nLab
Lie algebra

Idea

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

Definition

Ordinary definition

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

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

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

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

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 $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(C,\otimes,1)$ with the symmetry $\tau$, a Lie algebra in $(C,\otimes,1,\tau)$ is an object $L$ in $C$ together with a morphism $[-,-]: A\otimes A\to A$ such that the Jacobi identity

$$\left[-,\left[-,-\right]\right]+\left[-,\left[-,-\right]\right]\circ(id_L\otimes\tau_{L,L})\circ(\tau\otimes id_L)+\left[-,\left[-,-\right]\right]\circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0$$

and antisymmetry

$$[-,-]+[-,-]\circ \tau_{L,L} = 0$$

hold. If $k$ is the ring $\mathbb{Z}$ of integers, then we say (internal) Lie ring, and if $k$ is a field and $C=Vec$ then we say a Lie $k$-algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category $\mathbb{Z}_2-Vec$ 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-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

• extra property

  • abelian Lie algebra

  • simple Lie algebra

  • semisimple Lie algebra

  • reductive Lie algebra

• extra structure

  • super Lie algebra

• extra stuff

  • ∞-Lie algebra

    • simplicial Lie algebra

  • ∞-Lie algebroid

Properties

General

• PBW theorem

Cohomology

See Lie algebra cohomology.

Lie theory

See

• Lie theory

  • Lie integration

  • Lie's three theorems

Examples

• translation Lie algebra, line Lie algebra

• derivation Lie algebra

• endomorphism Lie algebra

• general linear Lie algebra

• orthogonal Lie algebra

• special orthogonal Lie algebra

• special unitary Lie algebra

• Poincare Lie algebra

• super Poincare Lie algebra

Related concepts

• Lie algebra

  • universal enveloping algebra, Cartan subalgebra

  • root (in representation theory), weight (in representation theory)

  • Lie algebra representation

  • Lie ideal

  • Kac-Moody algebra

  • complex Lie algebra

  • restricted Lie algebra

  • Lie group

  • Lie-Poisson structure

  • Lie algebra extension

    • semidirect product Lie algebra

  • Lie algebra contraction

• Lie 2-algebra, Lie 2-group

• L-∞ algebra, smooth ∞-group

• Lie algebroid, Lie groupoid

• L-∞ algebroid, smooth ∞-groupoid

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

• 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

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

Discussion in synthetic differential geometry is in

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