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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*{universal enveloping algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{for_lie_algebras}{For Lie algebras}\dotfill \pageref*{for_lie_algebras} \linebreak \noindent\hyperlink{for_algebras}{For $L_\infty$-algebras}\dotfill \pageref*{for_algebras} \linebreak \noindent\hyperlink{existence}{Existence}\dotfill \pageref*{existence} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{isomorphism_problem}{Isomorphism problem}\dotfill \pageref*{isomorphism_problem} \linebreak \noindent\hyperlink{poisson_algebra_structure_on_}{Poisson algebra structure on $U(\mathfrak{g})$}\dotfill \pageref*{poisson_algebra_structure_on_} \linebreak \noindent\hyperlink{hopf_algebra_structure_on_}{Hopf algebra structure on $U(\mathfrak{g})$}\dotfill \pageref*{hopf_algebra_structure_on_} \linebreak \noindent\hyperlink{pbw_theorem}{PBW theorem}\dotfill \pageref*{pbw_theorem} \linebreak \noindent\hyperlink{relation_to_formal_deformation_quantization}{Relation to formal deformation quantization}\dotfill \pageref*{relation_to_formal_deformation_quantization} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{universal_enveloping_of_a_tangent_lie_algebra}{Universal enveloping of a tangent Lie algebra}\dotfill \pageref*{universal_enveloping_of_a_tangent_lie_algebra} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{for_lie_algebras}{}\subsubsection*{{For Lie algebras}}\label{for_lie_algebras} Given a [[Lie algebra]] $L$ internal to some [[symmetric monoidal category|symmetric monoidal]] $k$-linear category $C = (C,\otimes, \mathbf{1},\tau)$, an \textbf{enveloping monoid} (or \textbf{enveloping algebra}) of $L$ in $C$ is any morphism $f: L\to Lie(A)$ of Lie algebras in $C$ where $A$ is a monoid (= algebra) in $C$, and $Lie(A)$ is the underlying object of $A$ equipped with the Lie bracket $[,]_{Lie(A)}=\mu-\mu\circ\tau_{A,A}$. In further we will just write $A$ for $Lie(A)$. A morphism of enveloping algebras $\phi : (f:L\to A)\to (f':L\to A')$ is a morphism $g: A\to A'$ of monoids completing a commutative triangle of morphisms in $C$, i.e. $g\circ f = f'$. With an obvious composition of morphisms, the enveloping algebras of $L$ form a category. A \textbf{universal enveloping algebra} of $L$ in $C$ is any universal [[initial object]] $i_L:L\to U(L)$ in the category of enveloping algebras of $L$; it is of course unique up to an isomorphism if it exists. If it exists for all Lie algebras in $C$, then the rule $L\mapsto U(L)$ can be extended to a functor $U$ which is the left adjoint to the functor $Lie:A\mapsto Lie(A)$ defined above and the morphism $i_L:L\to U(L)$ is the unit of the adjunction. \hypertarget{for_algebras}{}\subsubsection*{{For $L_\infty$-algebras}}\label{for_algebras} In the more general context of [[higher algebra]] there is a notion of universal enveloping [[E-n algebra]] of an [[L-infinity algebra]] for all $n \in \mathbb{N}$ which generalizes the notion of universal associative algebra envelope of a Lie algebra. See at \emph{[[universal enveloping E-n algebra]]}. \hypertarget{existence}{}\subsection*{{Existence}}\label{existence} The existence of the universal enveloping algebra is easy in many concrete symmetric monoidal categories, e.g. the symmetric monoidal category of bounded chain complexes (giving the universal enveloping [[differential graded algebra|dg-algebra]] of a [[differential graded Lie algebra|dg-Lie algebra]]), but not true in general. First of all if $C$ admits countable coproducts, form the tensor algebra $TL=\coprod_{n=0}^\infty L^{\otimes n}$ on the object $L$; this is a monoid in $C$. In most standard cases, one can also form the smallest 2-sided ideal (i.e. $A$-subbimodule) $I$ in monoid $A$ among those ideals whose inclusion into $A$ is factorizing the map $([,]-m_{TL}+m_{TL}\circ\tau)\circ \otimes :L\otimes L\to TL$; if the coequalizers exist in $C$ then we can form the quotient object $TL/I$ and there is an induced monoid structure in it. Under mild conditions on $C$, the natural morphism $i_L:L\to TL/I$ is an universal enveloping monoid of $L$ in $C$. If $C$ is an abelian tensor category and some flatness conditions on the tensor product are satisfied, then the enveloping monoid $i_L:L\to TL/I$ is a monic morphism in $C$ and $U(L\coprod L)\cong U(L)\otimes U(L)$. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{isomorphism_problem}{}\subsubsection*{{Isomorphism problem}}\label{isomorphism_problem} The \emph{isomorphism problem} for enveloping algebras is about the fact that the universal enveloping monoids of two Lie algebras of $C$ are isomorphic as associative [[monoids]] in $C$, but this does not imply that the Lie algebras are isomorphic. This is even not true in general for the Lie $k$-algebras (in classical sense), even if $k$ is a [[field]] of characteristics zero. It is known however in that case that the dimension of the finite-dimensional Lie $k$-algebra $L$ can be read off from its universal enveloping $k$-algebra as its Gel'fand-Kirillov dimension $GK(U(L))$. \hypertarget{poisson_algebra_structure_on_}{}\subsubsection*{{Poisson algebra structure on $U(\mathfrak{g})$}}\label{poisson_algebra_structure_on_} The universal enveloping algebra $U(\mathfrak{g})$ of a [[Lie algebra]] is naturally a (non-commutative) [[Poisson algebra]] with the restriction of the Poisson bracket to generators being the original [[Lie bracket]] \hypertarget{hopf_algebra_structure_on_}{}\subsubsection*{{Hopf algebra structure on $U(\mathfrak{g})$}}\label{hopf_algebra_structure_on_} Suppose the universal enveloping algebras of Lie algebras exist in a $k$-linear symmetric monoidal category $C$ and the functorial choice $L\mapsto U(L)$ realizing the above construction with tensor products is fixed. For example, this is true in the category of $k$-[[modules]] where $k$ is a [[commutative ring]]. Then the projection $L\to 0$ (where $0$ is the trivial Lie algebra) induces the counit $\epsilon:U(L)\to U(0)=\mathbf{1}$. The coproduct $\Delta:U(L)\to U(L\times L)\cong U(L)\otimes U(L)$ is induced by the [[diagonal]] map $L\to L\times L$ whereas the antipode $S=U(-id):U(L)\to U(L)$. One checks that these morphisms make $U(L)$ into a [[Hopf algebra]] in $C$. (e.g \hyperlink{MilnorMoore65}{Milnor-Moore 65, section 5}) The \emph{[[Milnor-Moore theorem]]} states conditions under which the converse holds (hence under which a primitively generated Hopf algebra is a universal enveloping algebra of a Lie algebra). If the category is simply the vector spaces over a field $k$, then for $l\in L$, after we identify $L$ with its image in $U(L)$, $\Delta(l) = l\otimes 1 + 1\otimes l$, i.e. the elements in $L$ are the [[primitive element]]s in $U(L)$. \hypertarget{pbw_theorem}{}\subsubsection*{{PBW theorem}}\label{pbw_theorem} The [[Poincaré–Birkhoff–Witt theorem]] states that the [[associated graded]] algebra of an enveloping algebra $U(g)$ in [[characteristics zero]] is canonically isomorphic to a [[symmetric algebra]] $Sym(g)$, and $U(g)$ is isomorphic to $S(g)$ as a coalgebra, via the projection map $U(g)\to Gr U(g)$. \hypertarget{relation_to_formal_deformation_quantization}{}\subsubsection*{{Relation to formal deformation quantization}}\label{relation_to_formal_deformation_quantization} See at \emph{[[deformation quantization]]} the section \emph{\href{deformation+quantization#RelationToUniversalEnvelopingAlgebras}{Relation to universal enveloping algebras}}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{universal_enveloping_of_a_tangent_lie_algebra}{}\subsubsection*{{Universal enveloping of a tangent Lie algebra}}\label{universal_enveloping_of_a_tangent_lie_algebra} The universal enveloping algebra of the [[tangent Lie algebra]] of a finite-dimensional Lie group $G$ over real or complex numbers is canonically isomorphic to the algebra of the left invariant [[differential operators]] on $G$. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Hausdorff series]], [[Poincaré–Birkhoff–Witt theorem]], [[coexponential map]], [[Duflo isomorphism]] \item An [[oidification]] is the [[universal enveloping algebroid]]. \item [[universal enveloping E-n algebra]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[N. Bourbaki]], \emph{Lie groups and Lie algebras} \item [[Jacques Dixmier]]; 1974; \emph{[[Algèbres enveloppantes]]}, Cahiers Scientifique. English translation: 1996; \emph{Enveloping Algebras}, Graduate Studies in Mathematics 11, American Mathematical Society. \item [[John Milnor]], [[John Moore]], \emph{On the structure of Hopf algebras}, Annals of Math. \textbf{81} (1965), 211-264. (\href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf}{pdf}) \item wikipedia: \href{http://en.wikipedia.org/wiki/Universal_enveloping_algebra}{universal enveloping algebra}, \href{http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem}{PBW theorem} \item F. A. Berezin, \emph{Some remarks about the associated envelope of a Lie algebra}, Func. Analysis and Its Appl. 1967, 1:2, 91--102; Rus. original in Fun. Anal. Pril. 1:2 (1967) 1--14 \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2813&volume=1&year=1967&issue=2&fpage=1&what=fullt&option_lang=eng}{Rus. pdf} \href{http://www.ams.org/mathscinet-getitem?mr=219671}{MR219671} \item MathOverflow question \href{http://mathoverflow.net/questions/25020/what-is-the-universal-enveloping-algebra}{What is the universal enveloping algebra} which is looking for a rather general construction in a class of symmetric monoidal [[pseudoabelian category|pseudoabelian categories]]. \end{itemize} [[!redirects universal enveloping algebras]] \end{document}