of a tensor power we have an external power. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>HL</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>d</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HL^*(\mathfrak{g},M) = H^*(C^*(\mathfrak{g}, M),d^*)</annotation></semantics></math>.</p> <p>There are standard interpretations of cocycles in low dimensions. For example for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>HL</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HL^0(\mathfrak{g}, M)</annotation></semantics></math> is the submodule of invariants. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math> there is a natural projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>HL</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Der(\mathfrak{g},M)\to HL^1(\mathfrak{g},M)</annotation></semantics></math> whose kernel is generated by inner derivations.</p> <h3 id="relation_to_zinbiel_algebras">Relation to Zinbiel algebras</h3> <p>The Leibniz operad is quadratic Koszul algebra whose Koszul dual operad is called the operad of dual Leibniz algebras or of <a class="existingWikiWord" href="/nlab/show/Zinbiel+algebra">Zinbiel algebra</a>s, see there.</p> <h3 id="LieThirdTheorem">Lie’s third theorem for Leibniz algebras</h3> <p>In complete analogy to the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> and the category of <a class="existingWikiWord" href="/nlab/show/local+Lie+groups">local Lie groups</a> (<a class="existingWikiWord" href="/nlab/show/Lie%27s+third+theorem">Lie's third theorem</a>), the <a class="existingWikiWord" href="/nlab/show/category">category</a> of Leibniz algebras is equivalent to the category of local pointed augmented Lie <a class="existingWikiWord" href="/nlab/show/racks">racks</a>. See <a href="#Covez10">Covez 10</a>.</p> <p>This equivalence restricts to the equivalence between <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> and <a class="existingWikiWord" href="/nlab/show/local+Lie+groups">local Lie groups</a>.</p> <p>Here a local pointed augmented Lie <a class="existingWikiWord" href="/nlab/show/rack">rack</a> is a pointed augmented <a class="existingWikiWord" href="/nlab/show/rack">rack</a> object in the category of germs of pointed smooth manifolds. An <strong>augmented rack</strong> is a triple <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>X</mi><mo>,</mo><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,X,p\colon X\to G)</annotation></semantics></math>, where <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 a <a class="existingWikiWord" href="/nlab/show/group">group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/G-set">G-set</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">map of sets</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo>⋅</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">p(g\cdot x)=g p(x)g^{-1}</annotation></semantics></math>. An augmented rack is <strong>pointed</strong> if there is an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">1\in X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p(1)=1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>⋅</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g\cdot 1=1</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g\in G</annotation></semantics></math>.</p> <p>If <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>X</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,X,p)</annotation></semantics></math> is an augmented rack, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> can be made into a <a class="existingWikiWord" href="/nlab/show/rack">rack</a> as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>▹</mo><mi>y</mi><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x\triangleright y = p(x)\cdot y</annotation></semantics></math>. If <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>X</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,X,p)</annotation></semantics></math> is a pointed augmented rack, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a pointed <a class="existingWikiWord" href="/nlab/show/rack">rack</a>, meaning there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">1\in X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>▹</mo><mi>x</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">1\triangleright x=x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>▹</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x\triangleright 1=1</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <h3 id="basic_examples">Basic examples</h3> <ul> <li>Every <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> is a Leibniz algebra that happens to have skew-symmetric product. Conversely, a Leibniz algebra with skew-symmetric product is a Lie algebra.</li> </ul> <h3 id="FromLieModulesAndEmbeddingTensors">From Lie modules and embedding tensors</h3> <div class="num_prop" id="LeibnizAlgebraFromLieModuleAndEmbeddingTensor"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a> from <a class="existingWikiWord" href="/nlab/show/Lie+module">Lie module</a> with <a class="existingWikiWord" href="/nlab/show/embedding+tensor">embedding tensor</a>)</strong></p> <p>Let</p> <ul> <li> <p><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">\mathfrak{g}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>⊗</mo><mi>V</mi><mover><mo>⟶</mo><mi>ρ</mi></mover><mi>V</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g} \otimes V \overset{\rho}{\longrightarrow} V </annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+representation">Lie algebra representation</a> 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">\mathfrak{g}</annotation></semantics></math>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>⟶</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\Theta \colon V \longrightarrow \mathfrak{g}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/embedding+tensor">embedding tensor</a>, hence a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> such that the “quadratic constraint” is satisfied: for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v_1, v_2 \in V</annotation></semantics></math> we have</p> <div class="maruku-equation" id="eq:QuadraticConstraintForEmbeddingTenson"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>Θ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> [\Theta(v_1), \Theta(v_2)] \;=\; \Theta \big( \rho_{\Theta(v_1)}(v_2) \big) </annotation></semantics></math></div></li> </ul> <p>Then <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> becomes a <a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a> with product defined by</p> <div class="maruku-equation" id="eq:LeibnizProductFromEmbeddingTensor"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> v_1 \cdot v_2 \;\coloneqq\; \rho_{\Theta(v_1)}(v_2) </annotation></semantics></math></div> <p>and with respect to this the <em>quadratic constraint</em> <a class="maruku-eqref" href="#eq:QuadraticConstraintForEmbeddingTenson">(2)</a> becomes the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/Leibniz+algebras">Leibniz algebras</a>.</p> </div> <p>(<a href="#Lavau17">Lavau 17, Example 3</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>We directly check the Leibniz rule <a class="maruku-eqref" href="#eq:LeibnizRule">(1)</a> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>ρ</mi> <munder><munder><mrow><mo stretchy="false">[</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>+</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} v_1 \cdot (v_2 \cdot v_3) & = \rho_{\Theta(v_1)} \big( \rho_{\Theta(v_2)}(v_3) \big) \\ & = \rho_{ \underset{ = \rho_{\Theta(v_1)}(v_2) }{ \underbrace{ [\Theta(v_1), \Theta(v_2)] } } }(v_3) + \rho_{\Theta(v_2)} \big( \rho_{\Theta(v_1)}(v_3) \big) \\ & = (v_1 \cdot v_2) \cdot v_3 + v_2 \cdot (v_1 \cdot v_3) \end{aligned} </annotation></semantics></math></div> <p>Here the first line is the definition <a class="maruku-eqref" href="#eq:LeibnizProductFromEmbeddingTensor">(3)</a>, the second line is the <a class="existingWikiWord" href="/nlab/show/Lie+action+property">Lie action property</a> (<a href="Lie+algebra+representation#eq:LieActionProperty">here</a>), under the brace we use the quadratic constraint <a class="maruku-eqref" href="#eq:QuadraticConstraintForEmbeddingTenson">(2)</a> on the embedding tensor, and in the last line we observe again the definition <a class="maruku-eqref" href="#eq:LeibnizProductFromEmbeddingTensor">(3)</a>.</p> </div> <h3 id="LeibnizAlgebrasFromDgLieAlgebras">From dg-Lie algebras</h3> <div class="num_prop" id="LeibnizAlgebraFromdgLieAlgebra"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a> from <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>,</mo><mo>∂</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((V_\bullet, \partial), [-,-])</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> with underlying <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>,</mo><mo>∂</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V_\bullet, \partial)</annotation></semantics></math> and with <a class="existingWikiWord" href="/nlab/show/super+Lie+bracket">super Lie bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math>.</p> <p>On the <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> which is the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>V</mi></mstyle><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mo>⊕</mo><mi>n</mi></munder><msub><mi>V</mi> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> \mathbf{V} \;\coloneqq\; \underset{n}{\oplus} V_n \;\in\; Vect </annotation></semantics></math></div> <p>of all the component <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, consider the <a class="existingWikiWord" href="/nlab/show/magma">product</a> given by the formula</p> <div class="maruku-equation" id="eq:LeibnizProductFromdgLieAlgebra"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>V</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>V</mi></mstyle></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>v</mi><mo>⋅</mo><mi>w</mi><mpadded width="0"><mrow><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mo>∂</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{V} \otimes \mathbf{V} &\longrightarrow& \mathbf{V} \\ (v,w) &\mapsto& v \cdot w \mathrlap{ \;\coloneqq\; [\partial v, w] } } </annotation></semantics></math></div> <p>Then: Restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>⊂</mo><mstyle mathvariant="bold"><mi>V</mi></mstyle></mrow><annotation encoding="application/x-tex">V_1 \subset \mathbf{V}</annotation></semantics></math> this product gives a left Leibniz algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V_1, \cdot)</annotation></semantics></math> (Def. <a class="maruku-ref" href="#LeftLeibnizAlgebra"></a>), i.e. satisfies the Leibniz condition <a class="maruku-eqref" href="#eq:LeibnizRule">(1)</a>.