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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Lie algebra cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{as_extgroup_or_derived_functor}{As Ext-group or derived functor}\dotfill \pageref*{as_extgroup_or_derived_functor} \linebreak \noindent\hyperlink{via_resolutions}{Via resolutions}\dotfill \pageref*{via_resolutions} \linebreak \noindent\hyperlink{via_lie_algebras}{Via $\infty$-Lie algebras}\dotfill \pageref*{via_lie_algebras} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{whiteheads_lemma}{Whitehead's lemma}\dotfill \pageref*{whiteheads_lemma} \linebreak \noindent\hyperlink{van_est_isomorphism}{Van Est isomorphism}\dotfill \pageref*{van_est_isomorphism} \linebreak \noindent\hyperlink{HochschildSerreSpectralSequence}{Hochschild-Serre spectral sequence}\dotfill \pageref*{HochschildSerreSpectralSequence} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{extensions}{Extensions}\dotfill \pageref*{extensions} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{ordinary_lie_algebras}{Ordinary Lie algebras}\dotfill \pageref*{ordinary_lie_algebras} \linebreak \noindent\hyperlink{ReferencesSuperLieAlg}{Super Lie algebras}\dotfill \pageref*{ReferencesSuperLieAlg} \linebreak \noindent\hyperlink{extensions_2}{Extensions}\dotfill \pageref*{extensions_2} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Lie algebra cohomology} is the intrinsic notion of [[cohomology]] of [[Lie algebra]]s. There is a precise sense in which Lie algebras $\mathfrak{g}$ are [[infinitesimal object|infinitesimal]] [[Lie group]]s. Lie algebra cohomology is the restriction of the definition of [[Lie group cohomology]] to Lie algebras. In [[∞-Lie theory]] one studies the relation between the two via [[Lie integration]]. Lie algebra cohomology generalizes to [[nonabelian Lie algebra cohomology]] and to [[∞-Lie algebra cohomology]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} There are several different but equivalent definitions of the [[cohomology]] of a [[Lie algebra]]. \hypertarget{as_extgroup_or_derived_functor}{}\subsubsection*{{As Ext-group or derived functor}}\label{as_extgroup_or_derived_functor} The abelian [[cohomology]] of a $k$-[[Lie algebra]] $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U\mathfrak{g}$. In particular it is a derived functor. \hypertarget{via_resolutions}{}\subsubsection*{{Via resolutions}}\label{via_resolutions} Before this approach was advanced in Cartan-Eilenberg's \emph{Homological algebra}, Lie algebra cohomology and homology were defined by Chevalley-Eilenberg with a help of concrete Koszul-type resolution which is in this case a cochain complex \begin{displaymath} Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M), \end{displaymath} where the first argument $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is naturally equipped with a differential to start with (see below). WHERE BELOW? The first argument in the Hom, i.e. $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is sometimes called the Chevalley-Eilenberg chain complex (cf. Weibel); the [[Chevalley-Eilenberg cochain complex]] is the whole thing, i.e. \begin{displaymath} CE(\mathfrak{g},M) := Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M). \end{displaymath} If $M$ is a trivial module $k$ then $CE(\mathfrak{g}) := Hom_k(\Lambda^* \mathfrak{g},k)$ and if $\mathfrak{g}$ is finite-dimensional this equals $\Lambda^* \mathfrak{g}^*$ with an appropriate differential and the exterior multiplication gives it a dg-algebra structure. \hypertarget{via_lie_algebras}{}\subsubsection*{{Via $\infty$-Lie algebras}}\label{via_lie_algebras} As discussed at [[Chevalley-Eilenberg algebra]], we may identify [[Lie algebra]]s $\mathfrak{g}$ as the duals $CE(\mathfrak{g})$ of [[dg-algebra]]s whose underlying graded algebra is the [[Grassmann algebra]] on the vector space $\mathfrak{g}^*$. Similarly, a dg-algebra $CE(\mathfrak{h})$ whose underlying algebra is free on a [[graded vector space]] $\mathfrak{h}$ we may understand as exibiting an [[∞-Lie algebra]]-structure on $\mathfrak{h}$. Then a morphism $\mathfrak{g} \to \mathfrak{h}$ of these $\infty$-Lie