# Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.

The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.

In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays a decisive role. The universality of this construction of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra.

## Formal definition

$\rho \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)$ Explicitly, this means that

$\rho _{[x,y]}=[\rho _{x},\rho _{y}]=\rho _{x}\rho _{y}-\rho _{y}\rho _{x}\,$ for all x,y in ${\mathfrak {g}}$ . The vector space V, together with the representation ρ, is called a ${\mathfrak {g}}$ -module. (Many authors abuse terminology and refer to V itself as the representation).

The representation $\rho$ is said to be faithful if it is injective.

$[x,y]\cdot v=x\cdot (y\cdot v)-y\cdot (x\cdot v)$ for all x,y in ${\mathfrak {g}}$ and v in V. This is related to the previous definition by setting xv = ρx (v).

## Examples

{{#invoke:main|main}} The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra ${\mathfrak {g}}$ on itself:

${\textrm {ad}}:{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}}),\quad x\mapsto \operatorname {ad} _{x},\quad \operatorname {ad} _{x}(y)=[x,y].$ Indeed, by virtue of the Jacobi identity, $\operatorname {ad}$ is a Lie algebra homomorphism.

### Infinitesimal Lie group representations

A Lie algebra representation also arises in nature. If φ: GH is a homomorphism of (real or complex) Lie groups, and ${\mathfrak {g}}$ and ${\mathfrak {h}}$ are the Lie algebras of G and H respectively, then the differential $d_{e}\phi :{\mathfrak {g}}\to {\mathfrak {h}}$ on tangent spaces at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space V, a representation of Lie groups

$\phi :G\to \mathrm {GL} (V)\,$ determines a Lie algebra homomorphism

$d\phi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)$ from ${\mathfrak {g}}$ to the Lie algebra of the general linear group GL(V), i.e. the endomorphism algebra of V.

A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.

## Basic concepts

Let V be a ${\mathfrak {g}}$ -module. Then V is said to be semisimple or completely reducible if it satisfies the following equivalent conditions: (cf. semisimple module)

1. V is a direct sum of simple modules.
2. V is the sum of its simple submodules.
3. Every submodule of V is a direct summand: for every submodule W of V, there is a complement P such that V = W ⊕ P.

If ${\mathfrak {g}}$ is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple (Weyl's complete reducibility theorem). A Lie algebra is said to be reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive. An element v of V is said to be ${\mathfrak {g}}$ -invariant if $xv=0$ for all $x\in {\mathfrak {g}}$ . The set of all invariant elements is denoted by $V^{\mathfrak {g}}$ . $V\mapsto V^{\mathfrak {g}}$ is a left-exact functor.

## Basic constructions

If we have two representations, with V1 and V2 as their underlying vector spaces and ·[·]1 and ·[·]2 as the representations, then the product of both representations would have V1V2 as the underlying vector space and

$x[v_{1}\otimes v_{2}]=x[v_{1}]\otimes v_{2}+v_{1}\otimes x[v_{2}].$ If L is a real Lie algebra and ρ: L × VV is a complex representation of it, we can construct another representation of L called its dual representation as follows.

Let V be the dual vector space of V. In other words, V is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that $(z\omega )[X]={\bar {z}}\omega [X]$ for any z in C, ω in V and X in V. This is usually rewritten as a contraction with a sesquilinear form ⟨·,·⟩. i.e. ⟨ω,X⟩ is defined to be ω[X].

${\bar {\rho }}$ (A)[ω],X⟩ + ⟨ω, ρA[X]⟩ = 0,

for any A in L, ω in V and X in V. This defines ${\bar {\rho }}$ uniquely.

## Induced representation

$\operatorname {Hom} _{\mathfrak {g}}(\operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W,E)\simeq \operatorname {Hom} _{\mathfrak {h}}(W,\operatorname {Res} _{\mathfrak {h}}^{\mathfrak {g}}E)$ .

## Representations of a semisimple Lie algebra

Let ${\mathfrak {g}}$ be a finite-dimensional semisimple Lie algebra over a field of characteristic zero. (in the solvable or nilpotent case, one studies primitive ideals of the enveloping algebra; cf. Dixmier for the definitive account.)

The category of modules over ${\mathfrak {g}}$ turns out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory category O is a better place for the representation theory in the semisimple case in zero characteristic. For instance, the category O turned out to be of a right size to formulate the celebrated BGG reciprocity.

## (g,K)-module

{{#invoke:main|main}}

One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie group. The application is based on the idea that if $\pi$ is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification ${\mathfrak {g}}$ and the connected maximal compact subgroup K. The ${\mathfrak {g}}$ -module structure of $\pi$ allows algebraic especially homological methods to be applied and $K$ -module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.

## Classification

### Finite-dimensional representations of semisimple Lie algebras

Template:Expand section Template:Rellink Similarly to how semisimple Lie algebras can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified. This is a classical theory, widely regarded as beautiful, and a standard reference is Template:Harv.

Briefly, finite-dimensional representations of a semisimple Lie algebra are completely reducible, so it suffices to classify irreducible (simple) representations. Semisimple Lie algebras are classified in terms of the weights of the adjoint representation, the so-called root system; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights; see weight (representation theory) for details.

## Representation on an algebra

If we have a Lie superalgebra L, then a representation of L on an algebra is a (not necessarily associative) Z2 graded algebra A which is a representation of L as a Z2 graded vector space and in addition, the elements of L acts as derivations/antiderivations on A.

More specifically, if H is a pure element of L and x and y are pure elements of A,

H[xy] = (H[x])y + (−1)xHx(H[y])

Also, if A is unital, then

H = 0

Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors.

A Lie (super)algebra is an algebra and it has an adjoint representation of itself. This is a representation on an algebra: the (anti)derivation property is the superJacobi identity.

If a vector space is both an associative algebra and a Lie algebra and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra.