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Homotopy Lie algebra
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<h2 id="definition">Definition</h2> <p>There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations more than others. The most traditional definition is via symmetric multi-linear maps, but there also exists a more succinct geometric definition using the language of <a href="/facts/Formal_geometry/FCB3BMez">formal geometry</a>. Here the blanket assumption that the underlying field is of characteristic zero is made. </p> <h3>Geometric definition</h3> <p>A homotopy Lie algebra on a <a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a> V = ⨁ V i {\displaystyle V=\bigoplus V_{i}} is a continuous derivation, m {\displaystyle m} , of order > 1 {\displaystyle >1} that squares to zero on the formal manifold S ^ Σ V ∗ {\displaystyle {\hat {S}}\Sigma V^{*}} . Here S ^ {\displaystyle {\hat {S}}} is the completed <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a>, Σ {\displaystyle \Sigma } is the suspension of a graded vector space, and V ∗ {\displaystyle V^{*}} denotes the linear dual. Typically one describes ( V , m ) {\displaystyle (V,m)} as the homotopy Lie algebra and S ^ Σ V ∗ {\displaystyle {\hat {S}}\Sigma V^{*}} with the differential m {\displaystyle m} as its representing commutative differential graded algebra. </p><p>Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras, f : ( V , m V ) → ( W , m W ) {\displaystyle f\colon (V,m_{V})\to (W,m_{W})} , as a morphism f : S ^ Σ V ∗ → S ^ Σ W ∗ {\displaystyle f\colon {\hat {S}}\Sigma V^{*}\to {\hat {S}}\Sigma W^{*}} of their representing commutative differential graded algebras that commutes with the vector field, i.e., f ∘ m V = m W ∘ f {\displaystyle f\circ m_{V}=m_{W}\circ f} . Homotopy Lie algebras and their morphisms define a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a>. </p> <h3>Definition via multi-linear maps</h3> <p>The more traditional definition of a homotopy Lie algebra is through an infinite collection of symmetric multi-linear maps that is sometimes referred to as the definition via higher brackets. It should be stated that the two definitions are equivalent. </p><p>A homotopy Lie algebra<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> on a <a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a> V = ⨁ V i {\displaystyle V=\bigoplus V_{i}} is a collection of symmetric multi-linear maps l n : V ⊗ n → V {\displaystyle l_{n}\colon V^{\otimes n}\to V} of degree n − 2 {\displaystyle n-2} , sometimes called the n {\displaystyle n} -ary bracket, for each n ∈ N {\displaystyle n\in \mathbb {N} } . Moreover, the maps l n {\displaystyle l_{n}} satisfy the generalised Jacobi identity: </p> ∑ i + j = n + 1 ∑ σ ∈ U n S h u f f ( i , n − i ) χ ( σ , v 1 , … , v n ) ( − 1 ) i ( j − 1 ) l j ( l i ( v σ ( 1 ) , … , v σ ( i ) ) , v σ ( i + 1 ) , … , v σ ( n ) ) = 0 , {\displaystyle \sum _{i+j=n+1}\sum _{\sigma \in \mathrm {UnShuff} (i,n-i)}\chi (\sigma ,v_{1},\dots ,v_{n})(-1)^{i(j-1)}l_{j}(l_{i}(v_{\sigma (1)},\dots ,v_{\sigma (i)}),v_{\sigma (i+1)},\dots ,v_{\sigma (n)})=0,} <p>for each n. Here the inner sum runs over ( i , j ) {\displaystyle (i,j)} -unshuffles and χ {\displaystyle \chi } is the signature of the permutation. The above formula have meaningful interpretations for low values of n {\displaystyle n} ; for instance, when n = 1 {\displaystyle n=1} it is saying that l 1 {\displaystyle l_{1}} squares to zero (i.e., it is a differential on V {\displaystyle V} ), when n = 2 {\displaystyle n=2} it is saying that l 1 {\displaystyle l_{1}} is a derivation of l 2 {\displaystyle l_{2}} , and when n = 3 {\displaystyle n=3} it is saying that l 2 {\displaystyle l_{2}} satisfies the Jacobi identity up to an exact term of l 3 {\displaystyle l_{3}} (i.e., it holds up to homotopy). Notice that when the higher brackets l n {\displaystyle l_{n}} for n ≥ 3 {\displaystyle n\geq 3} vanish, the definition of a <a href="/facts/Differential_graded_Lie_algebra/KfMGlkQr">differential graded Lie algebra</a> on V {\displaystyle V} is recovered. </p><p>Using the approach via multi-linear maps, a morphism of homotopy Lie algebras can be defined by a collection of symmetric multi-linear maps f n : V ⊗ n → W {\displaystyle f_{n}\colon V^{\otimes n}\to W} which satisfy certain conditions. </p> <h3>Definition via operads</h3> <p>There also exists a more abstract definition of a homotopy algebra using the theory of <a href="/facts/Operad/RfUyH38R">operads</a>: that is, a homotopy Lie algebra is an <a href="/facts/Algebra_over_an_operad/tHDxMG7R">algebra over an operad</a> in the category of chain complexes over the L ∞ {\displaystyle L_{\infty }} operad. </p> <h2 id="quasi-isomorphisms-and-minimal-models">(Quasi) isomorphisms and minimal models</h2> <p>A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component f : V → W {\displaystyle f\colon V\to W} is a (quasi) isomorphism, where the differentials of V {\displaystyle V} and W {\displaystyle W} are just the linear components of m V {\displaystyle m_{V}} and m W {\displaystyle m_{W}} . </p><p>An important special class of homotopy Lie algebras are the so-called minimal homotopy Lie algebras, which are characterized by the vanishing of their linear component l 1 {\displaystyle l_{1}} . This means that any quasi isomorphism of minimal homotopy Lie algebras must be an isomorphism. Any homotopy Lie algebra is quasi-isomorphic to a minimal one, which must be unique up to isomorphism and it is therefore called its minimal model. </p> <h2 id="examples">Examples</h2> <p>Because L ∞ {\displaystyle L_{\infty }} -algebras have such a complex structure describing even simple cases can be a non-trivial task in most cases. Fortunately, there are the simple cases coming from differential graded Lie algebras and cases coming from finite dimensional examples. </p> <h3>Differential graded Lie algebras</h3> <p>One of the approachable classes of examples of L ∞ {\displaystyle L_{\infty }} -algebras come from the embedding of differential graded Lie algebras into the category of L ∞ {\displaystyle L_{\infty }} -algebras. This can be described by l 1 {\displaystyle l_{1}} giving the derivation, l 2 {\displaystyle l_{2}} the Lie algebra structure, and l k = 0 {\displaystyle l_{k}=0} for the rest of the maps. </p> <h3>Two term L∞ algebras</h3> <h4>In degrees 0 and 1</h4> <p>One notable class of examples are L ∞ {\displaystyle L_{\infty }} -algebras which only have two nonzero underlying vector spaces V 0 , V 1 {\displaystyle V_{0},V_{1}} . Then, cranking out the definition for L ∞ {\displaystyle L_{\infty }} -algebras this means there is a linear map </p> d : V 1 → V 0 {\displaystyle d\colon V_{1}\to V_{0}} , <p>bilinear maps </p> l 2 : V i × V j → V i + j {\displaystyle l_{2}\colon V_{i}\times V_{j}\to V_{i+j}} , where 0 ≤ i + j ≤ 1 {\displaystyle 0\leq i+j\leq 1} , <p>and a trilinear map </p> l 3 : V 0 × V 0 × V 0 → V 1 {\displaystyle l_{3}\colon V_{0}\times V_{0}\times V_{0}\to V_{1}} <p>which satisfy a host of identities.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> pg 28 In particular, the map l 2 {\displaystyle l_{2}} on V 0 × V 0 → V 0 {\displaystyle V_{0}\times V_{0}\to V_{0}} implies it has a lie algebra structure up to a homotopy. This is given by the differential of l 3 {\displaystyle l_{3}} since the gives the L ∞ {\displaystyle L_{\infty }} -algebra structure implies </p> d l 3 ( a , b , c ) = − [ [ a , b ] , c ] + [ [ a , c ] , b ] + [ a , [ b , c ] ] {\displaystyle dl_{3}(a,b,c)=-[[a,b],c]+[[a,c],b]+[a,[b,c]]} , <p>showing it is a higher Lie bracket. In fact, some authors write the maps l n {\displaystyle l_{n}} as [ − , ⋯ , − ] n : V ∙ → V ∙ {\displaystyle [-,\cdots ,-]_{n}:V_{\bullet }\to V_{\bullet }} , so the previous equation could be read as </p> d [ a , b , c ] 3 = − [ [ a , b ] , c ] + [ [ a , c ] , b ] + [ a , [ b , c ] ] {\displaystyle d[a,b,c]_{3}=-[[a,b],c]+[[a,c],b]+[a,[b,c]]} , <p>showing that the differential of the 3-bracket gives the failure for the 2-bracket to be a Lie algebra structure. It is only a Lie algebra up to homotopy. If we took the complex H ∗ ( V ∙ , d ) {\displaystyle H_{*}(V_{\bullet },d)} then H 0 ( V ∙ , d ) {\displaystyle H_{0}(V_{\bullet },d)} has a structure of a Lie algebra from the induced map of [ − , − ] 2 {\displaystyle [-,-]_{2}} . </p> <h4>In degrees 0 and n</h4> <p>In this case, for n ≥ 2 {\displaystyle n\geq 2} , there is no differential, so V 0 {\displaystyle V_{0}} is a Lie algebra on the nose, but, there is the extra data of a vector space V n {\displaystyle V_{n}} in degree n {\displaystyle n} and a higher bracket </p> l n + 2 : ⨁ n + 2 V 0 → V n . {\displaystyle l_{n+2}\colon \bigoplus ^{n+2}V_{0}\to V_{n}.