nLab
Lie algebra

Context

\infty-Lie theory

∞-Lie theory

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} \vee \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} : [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 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.

In a general linear category

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

Given a commutative unital ring kk, and a (strict for simplicity) symmetric monoidal kk-linear category (C,,1)(C,\otimes,1) with the symmetry τ\tau, a Lie algebra in (C,,1,τ)(C,\otimes,1,\tau) is an object LL in CC together with a morphism [,]:AAA[,]: A\otimes A\to A such that the Jacobi identity

[,[,]]+[,[,]](id Lτ L,L)(τid L)+[,[,]](τ L,Lid L)(id Lτ L,L)=0[,[,]]+[,[,]]\circ(id_L\otimes\tau_{L,L})\circ(\tau\otimes id_L)+[,[,]]\circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0

and antisymmetry

[,]+[,]τ L,L=0[,]+[,]\otimes\tau_{L,L} = 0

hold. If kk is the ring \mathbb{Z} of integers, then we say (internal) Lie ring, and if kk is a field and C=VecC=Vec then we say a Lie kk-algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category 2Vec\mathbb{Z}_2-Vec of supervector spaces and the Lie algebras in the Loday-Pirashvili tensor category of linear maps.

Alternatively, Lie algebras are the algebras over certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.

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), 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

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)

Revised on July 25, 2014 14:49:43 by David Roberts (203.134.149.74)