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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*{differential graded Lie algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory} [[!include differential graded objects - 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{DirectExplicitDefinition}{Direct explicit definition}\dotfill \pageref*{DirectExplicitDefinition} \linebreak \noindent\hyperlink{AsGradedLieAlgebrasWithDerivations}{As graded Lie algebras with nilpotent derivations}\dotfill \pageref*{AsGradedLieAlgebrasWithDerivations} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak \noindent\hyperlink{RelationToLInfinityAlgebras}{Relation to $L_\infty$-algebras}\dotfill \pageref*{RelationToLInfinityAlgebras} \linebreak \noindent\hyperlink{relation_to_dgcoalgebras}{Relation to dg-coAlgebras}\dotfill \pageref*{relation_to_dgcoalgebras} \linebreak \noindent\hyperlink{relation_to_simplicial_lie_algebras}{Relation to simplicial Lie algebras}\dotfill \pageref*{relation_to_simplicial_lie_algebras} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{differential graded Lie algebra}, or \emph{dg-Lie algebra} for short, is equivalently \begin{itemize}% \item a graded [[Lie algebra]] equipped with a [[differential]] that acts as a graded [[derivation]] with respect to the Lie bracket; \item a [[Lie algebra]] [[internalization|internal to]] the [[category of chain complexes]]; \item a \emph{strict} [[L-∞-algebra]], i.e. an $L_\infty$-algebra in which only the unary and the binary brackets may be nontrivial. \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{DirectExplicitDefinition}{}\subsubsection*{{Direct explicit definition}}\label{DirectExplicitDefinition} \begin{defn} \label{}\hypertarget{}{} A \emph{dg-Lie algebra} $(\mathfrak{g},\partial,[-,-])$ is \begin{enumerate}% \item a $\mathbb{Z}$-[[graded vector space]] $\mathfrak{g} = \bigoplus_{i} \mathfrak{g}_i$; \item a [[linear map]] $\partial \colon \mathfrak{g} \longrightarrow \mathfrak{g}$; \item a [[bilinear map]] $[-,-] \colon \mathfrak{g}\otimes\mathfrak{g} \longrightarrow \mathfrak{g}$, the \emph{bracket}; \end{enumerate} such that (all conditions are expressed for homogeneously graded elements $x_i \in \mathfrak{g}_{\vert x_i \vert}$): \begin{enumerate}% \item $\partial$ is a [[differential]] that makes $(\mathfrak{g},\partial)$ into a [[chain complex]], i.e. \begin{enumerate}% \item it is of degree -1, $\partial \colon \mathfrak{g}_{i} \to \mathfrak{g}_{i-1}$; \item it squares to zero, $\partial \circ \partial = 0$; \end{enumerate} \item $\partial$ is a graded [[derivation]] of the bilinear pairing, i.e. \begin{displaymath} \partial [x_1,x_2] = [\partial x_1, x_2] + (-1)^{\vert x_1 \vert} [x_1, \partial x_2] \,, \end{displaymath} \item the bilinear pairing is graded skew-symmetric, i.e. \begin{displaymath} [x_1, x_2] = -(-1)^{\vert x_1\vert \vert x_2 \vert} [x_2,x_1] \,, \end{displaymath} \item the bilinear pairing satisfies the graded [[Jacobi identity]] (saying that $[x,-]$ is a graded [[derivation]]) \begin{displaymath} [x_1,[x_2,x_3]] = [[x_1,x_2], x_3] + (-1)^{\vert x_1 \vert \vert x_2 \vert} [x_2, [x_1, x_3]] \,. \end{displaymath} \end{enumerate} \end{defn} \hypertarget{AsGradedLieAlgebrasWithDerivations}{}\subsubsection*{{As graded Lie algebras with nilpotent derivations}}\label{AsGradedLieAlgebrasWithDerivations} \begin{defn} \label{}\hypertarget{}{} A \emph{pre-graded Lie algebra} (pre-gla) is a [[graded vector space|pre-gvs]], $L$, together with a bilinear map of degree zero \begin{displaymath} [\quad,\quad ] : L\otimes L \to L, \end{displaymath} such that \begin{displaymath} [x,y] = (-1)^{|x||y|+1}[y,x] \end{displaymath} and \begin{displaymath} (-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0 \end{displaymath} for every triple $(x,y,z)$ of homogeneous elements in $L$. \end{defn} (The first property is call \emph{antisymmetry}, the second \emph{the Jacobi identity}.) A \emph{morphism} $f: L \to L'$ of pre-glas is a linear map of degree zero, that is compatible with the brackets, \begin{displaymath} f[x,y] = [f(x),f(y)]. \end{displaymath} \begin{example} \label{}\hypertarget{}{} To any augmented pre-ga $A$, one can associate a pre-gla, denoted $\bar{A}_L$, with underlying gvs $\bar{A}$ and with bracket, the commutator, $[x,y] = xy -(-1)^{|x||y|}yx$ for each pair $(x,y)$ of homogeneous elements. $\bar{A}_L$ is abelian (i.e. with trivial bracket) if and only if $A$ is graded commutative. \end{example} \begin{example} \label{}\hypertarget{}{} If $A$ is a pre-cga and $L$ is a pre-gla, the tensor product $A\otimes L$ has a pre-gla structure with bracket \begin{displaymath} [a\otimes l,a'\otimes l'] = (-1)^{|a'||l|}aa' \otimes [l,l'] \end{displaymath} for $a,a', l, l'$ homogeneous. \end{example} \begin{defn} \label{}\hypertarget{}{} Let $L$ be a pre-gla. A \emph{[[derivation]]} of gla-s, of degree $p\in \mathbb{Z}$, is a linear mapping $\theta \in Hom_p(L,L)$ such that \begin{displaymath} \theta[x,y] = [\theta{x},y] + (-1)^{p|x|}[x,\theta(y)] \end{displaymath} for any pair $x,y$ of homogeneous elements of $L$. We denote by $Der_p(L)$, the vector space of degree $p$ derivations of the gla, $L$. \end{defn} \begin{defn} \label{}\hypertarget{}{} A differential $\partial$ of a pre-gla is a Lie algebra derivation of degree -1 such that $\partial\circ \partial = 0$. The pair $(L,\partial)$ is then called a \emph{differential pre-graded Lie algebra} (pre-dgla); its homology $H(L,\partial)$, is a pre-gla. A morphism of pre-dglas is a morphism for both the underlying pre-gla and the pre-dgvs. We denote the corresponding category by $pre DGLA$. \end{defn} This means that a differential graded Lie algebra is an internal Lie algebra in the symmetric monoidal category of chain complexes with tensor product given as in [[differential graded vector space|differential graded vector spaces]]. \begin{example} \label{}\hypertarget{}{} If $(A,\partial)$ is an augmented pre-dga, $(\bar{A}_L,\partial)$ is a pre-dgla. \end{example} \begin{example} \label{}\hypertarget{}{} If $(A,\partial)$ is a pre-cdga and $(L,\partial')$, a pre-dgla, $A \otimes L$, together with the tensor product differential, is a pre-dgla. \end{example} \begin{example} \label{}\hypertarget{}{} Let $(V,\partial)$ be a pre-dgvs, then the pre-dgvs, $(Hom(V,V),D)$, constructed earlier is a pre-dga for the multiplication law given by composition of mappings. Its associated pre-dgla has \begin{displaymath} [f,g] = f\circ g - (-1)^{|f||g|}g\circ f \end{displaymath} and \begin{displaymath} Df = [\partial,f]. \end{displaymath} In particular, if $(V,\partial) = (A,d)$ is a cdga (resp. $(V,\partial) = (L,\partial)$ is dgla), then $Der(A,d) = (\bigoplus_p Der_p(A),D)$, (resp. $Der(L,\partial) = (Der_p(L),D)$) is a sub-pre-dgl of $(Hom(V,V),D)$. \end{example} \begin{defn} \label{}\hypertarget{}{} A dgla is a pre-dgla with a lower grading; explicitly: A \textbf{differential graded Lie algebra}, $(L,\partial)$, is a graded vector space $L = \bigoplus_{p\geq 0}L_p$, together with a bilinear map of degree 0 \begin{displaymath} [\quad ,\quad] : L\otimes L \to L, \end{displaymath} and a differential $\partial$ satisfying \begin{displaymath} \partial L_p \subseteq L_{p-1}, \quad [x,y] = (-1)^{|x||y|+1}[y,x], \end{displaymath} \begin{displaymath} (-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0 \end{displaymath} and \begin{displaymath} \partial[x,y] = [\partial x,y] + (-1)^{|x|}[x,\partial y] \end{displaymath} for every triple $(x,y,z)$ of homogeneous elements in $L$. \end{defn} Let $DGLA$ be the corresponding category. \begin{defn} \label{}\hypertarget{}{} A dgla is $n$-\emph{reduced} (resp. \emph{homologically $n$-reduced}) if $L_p = 0$ (resp. $H_p(L,\partial) = 0$) for all $p\lt n$. Denote by $DGLA_n$ (resp. $DGLA_{hn}$), the corresponding categories. \end{defn} \begin{defn} \label{}\hypertarget{}{} If $(L,\partial)$ is a pre-dgla, a \emph{gla-filtration} of $L$ (resp. a \emph{dgla-filtration} of $( L,\partial)$ ) is a family of subgraded vector spaces $F_p L$, $p\in \mathbb{Z}$, such that $F_p L\subseteq F_{p+1}L$, $[F_p L,F_n L]\subseteq F_{p+n} L$, (resp. and $\partial F_p L\subseteq F_p L$). \end{defn} \begin{defn} \label{}\hypertarget{}{} Let $L$ be a pre-gla. Its \emph{bracket length filtration} is obtained from the descending central series: \begin{displaymath} F^1 L = L; \quad F^p L = [L,F^{p-1} L] \quad if \quad p\geq 2. \end{displaymath} It is a gla-filtration. $Q(L) = L/F^2L$ is called the \emph{space of indecomposables} of $L$. If $(L,\partial)$ is a pre-dgla, $F^p L$ is stable by $\partial$. Letting $Q(\partial)$ be the induced differential on $Q(L)$, $Q$ then defines a functor \begin{displaymath} Q : pre DGLA \to pre DGVS. \end{displaymath} \end{defn} \begin{example} \label{}\hypertarget{}{} \textbf{Free Lie algebra, $\mathbb{L}(V)$} Let $V$ be a pre-gvs, $T(V)$, the tensor algebra on $V$ with augmentation ideal $\overline{T(V)}$ (recall $T(V) = \bigoplus_{n\geq 0} V^{\otimes n}$ and the augmentation sends $V (= V^{\otimes 1}$ to 0). Let $\overline{T(V)}_L$ be $\overline{T(V)}$ with the pre-gla structure given by the commutators. We denote by $\mathbb{L}(V)$, the Lie subalgebra of $\overline{T(V)}_L$ generated by $V$. [[Tim Porter|Tim]]: A more explicit description may help here, cf. Quillen, Rational Homotopy theory (p.281) or MacLane, Homology. If $L$ is a pre-gla, any morphism of pre-gvs $f: V\to L$ has a unique extension to a pre-gla morphism $\hat{f} :\mathbb{L}(V)\to L$. If $(e_\alpha)_{\alpha\in I}$ is a homogeneous basis for $V$, $\mathbb{L}(V)$ may be denoted $\mathbb{L}((e_\alpha)_{\alpha\in I})$. On the free Lie algebra $\mathbb{L}(V)$, the bracket length filtration comes from a gradation $\mathbb{L}(V) = \bigoplus_j\mathbb{L}^j(V)$, where $\mathbb{L}^j(V)$ is the subspace generated by the brackets of elements of $V$ of length $j$. The inclusion $\mathbb{L}(V)\hookrightarrow T(V)$ identifies $\mathbb{L}^j(V)$ with $\mathbb{L}(V)\cap T^j(V)$. If $\mathbb{L}(V), \partial)$ is a dgla, free as a gla, with $V$ fixed, $\partial$ is the sum of derivations $\partial_k$ defined by : $\partial_k \subset \mathbb{L}^k(V)$. The isomorphism between $V$ and $Q\mathbb{L}(V)$ identifies $\partial_1V$ with $Q(\partial)$. $\partial_1$ (resp. $\partial_2$) is called the \emph{linear part} (resp, the \emph{quadratic part}) of $\partial$. \end{example} \begin{defn} \label{}\hypertarget{}{} Let $(L,\partial)$ and $(L',\partial')$ be two dglas. Their \emph{product} $(L,\partial)\times(L',\partial')$ in $DGLA$ is defined by: \begin{itemize}% \item the underlying vector space is the direct sum $L\oplus L'$; \item $(L,\partial)$ and $(L',\partial')$ are two sub differential graded Lie algebras of $(L,\partial)\times(L',\partial')$; \item if $x\in L$ and $x' \in L'$, then $[x,x'] = 0$. \end{itemize} \end{defn} Their \emph{coproduct} or \emph{sum} $(L,\partial)\star(L',\partial')$ is often called their \emph{free product}. \begin{example} \label{}\hypertarget{}{} \begin{displaymath} \mathbb{L}(V)\star \mathbb{L}(V') \cong \mathbb{L}(V\oplus V'). \end{displaymath} More generally if $L$ and $L'$ are given by generators and relations \begin{displaymath} L = \mathbb{L}(V)/I , \quad L' = \mathbb{L}(V')/I' , \end{displaymath} \begin{displaymath} L\star L' = \mathbb{L}(V\oplus V')/{I,I'}. \end{displaymath} The differential on $L\star L'$ is the unique Lie algebra derivation extending $\partial$ and $\partial'$. \end{example} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure} \begin{itemize}% \item [[model structure on dg-Lie algebras]] \end{itemize} \hypertarget{RelationToLInfinityAlgebras}{}\subsubsection*{{Relation to $L_\infty$-algebras}}\label{RelationToLInfinityAlgebras} Every dg-Lie algebra is in an evident way an [[L-infinity algebra]]. Dg-Lie algebras are precisely those $L_\infty$-algebras for which all $n$-ary brackets for $n \gt 2$ are trivial. These may be thought of as the \emph{strict} $L_\infty$-algebras: those for which the [[Jacobi identity]] holds on the nose and all its possible higher coherences are trivial. \begin{theorem} \label{}\hypertarget{}{} Let $k$ be a [[field]] of [[characteristic]] 0 and write $L_\infty Alg_k$ for the [[category]] of [[L-infinity algebras]] over $k$. Then every object of $L_\infty Alg_k$ is [[quasi-isomorphism|quasi-isomorphic]] to a [[dg-Lie algebra]]. Moreover, one can find a functorial replacement: there is a [[functor]] \begin{displaymath} W : L_\infty Alg_k \to L_\infty Alg_k \end{displaymath} such that for each $\mathfrak{g} \in L_\infty Alg_k$ \begin{enumerate}% \item $W(\mathfrak{k})$ is a [[dg-Lie algebra]]; \item there is a [[quasi-isomorphism]] \begin{displaymath} \mathfrak{g} \stackrel{\simeq}{\to} W(\mathfrak{g}) \,. \end{displaymath} \end{enumerate} \end{theorem} This appears for instance as (\hyperlink{KrizMay}{KrizMay, cor. 1.6}). For more see at \begin{itemize}% \item \emph{[[model structure on dg-Lie algebras]]} the section \emph{\href{model+structure+on+dg-Lie+algebras#RectificationResolution}{Relation to L-infinity algebras}}. \item [[model structure for L-infinity algebras]], the section \emph{\href{model+structure+for+L-infinity+algebras#OndgLieAlgebras}{on dg-Lie algebras}}. \end{itemize} \hypertarget{relation_to_dgcoalgebras}{}\subsubsection*{{Relation to dg-coAlgebras}}\label{relation_to_dgcoalgebras} Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by adjunction to [[dg-coalgebras]] \begin{displaymath} dgLieAlg_k \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCoAlg_k \end{displaymath} Here \begin{itemize}% \item $CE$ is the [[Chevalley-Eilenberg algebra]] functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to \begin{displaymath} CE(\mathfrak{g},\partial,[-,-]) \;\coloneqq\; \left( \vee^\bullet \mathfrak{g}[1] ,\; D = \partial + [-,-] \right) \,, \end{displaymath} where on the right the extension of $\partial$ and $[-,-]$ to graded derivations is understood. \item For $(X,D)$ a dg-coalgebra, then \begin{displaymath} \mathcal{L}(X,D) \coloneqq \left( F(\overline{X}[-1]),\; \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1) \right) \end{displaymath} where \begin{enumerate}% \item $\overline{X} \coloneqq ker(\epsilon)$ is the [[kernel]] of the [[counit]], regarded as a [[chain complex]]; \item $F$ is the [[free Lie algebra]] functor (as graded Lie algebras); \item on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra [[derivation]] \end{enumerate} \end{itemize} Moreover \begin{displaymath} Hom(\mathcal{L}(X), \mathfrak{g}) \simeq