algebras is by definition just a morphism $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$ of dg-algebras. Such a morphis may be thought of as a cocycle in [[nonabelian Lie algebra cohomology]] $H(\mathfrak{g}, \mathfrak{h})$. Specifically, write $b^{n-1} \mathbb{R}$ for the [[line Lie n-algebra]], the $\infty$-Lie algebra given by the fact that $CE(b^{n-1}\mathbb{R})$ has a single generator in degree $n$ and vanishing differential. Then a morphism \begin{displaymath} \mu : \mathfrak{g} \to b^{n-1} \mathbb{R} \end{displaymath} is a cocycle in the abelian Lie algebra cohomology $H^n(\mathfrak{g}, \mathbb{R})$. Notice that dually, by definition, this is a morphism of dg-algebras \begin{displaymath} CE(\mathfrak{g}) \leftarrow CE(b^{n-1} \mathbb{R}) : \mu \,. \end{displaymath} Since on the right we only have a single closed degree-$n$ generator, such a morphism is precily a closed degree $n$-element \begin{displaymath} \mu \in CE(\mathfrak{g}) \,. \end{displaymath} This way we recover the above definition of Lie algebra cohomology (with coefficient in the trivial module) in terms of the cochain complex cohomology of the CE-algebra. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{whiteheads_lemma}{}\subsubsection*{{Whitehead's lemma}}\label{whiteheads_lemma} The following lemma asserts that for semisimple Lie algebras $\mathfrak{g}$ only the cohomology $\mathfrak{g} \to b^{n-1} \mathbb{R}$ with coefficients in the trivial module is nontrivial. \begin{prop} \label{}\hypertarget{}{} \textbf{(Whitehead's lemma)} For $\mathfrak{g}$ a finite dimensional [[semisimple Lie algebra]] over a [[field]] of [[characteristic]] 0, and for $V$ a non-trivial finite-dimensional [[irreducible representation]], we have \begin{displaymath} H^p(\mathfrak{g}, V) = 0 \;\;\; for\;p \gt 0 \,. \end{displaymath} \end{prop} \hypertarget{van_est_isomorphism}{}\subsubsection*{{Van Est isomorphism}}\label{van_est_isomorphism} The content of a [[van Est isomorphism]] is that the canonical comparison map from [[Lie group cohomology]] to Lie algebra cohomology (by [[differentiation]]) is an [[isomorphism]] whenever the Lie group is sufficiently connected. \hypertarget{HochschildSerreSpectralSequence}{}\subsubsection*{{Hochschild-Serre spectral sequence}}\label{HochschildSerreSpectralSequence} \begin{defn} \label{RelativeLieAlgebraCohomologyWithCoefficients}\hypertarget{RelativeLieAlgebraCohomologyWithCoefficients}{} \textbf{(relative Lie algebra cohomology with coefficents)} Let \begin{enumerate}% \item $(\mathfrak{g}, [-,-])$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]]; \item $(V, \rho)$ a $\mathfrak{g}$-[[Lie algebra module]] of [[finite number|finite]] [[dimension]]; \item $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra. \end{enumerate} Consider the $\mathbb{N}$-[[graded vector space]] \begin{displaymath} C^\bullet(\mathfrak{g}, \mathfrak{h}, V) \;\coloneqq\; \left( ( \wedge^\bullet (\mathfrak{g}/\mathfrak{h})^\ast ) \otimes V \right)^{\mathfrak{h}} \end{displaymath} consisting of the $\mathfrak{h}$-[[invariant|invariant elements]] in the [[tensor product]] of $V$ with the [[exterior algebra]] of the [[coset]] $\mathfrak{g}/\mathfrak{h}$. On this graded vector space, the [[dual linear maps]] of the [[Lie bracket]] $[-,-]$, extended as a graded [[derivation]] to the exterior algebra, and the [[Lie algebra action]] $\rho$ define a [[differential]] \begin{displaymath} d_{CE} \coloneqq \rho^\ast + [-,-]^\ast \,. \end{displaymath} The resulting [[cochain complex]] is the \emph{[[Chevalley-Eilenberg complex]] of $\mathfrak{g}$ relative $\mathfrak{h}$ with [[coefficients]] in $V$}. Its [[cochain cohomology]] \begin{displaymath} H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \;\coloneqq\; \left( \left( ( \wedge^\bullet (\mathfrak{g}/\mathfrak{h})^\ast ) \otimes V \right)^{\mathfrak{h}} , d_{CE} \right) \end{displaymath} is the \emph{Lie algebra cohomology of $\mathfrak{g}$ relative $\mathfrak{h}$ with [[coefficients]] in $V$}. \end{defn} (e.g. \hyperlink{Solleveld02}{Solleveld 02, def. 2.13 and def. 2.17}) If in def. \ref{RelativeLieAlgebraCohomologyWithCoefficients} $\mathfrak{h} = 0$ then the