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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.

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 $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(\mathcal{C},\otimes,1)$ with braiding $\tau$.

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

1. an object

   $$L \in \mathcal{C}$$

2. morphism (the Lie bracket)

   $$[-,-] \;\colon\; L \otimes L \to L$$

such that the following conditions hold:

1. Jacobi identity:

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

2. skew-symmetry:

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

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 $k$ is the ring $\mathbb{Z}$ of integers and $\mathcal{C} =$ $k$Mod = 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 $k$ is a field and $\mathcal{C} =$ Vect is the category of vector spaces over $k$ equipped with the tensor product of vector spaces then a Lie algebra object is an ordinary_Lie k-algebra.

If $k$ is a field and $\mathcal{C}$ = sVect is the category of super vector spaces over $k$, 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

• 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

  • metric 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

• Lie coalgebra

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

• 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

• 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)