} <p>It turns out this higher bracket is in fact a higher cocyle in <a href="/facts/Lie_algebra_cohomology/CHGMUaaU">Lie algebra cohomology</a>. More specifically, if we rewrite V 0 {\displaystyle V_{0}} as the Lie algebra g {\displaystyle {\mathfrak {g}}} and V n {\displaystyle V_{n}} and a Lie algebra representation V {\displaystyle V} (given by structure map ρ {\displaystyle \rho } ), then there is a <a href="/facts/Bijection/j4gTuTmW">bijection</a> of quadruples </p> ( g , V , ρ , l n + 2 ) {\displaystyle ({\mathfrak {g}},V,\rho ,l_{n+2})} where l n + 2 : g ⊗ n + 2 → V {\displaystyle l_{n+2}\colon {\mathfrak {g}}^{\otimes n+2}\to V} is an ( n + 2 ) {\displaystyle (n+2)} -cocycle <p>and the two-term L ∞ {\displaystyle L_{\infty }} -algebras with non-zero vector spaces in degrees 0 {\displaystyle 0} and n {\displaystyle n} .<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>pg 42 Note this situation is highly analogous to the relation between <a href="/facts/Group_cohomology/ncLTIHj4">group cohomology</a> and the structure of <a href="/facts/N-group_(category_theory)/h2XalQ26">n-groups</a> with two non-trivial homotopy groups. For the case of term term L ∞ {\displaystyle L_{\infty }} -algebras in degrees 0 {\displaystyle 0} and 1 {\displaystyle 1} there is a similar relation between Lie algebra cocycles and such higher brackets. Upon first inspection, it's not an obvious results, but it becomes clear after looking at the homology complex </p> H ∗ ( V 1 → d V 0 ) {\displaystyle H_{*}(V_{1}\xrightarrow {d} V_{0})} , <p>so the differential becomes trivial. This gives an equivalent L ∞ {\displaystyle L_{\infty }} -algebra which can then be analyzed as before. </p> <h4>Example in degrees 0 and 1</h4> <p>One simple example of a Lie-2 algebra is given by the L ∞ {\displaystyle L_{\infty }} -algebra with V 0 = ( R 3 , × ) {\displaystyle V_{0}=(\mathbb {R} ^{3},\times )} where × {\displaystyle \times } is the cross-product of vectors and V 1 = R {\displaystyle V_{1}=\mathbb {R} } is the trivial representation. Then, there is a higher bracket l 3 {\displaystyle l_{3}} given by the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of vectors </p> l 3 ( a , b , c ) = a ⋅ ( b × c ) . {\displaystyle l_{3}(a,b,c)=a\cdot (b\times c).} <p>It can be checked the differential of this L ∞ {\displaystyle L_{\infty }} -algebra is always zero using basic linear algebra<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>pg 45. </p> <h3>Finite dimensional example</h3> <p>Coming up with simple examples for the sake of studying the nature of L ∞ {\displaystyle L_{\infty }} -algebras is a complex problem. For example,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> given a graded vector space V = V 0 ⊕ V 1 {\displaystyle V=V_{0}\oplus V_{1}} where V 0 {\displaystyle V_{0}} has basis given by the vector w {\displaystyle w} and V 1 {\displaystyle V_{1}} has the basis given by the vectors v 1 , v 2 {\displaystyle v_{1},v_{2}} , there is an L ∞ {\displaystyle L_{\infty }} -algebra structure given by the following rules </p> l 1 ( v 1 ) = l 1 ( v 2 ) = w l 2 ( v 1 ⊗ v 2 ) = v 1 , l 2 ( v 1 ⊗ w ) = w l n ( v 2 ⊗ w ⊗ n − 1 ) = C n w for n ≥ 3 , {\displaystyle {\begin{aligned}&l_{1}(v_{1})=l_{1}(v_{2})=w\\&l_{2}(v_{1}\otimes v_{2})=v_{1},l_{2}(v_{1}\otimes w)=w\\&l_{n}(v_{2}\otimes w^{\otimes n-1})=C_{n}w{\text{ for }}n\geq 3\end{aligned}},} <p>where C n = ( − 1 ) n − 1 ( n − 3 ) C n − 1 , C 3 = 1 {\displaystyle C_{n}=(-1)^{n-1}(n-3)C_{n-1},C_{3}=1} . Note that the first few constants are </p> C 3 C 4 C 5 C 6 1 − 1 − 2 12 {\displaystyle {\begin{matrix}C_{3}&C_{4}&C_{5}&C_{6}\\1&-1&-2&12\end{matrix}}} <p>Since l 1 ( w ) {\displaystyle l_{1}(w)} should be of degree − 1 {\displaystyle -1} , the axioms imply that l 1 ( w ) = 0 {\displaystyle l_{1}(w)=0} . There are other similar examples for super<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Lie algebras.