Hom(X, CE(\mathfrak{g})) \simeq MC(Hom(\overline{X},\mathfrak{g})) \end{displaymath} is the [[Maurer-Cartan elements]] in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$. For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (\hyperlink{Quillen}{Quillen 69, appendix B, prop 6.1, 6.2}). For unbounded dg-algebras, this is due to (\hyperlink{Hinich98}{Hinich 98, 2.2}). For more see at \emph{[[model structure on dg-Lie algebras]]}. \hypertarget{relation_to_simplicial_lie_algebras}{}\subsubsection*{{Relation to simplicial Lie algebras}}\label{relation_to_simplicial_lie_algebras} \begin{theorem} \label{}\hypertarget{}{} There is an [[adjunction]] \begin{displaymath} (N^* \dashv N) : LieAlg^\Delta \stackrel{\overset{N^*}{\leftarrow}}{\underset{N}{\to}} dgLieAlg \end{displaymath} between [[simplicial Lie algebras]] and [[dg-Lie algebra]]s, where $N$ acts on the underlying simplicial vector spaces as the [[Moore complex]] functor. \end{theorem} This is (\hyperlink{Quillen}{Quillen, prop. 4.4}). For more see at \emph{[[simplicial Lie algebra]]}. \begin{theorem} \label{}\hypertarget{}{} This adjunction is a [[Quillen adjunction]] with respect to the projective [[model structure on dg-Lie algebras]] and the projective [[model structure on simplicial Lie algebras]] (\href{model+structure+on+simplicial+Lie+algebras#QuillenAdjunctionTodgLieAlgebras}{this prop.}). The corresponding [[derived functors]] constitute an [[equivalence of categories]] between the corresponding [[homotopy categories]] \begin{displaymath} (L N^* \dashv \tilde N) : Ho(LieAlg^\Delta)_1 \stackrel{\overset{L N^*}{\leftarrow}}{\underset{\tilde N}{\to}} Ho(dgLieAlg)_1 \end{displaymath} of [[1-connected]] objects on both sides. \end{theorem} This is in the proof of (\hyperlink{Quillen}{Quillen, theorem. 4.4}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[crossed complex]] $\Leftarrow$ [[crossed module]] $\Rightarrow$ [[2-crossed module]] \item \textbf{dg-Lie algebra} $\Leftarrow$ [[differential crossed module]] $\Rightarrow$ [[differential 2-crossed module]] \item [[L-∞ algebra]] \begin{itemize}% \item [[model structure for L-∞ algebras]] \item [[sheaf of L-∞ algebras]] \end{itemize} \item [[formal moduli problem]], [[deformation theory]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} A standard reference in the context of [[rational homotopy theory]] is \begin{itemize}% \item [[Dan Quillen]], \emph{Rational homotopy theory}, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (\href{http://www.jstor.org/stable/1970725}{JSTOR}) \end{itemize} For the unbounded case there is general discussion in \begin{itemize}% \item [[Vladimir Hinich]], \emph{Homological algebra of homotopy algebras}, Comm. in algebra, 25(10)(1997), 3291--3323 (\href{http://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}, \emph{Erratum} (\href{http://arxiv.org/abs/math/0309453}{arXiv:math/0309453})) \end{itemize} The relation to $L_\infty$-algebras is discussed for instance in \begin{itemize}% \item [[Igor Kriz]], [[Peter May]], \emph{Operads, algebras, modules and motives} (\href{http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf}{pdf}) \end{itemize} See also the references at [[model structure on dg-Lie algebras]]. A discussion of how formal neighbourhoods of points in [[infinity-stack]]s are governed by dg-Lie algebras: \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Moduli Problems and DG-Lie Algebras]]} \end{itemize} [[!redirects differential graded Lie algebras]] [[!redirects dg-Lie algebra]] [[!redirects Lie dg-algebra]] [[!redirects dg-Lie algebras]] [[!redirects Lie dg-algebras]] \end{document}