nLab
Lie algebra

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A Lie algebra is the infinitesimal approximation to a Lie group.

Definition

Ordinary definition

A Lie algebra is a vector space 𝔤\mathfrak{g} equipped with a bilinear skew-symmetric map [,]:𝔤𝔤𝔤[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g} which satisfies the Jacobi identity:

x,y,z𝔤:[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0. \forall x,y,z \in \mathfrak{g} : \left[x,\left[y,z\right]\right] + \left[z,\left[x,y\right]\right] + \left[y,\left[z,x\right]\right] = 0 \,.

A homomorphism of Lie algebras is a linear map ϕ:𝔤𝔥\phi : \mathfrak{g} \to \mathfrak{h} such that for all x,y𝔤x,y \in \mathfrak{g} we have

ϕ([x,y] 𝔤)=[ϕ(x),ϕ(y)] 𝔥. \phi([x,y]_{\mathfrak{g}}) = [\phi(x),\phi(y)]_{\mathfrak{h}} \,.

This defines the category LieAlg of Lie algebras.

Internal to a general linear category

The notion of Lie algebra may be formulated internal to any linear category. This general definition subsumes variants of Lie algebras such as super Lie algebras.

Consider a commutative unital ring kk, and a (strict for simplicity) symmetric monoidal kk-linear category (𝒞,,1)(\mathcal{C},\otimes,1) with braiding τ\tau.

A Lie algebra object in (𝒞,,1,τ)(\mathcal{C},\otimes,1,\tau) is

  1. an object

    L𝒞 L \in \mathcal{C}
  2. morphism (the Lie bracket)

    [,]:LLL [-,-] \;\colon\; L \otimes L \to L

such that the following conditions hold:

  1. Jacobi identity:

    [,[,]]+[,[,]](id Lτ L,L)(τid L)+[,[,]](τ L,Lid L)(id Lτ L,L)=0 \left[-,\left[-,-\right]\right] + \left[-,\left[-,-\right]\right] \circ(id_L\otimes\tau_{L,L}) \circ(\tau\otimes id_L) + \left[-,\left[-,-\right]\right] \circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0
  2. skew-symmetry:

    +[,] +[,]τ L,L =+0 \begin{aligned} & \phantom{+} [-,-] \\ & + [-,-]\circ \tau_{L,L} \\ & = \phantom{+} 0 \end{aligned}

Equivalently, Lie algebra objects are the algebras over an operad over a certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.

Examples of types of Lie algebra objects:

If kk is the ring \mathbb{Z} of integers and 𝒞=\mathcal{C} = kkMod = Ab is the category of abelian groups equipped with the tensor product of abelian groups, then a Lie algebra object is called a Lie ring.

If kk is a field and 𝒞=\mathcal{C} = Vect is the category of vector spaces over kk equipped with the tensor product of vector spaces then a Lie algebra object is an ordinary_Lie k-algebra.

If kk is a field and 𝒞\mathcal{C} = sVect is the category of super vector spaces over kk, then a Lie algebra object is a super Lie algebra.

General abstract perspective

Lie algebras are equivalently groups in “infinitesimal geometry”.

For instance in synthetic differential geometry then a Lie algebra of a Lie group is just the first-order infinitesimal neighbourhood of the unit element (e.g. Kock 09, section 6).

More generally in geometric homotopy theory, Lie algebras, being 0-truncated L-∞ algebras are equivalently “infinitesimal ∞-group geometric ∞-stacks” (e.g. here-topos#FormalModuliProblems)), also called formal moduli problems (see there for more).

Extra stuff, structure, properties

Notions of Lie algebras with extra stuff, structure, property includes

Properties

General

Cohomology

See Lie algebra cohomology.

Lie theory

See

Examples

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

  • A. L. Onishchik (ed.) Lie Groups and Lie Algebras

    • I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,

    • II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups

    Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

  • Eckhard Meinrenken, Lie groups and Lie algebas, Lecture notes 2010 (pdf)

Discussion with a view towards Chern-Weil theory is in chapter IV in vol III of

Discussion in synthetic differential geometry is in

  • Anders Kock, section 6 of Synthetic Geometry of Manifolds, 2009 (pdf)

Last revised on December 1, 2019 at 04:57:47. See the history of this page for a list of all contributions to it.