</p> </div> <p>This statement is highlighted in <a href="#LavauPalmkvist19">Lavau-Palmkvist 19, 2.1</a>.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We directly compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mover><mover><mrow><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mo>⏞</mo></mover><mrow><mo>=</mo><mn>0</mn></mrow></mover></msup><mo maxsize="1.2em" minsize="1.2em">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo>∂</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><msub><mi>v</mi> <mn>2</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} v_1 \cdot (v_2 \cdot v_3) & = \big[ \partial v_1 , [ \partial v_2, v_3 ] \big] \\ & = \big[ [ \partial v_1 , \partial v_2 ], v_3 \big] \;+\; (-1)^{ \overset{= 0}{ \overbrace{ (deg(v_1)-1) (deg(v_2)-2) } } } \big[ \partial v_2, [\partial v_1, v_3] \big] \\ & = \big[ \partial [ v_1 , \partial v_2 ], v_3 \big] \;+\; \big[ \partial v_2, [\partial v_1, v_3] \big] \\ & = (v_1 \cdot v_2) \cdot v_3 \;+\; v_2 \cdot (v_1 \cdot v_3) \,. \end{aligned} </annotation></semantics></math></div> <p>Here the first line is the definition <a class="maruku-eqref" href="#eq:LeibnizProductFromdgLieAlgebra">(4)</a>, the second line is the <a href="super+Lie+algebra#eq:GradedJacobiIdentity">graded Jacobi identity</a>, the third line uses the <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>-property and the nilpotency of the <a class="existingWikiWord" href="/nlab/show/differential">differential</a>, and the last line invokes again the definition <a class="maruku-eqref" href="#eq:LeibnizProductFromdgLieAlgebra">(4)</a>. Over the brace we used the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_i \in V_1</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The construction in Prop. <a class="maruku-ref" href="#LeibnizAlgebraFromdgLieAlgebra"></a> evidently extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgLieAlg</mi></mrow><annotation encoding="application/x-tex">dgLieAlg</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> to the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LeibAlg</mi></mrow><annotation encoding="application/x-tex">LeibAlg</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Leibniz+algebras">Leibniz algebras</a> (both over the given <a class="existingWikiWord" href="/nlab/show/ground+ring">ground ring</a>/<a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>):</p> <div class="maruku-equation" id="eq:FunctorFromdgLieAlgebrasToLeibnizAlgebras"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>dgLieAlg</mi><mo>⟶</mo><mi>LeibAlg</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-)_{1} \;\colon\; dgLieAlg \longrightarrow LeibAlg \,. </annotation></semantics></math></div> <p>Notice the analogy to the evident functor that extract the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> in degree 0:</p> <div class="maruku-equation" id="eq:FunctorFromdgLieAlgebrasToLeibnizAlgebras"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>dgLieAlg</mi><mo>⟶</mo><mi>LieAlg</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-)_{0} \;\colon\; dgLieAlg \longrightarrow LieAlg \,. </annotation></semantics></math></div></div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Named after <a class="existingWikiWord" href="/nlab/show/G.+W.+Leibniz">G. W. Leibniz</a>.</p> <p>The idea of <em>Leibniz algebra</em>, though not by this name, is given already in</p> <ul> <li id="Blokh65">A. Blokh, <em>A generalization of the concept of Lie algebra</em>, Dokl. Akad. Nauk SSSR, 165:471–473 (1965) (<a href="http://mi.mathnet.ru/eng/dan31825">mathrunet:dan31825</a>)</li> </ul> <p>The concept was revived (and apparently the name <em>Leibniz algebra</em> was first chosen) in</p> <ul> <li id="Loday93"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <em>Une version non commutative des algèbres de Lie: les algèbres de Leibniz</em>, Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Volume 44 (1993), Talk no. 5, 25 p. (<a href="http://www.numdam.org/item/?id=RCP25_1993__44__127_0">numdam:RCP25_1993__44__127_0</a>)</p> </li> <li id="LodayPirashvili93"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <a class="existingWikiWord" href="/nlab/show/Teimuraz+Pirashvili">Teimuraz Pirashvili</a>, <em>Universal enveloping algebras of Leibniz algebras and (co)homology</em>, Math. Ann. <strong>296</strong>, 139-158 (1993) (<a