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> Furthermore, L ∞ {\displaystyle L_{\infty }} structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Homotopy_associative_algebra/wV9wCHGJ">Homotopy associative algebra</a></li> <li><a href="/facts/Differential_graded_algebra/zySud4Kl">Differential graded algebra</a></li> <li><a href="/facts/Batalin%25E2%2580%2593Vilkovisky_formalism/E6b8pdmw">BV formalism</a></li> <li><a href="/facts/Simplicial_Lie_algebra/VCkH4fJV">Simplicial Lie algebra</a></li> <li><a href="/facts/Hochschild_homology/0phwDIyX">Hochschild homology</a></li> <li><a href="/facts/Deformation_quantization/4lSVKfhv">Deformation quantization</a></li> <li><a href="/facts/Lie_n-algebra/lfBzYQ0P">Lie n-algebra</a></li></ul> <h3>Introduction</h3> <ul><li>Deformation Theory (lecture notes) - gives an excellent overview of homotopy Lie algebras and their relation to deformation theory and deformation quantization</li> <li>Lada, Tom; Stasheff, Jim (1993). "Introduction to sh Lie algebras for physicists". <i><a href="/facts/International_Journal_of_Theoretical_Physics/IxfPoCld">International Journal of Theoretical Physics</a></i>. 32 (7): 1087–1104. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9209099">hep-th/9209099</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1993IJTP...32.1087L">1993IJTP...32.1087L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00671791">10.1007/BF00671791</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:16456088">16456088</a>.</li></ul> <h3>In physics</h3> <ul><li>Arvanitakis, Alex S. (2019). "The L∞-algebra of the S-matrix". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1903.05643">1903.05643</a> [<a href="https://arxiv.org/archive/hep-th">hep-th</a>].</li> <li>Hohm, Olaf; Zwiebach, Barton (2017). "L∞ Algebras and Field Theory". <i>Fortschr. Phys</i>. 65 (3–4): 1700014. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1701.08824">1701.08824</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2017ForPh..6500014H">2017ForPh..6500014H</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fprop.201700014">10.1002/prop.201700014</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:90628041">90628041</a>. — Towards classification of perturbative gauge invariant classical fields.</li></ul> <h3>In deformation and string theory</h3> <ul><li>Pridham, Jonathan P. (2015). "Derived deformations of Artin stacks". <i>Communications in Analysis and Geometry</i>. 23 (3): 419–477. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0805.3130">0805.3130</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2FCAG.2015.v23.n3.a1">10.4310/CAG.2015.v23.n3.a1</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3310522">3310522</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14505074">14505074</a>.</li> <li>Pridham, Jonathan P. (2010). <a href="https://doi.org/10.1016%2Fj.aim.2009.12.009">"Unifying derived deformation theories"</a>. <i><a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a></i>. 224 (3): 772–826. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0705.0344">0705.0344</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.aim.2009.12.009">10.1016/j.aim.2009.12.009</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2628795">2628795</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14136532">14136532</a>.</li> <li>Hu, Po; Kriz, Igor; Voronov, Alexander A. (2006). "On Kontsevich's Hochschild cohomology conjecture". <i><a href="/facts/Compositio_Mathematica/Igc7zupB">Compositio Mathematica</a></i>. 142 (1): 143–168. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0309369">math/0309369</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2FS0010437X05001521">10.1112/S0010437X05001521</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2197407">2197407</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:15153116">15153116</a>.</li></ul> <h3>Related ideas</h3> <ul><li>Roberts, Justin; Willerton, Simon (2010). <a href="http://www.msp.org/agt/2010/10-3/p09.xhtml">"On the Rozansky–Witten weight systems"</a>. <i><a href="/facts/Algebraic_%2526_Geometric_Topology/Mgmju995">Algebraic & Geometric Topology</a></i>. 