definition reduces to that of ordinary Lie algebra cohomology with coefficients: \begin{displaymath} H^\bullet(\mathfrak{g}, 0; V) = H^\bullet(\mathfrak{g}; V) \,. \end{displaymath} \begin{defn} \label{LieAlgebraReductiveInAmbientLieAlgebra}\hypertarget{LieAlgebraReductiveInAmbientLieAlgebra}{} \textbf{([[reductive Lie algebra|Lie algebra reductive]] in ambient Lie algebra)} A sub-[[Lie algebra]] \begin{displaymath} \mathfrak{h} \hookrightarrow \mathfrak{g} \end{displaymath} is called \emph{reductive} if the [[adjoint representation|adjoint]] [[Lie algebra representation]] of $\mathfrak{h}$ on $\mathfrak{g}$ is [[reducible representation|reducible]]. \end{defn} (\hyperlink{Koszul50}{Koszul 50}, recalled in e.g. \hyperlink{Solleveld02}{Solleveld 02, def. 2.27}) \begin{prop} \label{InvariantsInLieAlgebraCohomologyComputedByRelativeLieAlgebraCohomology}\hypertarget{InvariantsInLieAlgebraCohomologyComputedByRelativeLieAlgebraCohomology}{} \textbf{(invariants in Lie algebra cohomology computed by relative Lie algebra cohomology)} Let \begin{enumerate}% \item $(\mathfrak{g}, [-,-])$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]]; \item $(V, \rho)$ a $\mathfrak{g}$-[[Lie algebra module]] of [[finite number|finite]] [[dimension]], which is [[reducible representation|reducible]]; \item $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is [[reductive Lie algebra|reductive]] in $\mathfrak{g}$ (Def. \ref{LieAlgebraReductiveInAmbientLieAlgebra}) in that its [[adjoint representation]] on $\mathfrak{g}$ is [[reducible representation|reducible]]. \item such that \begin{displaymath} \mathfrak{g} = \mathfrak{h} \ltimes \mathfrak{a} \end{displaymath} is a [[semidirect product Lie algebra]] (hence $\mathfrak{a}$ a [[Lie ideal]]). \end{enumerate} Then the [[invariants]] in the Lie algebra cohomology of $\mathfrak{a}$ (either with respect to $\mathfrak{h}$ or all of $\mathfrak{g}$) coincide with the relative Lie algebra cohomology (Def. \ref{RelativeLieAlgebraCohomologyWithCoefficients}, using the invariant subcomplex!): \begin{displaymath} H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}} \;\simeq\; H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \,. \end{displaymath} \end{prop} (\hyperlink{Solleveld02}{Solleveld 02, theorem 2.28}) Proof via the [[Hochschild-Serre spectral sequence]]. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} Every [[invariant polynomial]] $\langle - \rangle \in W(\mathfrak{g})$ on a Lie algebra has a \emph{transgression} to a cocycle on $\mathfrak{g}$. See [[∞-Lie algebra cohomology]] for more. For instance for $\mathfrak{g}$ a [[semisimple Lie algebra]], there is the [[Killing form]] $\langle - ,- \rangle$. The corresponding 3-cocycle is \begin{displaymath} \mu = \langle -, [-,-] \rangle : CE(\mathfrak{g}) \,, \end{displaymath} that is: the function that sends three Lie algebra elements $x, y, z$ to the number $\mu(x,y,z) = \langle x, [y,z]\rangle$. On the [[super Poincare Lie algebra]] in dimension (10,1) there is a 4-cocycle \begin{displaymath} \mu_4 = \bar \psi \wedge \Gamma^{a b} \Psi\wedge e_a \wedge e_b \in CE(\mathfrak{siso}(10,1)) \end{displaymath} \hypertarget{extensions}{}\subsection*{{Extensions}}\label{extensions} Every Lie algebra degree $n$ cocycle $\mu$ (with values in the trivial model) gives rise to an extension \begin{displaymath} b^{n-2} \mathbb{R} \to \mathfrak{g}_{\mu} \to \mathfrak{g} \,. \end{displaymath} In the language of [[∞-Lie algebra]]s this was observed in (\hyperlink{BaezCrans}{BaezCrans Theorem 55}). In the dual [[dg-algebra]] language the extension is lust the relative [[Sullivan algebra]] \begin{displaymath} CE(\mathfrak{g}_\mu) \leftarrow CE(\mathfrak{g}) \end{displaymath} obtained by gluing on a rational $n$-sphere. By this kind of translation between familiar statements in [[rational homotopy theory]] dually into the language of [[∞-Lie algebra]]s many useful statements in [[∞-Lie theory]] are obtained. \textbf{Examples} \begin{itemize}% \item The [[string Lie 2-algebra]] is the extension of a [[semisimple Lie algebra]] induced by the canonical 3-cocycle coming from the [[Killing form]]. \item The [[supergravity Lie 3-algebra]] is