href="https://doi.org/10.1007/BF01445099">doi:10.1007/BF01445099</a>, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/93LodayPira%28Leibniz%29.pdf">pdf</a>)</p> </li> </ul> <p>Early review is in</p> <ul> <li id="Cuvier94"><a class="existingWikiWord" href="/nlab/show/Christian+Cuvier">Christian Cuvier</a>, <em>Algèbres de Leibnitz: définitions, propriétés</em>, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 27 (1994) no. 1, p. 1-45 (<a href="https://doi.org/10.24033/asens.1687">doi:10.24033/asens.1687</a>)</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Leibniz_algebra">Leibniz algebra</a></em></li> </ul> <p>Relaization of Leibniz algebras as <a class="existingWikiWord" href="/nlab/show/Lie+algebra+objects">Lie algebra objects</a> in a suitable <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>:</p> <ul> <li id="LodayPirashvili98"><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <a class="existingWikiWord" href="/nlab/show/Teimuraz+Pirashvili">Teimuraz Pirashvili</a>, <em>The tensor category of linear maps and Leibniz algebras</em>, Georg. Math. J. vol. 5, n.3 (1998) 263–276 (<a href="https://link.springer.com/article/10.1023/B:GEOR.0000008125.26487.f3">doi:10.1023/B:GEOR.0000008125.26487.f3</a>)</li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>:</p> <ul> <li id="Cuvier91"> <p><a class="existingWikiWord" href="/nlab/show/Christian+Cuvier">Christian Cuvier</a>, <em>Homologie de Leibniz et homologie de Hochschild</em>, C.R. Acad. Sci. Paris, Ser. A-B313, 569-572 (1991)</p> </li> <li> <p>Jerry M. Lodder, <em>Leibniz homology, characteristic classes and K-theory, <a href="http://www.math.uiuc.edu/K-theory/0493">K-theory archive/0493</a>;</em>Leibniz cohomology and the calculus of variations_ (<a href="http://arxiv.org/abs/math/9808036">arXiv:math/9808036</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <em>Algebraic K-theory and the conjectural Leibniz K-theory</em>, K-Theory 09/2003; 30(2):105-127, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/2003Loday%28LeibnizConj%29.pdf">pdf</a> <a href="http://dx.doi.org/10.1023/B:KTHE.0000018382.90150.ce">doi</a></p> </li> </ul> <p>This is partly based on earlier insights of Kinyon and Weinstein:</p> <ul> <li>Michael K. Kinyon, <em>Leibniz algebras, Lie racks, and digroups</em>, J. Lie Theory <strong>17</strong>:1 (2007) 099–114, <a href="http://arxiv.org/abs/math/0403509">arxiv:math.GR/0403509</a></li> </ul> <h3 id="relation_to_dglie_algebras_and_tensor_hierarchies">Relation to dg-Lie algebras and tensor hierarchies</h3> <p>Relation of <a class="existingWikiWord" href="/nlab/show/Leibniz+algebras">Leibniz algebras</a> to <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> such as the <a class="existingWikiWord" href="/nlab/show/tensor+hierarchies">tensor hierarchies</a> in <a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a>:</p> <ul> <li id="Lavau17"> <p><a class="existingWikiWord" href="/nlab/show/Sylvain+Lavau">Sylvain Lavau</a>, <em>Tensor hierarchies and Leibniz algebras</em>, J. Geom. Phys. 144:147-189 (2019) (<a href="https://arxiv.org/abs/1708.07068">arXiv:1708.07068</a>)</p> </li> <li id="LavauPalmkvist19"> <p><a class="existingWikiWord" href="/nlab/show/Sylvain+Lavau">Sylvain Lavau</a>, <a class="existingWikiWord" href="/nlab/show/Jakob+Palmkvist">Jakob Palmkvist</a>, <em>Infinity-enhancing of Leibniz algebras</em> (<a href="https://arxiv.org/abs/1907.05752">arXiv:1907.05752</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sylvain+Lavau">Sylvain Lavau</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra extensions of Leibniz algebras</em> (<a href="https://arxiv.org/abs/2003.07838">arXiv:2003.07838</a>)</p> </li> </ul> <h3 id="lie_integration">Lie integration</h3> <p>A generalization of <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> to conjectural Leibniz groups has been conjectured by <a class="existingWikiWord" href="/nlab/show/J-L.+Loday">J-L. Loday</a>. A local version via local Lie <a class="existingWikiWord" href="/nlab/show/racks">racks</a> has been proposed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Covez">Simon Covez</a>, <em>L’intégration locale des algèbres de Leibniz</em>, Thesis (2010) (<a href="http://tel.archives-ouvertes.fr/docs/00/49/54/69/PDF/THESE_Simon_Covez.pdf">pdf</a>)</p> </li> <li id="Covez10"> <p><a class="existingWikiWord" href="/nlab/show/Simon+Covez">Simon Covez</a>, <em>The local integration of Leibniz algebras</em>, Annales de l’Institut Fourier, Volume 63 (2013) no. 1, p. 1-35 (<a href="http://arxiv.org/abs/1011.4112">arXiv:1011.4112</a>, <a href="https://doi.org/10.5802/aif.2754">doi:10.5802/aif.2754</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Covez">Simon Covez</a>, <em>On the conjectural cohomology for groups</em>, Journal of K-theory <strong>10</strong>:03, Dec 2012, pp 519-563 (<a href="http://arxiv.org/abs/1202.2269">arXiv:1202.2269</a>, <a href="http://dx.doi.org/10.1017/is011011011jkt195">doi:10.1017/is011011011jkt195</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 18, 2021 at 17:47:58. 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