10 (3): 1455–1519. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0602653">math/0602653</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fagt.2010.10.1455">10.2140/agt.2010.10.1455</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2661534">2661534</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:17829444">17829444</a>. (Lie algebras in the <a href="/facts/Derived_category/pBQHnFYo">derived category</a> of coherent sheaves.)</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.mpim-bonn.mpg.de/node/8756">"Learning seminar on deformation theory"</a>. Max Planck Institute for Mathematics. 2018. Discusses deformation theory in the context of L ∞ {\displaystyle L_{\infty }} -algebras.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lurie, Jacob. "Derived Algebraic Geometry X: Formal Moduli Problems" (PDF). p. 31, Theorem 2.0.2. <a href="/wiki/Jacob_Lurie" target="_blank">/wiki/Jacob_Lurie</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Pridham, Jonathan Paul (2012). "Derived deformations of schemes". Communications in Analysis and Geometry. 20 (3): 529–563. arXiv:0908.1963. doi:10.4310/CAG.2012.v20.n3.a4. MR 2974205. <a href="http://www.intlpress.com/site/pub/pages/journals/items/cag/content/vols/0020/0003/a004/" target="_blank">http://www.intlpress.com/site/pub/pages/journals/items/cag/content/vols/0020/0003/a004/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Daily, Marilyn Elizabeth (2004-04-14). L ∞ {\displaystyle L_{\infty }} Structures on Spaces of Low Dimension (PhD). hdl:1840.16/5282. <a href="https://repository.lib.ncsu.edu/handle/1840.16/5282" target="_blank">https://repository.lib.ncsu.edu/handle/1840.16/5282</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Baez, John C.; Crans, Alissa S. (2010-01-24). "Higher-Dimensional Algebra VI: Lie 2-Algebras". Theory and Applications of Categories. 12: 492–528. arXiv:math/0307263. <a href="/wiki/John_C._Baez" target="_blank">/wiki/John_C._Baez</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Baez, John C.; Crans, Alissa S. (2010-01-24). "Higher-Dimensional Algebra VI: Lie 2-Algebras". Theory and Applications of Categories. 12: 492–528. arXiv:math/0307263. <a href="/wiki/John_C._Baez" target="_blank">/wiki/John_C._Baez</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Baez, John C.; Crans, Alissa S. (2010-01-24). "Higher-Dimensional Algebra VI: Lie 2-Algebras". Theory and Applications of Categories. 12: 492–528. arXiv:math/0307263. <a href="/wiki/John_C._Baez" target="_blank">/wiki/John_C._Baez</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Daily, Marilyn; Lada, Tom (2005). "A finite dimensional L ∞ {\displaystyle L_{\infty }} algebra example in gauge theory". Homology, Homotopy and Applications. 7 (2): 87–93. doi:10.4310/HHA.2005.v7.n2.a4. MR 2156308. <a href="https://projecteuclid.org/euclid.hha/1139839375" target="_blank">https://projecteuclid.org/euclid.hha/1139839375</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Fialowski, Alice; Penkava, Michael (2002). "Examples of infinity and Lie algebras and their versal deformations". Banach Center Publications. 55: 27–42. arXiv:math/0102140. doi:10.4064/bc55-0-2. MR 1911978. S2CID 14082754. <a href="http://www.impan.pl/get/doi/10.4064/bc55-0-2" target="_blank">http://www.impan.pl/get/doi/10.4064/bc55-0-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Fialowski, Alice; Penkava, Michael (2005). "Strongly homotopy Lie algebras of one even and two odd dimensions". Journal of Algebra. 283 (1): 125–148. arXiv:math/0308016. doi:10.1016/j.jalgebra.2004.08.023. MR 2102075. S2CID 119142148. <a href="https://linkinghub.elsevier.com/retrieve/pii/S0021869304004818" target="_blank">https://linkinghub.elsevier.com/retrieve/pii/S0021869304004818</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Daily, Marilyn Elizabeth (2004-04-14). L ∞ {\displaystyle L_{\infty }} Structures on Spaces of Low Dimension (PhD). hdl:1840.16/5282. <a href="https://repository.lib.ncsu.edu/handle/1840.16/5282" target="_blank">https://repository.lib.ncsu.edu/handle/1840.16/5282</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>