the extension of the [[super Poincare Lie algebra]] by a 4-cocycle. \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Hochschild-Serre spectral sequence]] \item [[signs in supergeometry]] \item [[infinity-Lie algebra cohomology]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{ordinary_lie_algebras}{}\subsubsection*{{Ordinary Lie algebras}}\label{ordinary_lie_algebras} Accounts of the standard theory of Lie algebra cohomology include \begin{itemize}% \item [[Jean-Louis Koszul]], \emph{Homologie et cohomologie des alg\`e{}bres de Lie}, Bull. Soc. Math. France 78 (1950), 65-127 \item [[Gerhard Hochschild]], [[Jean-Pierre Serre]], \emph{Cohomology of Lie algebras}, Annals of Mathematics, Second Series, Vol. 57, No. 3 (May, 1953), pp. 591-603 (\href{http://www.jstor.org/stable/1969740}{JSTOR}) \item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], in chapter V in vol III of \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973) \item [[Charles Weibel]], chapter 7 of \emph{An introduction to homological algebra}, Cambridge Studies in Adv. Math. \textbf{38}, CUP 1994 \item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, section 6 of \emph{[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]}, Cambridge monographs of mathematical physics, (1995) \item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, J. C. Perez Bueno, \emph{An introduction to some novel applications of Lie algebra cohomology and physics} (\href{http://arxiv.org/abs/physics/9803046}{arXiv:physics/9803046}) \item [[Maarten Solleveld]], \emph{Lie algebra cohomology and Macdonald’s conjectures}, 2002 (\href{https://www.math.ru.nl/~solleveld/scrip.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item [[eom]] \emph{\href{http://eom.springer.de/C/c023140.htm}{Lie algebra cohomology}} \item \href{http://en.wikipedia.org/wiki/Lie_algebra_cohomology}{wikipedia} \item scholarpedia: \href{http://www.scholarpedia.org/article/An_introduction_to_Lie_algebra_cohomology}{An introduction to Lie algebra cohomology} \end{itemize} \hypertarget{ReferencesSuperLieAlg}{}\subsubsection*{{Super Lie algebras}}\label{ReferencesSuperLieAlg} The cohomology of [[super Lie algebra]]s is analyzed via [[normed division algebra]]s in \begin{itemize}% \item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry I} (\href{http://arxiv.org/abs/0909.0551}{arXiv:0909.0551}) \item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry II} (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.34360}) \end{itemize} See also [[division algebra and supersymmetry]]. This subsumes some of the results in \begin{itemize}% \item [[José de Azcárraga]], [[Paul Townsend]], \emph{Superspace geometry and classification of supersymmetric extended objects}, Phys. Rev. Lett. 62, 2579--2582 (1989) \end{itemize} The cohomology of the [[super Poincare Lie algebra]] in low dimensions $\leq 5$ is analyzed in \begin{itemize}% \item [[Friedemann Brandt]], \emph{Supersymmetry algebra cohomology I: Definition and general structure} J. Math. Phys.51:122302, 2010, \href{http://arxiv.org/abs/0911.2118}{arXiv} \emph{Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions} J. Math. Phys. 51 (2010) 112303 (\href{http://arxiv.org/abs/1004.2978}{arXiv}) \emph{Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions} (\href{http://arxiv.org/abs/1005.2102}{arXiv}) \end{itemize} and in higher dimensions more generally in \begin{itemize}% \item Michael Movshev, [[Albert Schwarz]], Renjun Xu, \emph{Homology of Lie algebra of supersymmetries} (\href{http://arxiv.org/abs/1011.4731}{arXiv}) . \end{itemize} \hypertarget{extensions_2}{}\subsubsection*{{Extensions}}\label{extensions_2} The [[∞-Lie algebra]] [[∞-Lie algebra cohomology|extensions]] $b^{n-2} \to \mathfrak{g}_\mu \to \mathfrak{g}$ induced by a degree $n$-cocycle are considered \begin{itemize}% \item [[John Baez]], [[Alissa Crans]], around theorem 55 in \emph{Higher-Dimensional Algebra VI: Lie 2-Algebras}, Theory and Applications of Categories 12 (2004), 492-528. \href{http://arxiv.org/abs/math.QA/0307263}{arXiv} \end{itemize} [[!redirects Lie algebra cocycle]] [[!redirects Lie algebra cocycles]] \end{document}