# 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

${\displaystyle \rho \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)}$

from ${\displaystyle {\mathfrak {g}}}$ to the Lie algebra of endomorphisms on a vector space V (with the commutator as the Lie bracket), sending an element x of ${\displaystyle {\mathfrak {g}}}$ to an element ρx of ${\displaystyle {\mathfrak {gl}}(V)}$.

Explicitly, this means that

${\displaystyle \rho _{[x,y]}=[\rho _{x},\rho _{y}]=\rho _{x}\rho _{y}-\rho _{y}\rho _{x}\,}$

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

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

One can equivalently define a ${\displaystyle {\mathfrak {g}}}$-module as a vector space V together with a bilinear map ${\displaystyle {\mathfrak {g}}\times V\to V}$ such that

${\displaystyle [x,y]\cdot v=x\cdot (y\cdot v)-y\cdot (x\cdot v)}$

for all x,y in ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ on itself:

${\displaystyle {\textrm {ad}}:{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}}),\quad x\mapsto \operatorname {ad} _{x},\quad \operatorname {ad} _{x}(y)=[x,y].}$

Indeed, by virtue of the Jacobi identity, ${\displaystyle \operatorname {ad} }$ 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 ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle {\mathfrak {h}}}$ are the Lie algebras of G and H respectively, then the differential ${\displaystyle d_{e}\phi :{\mathfrak {g}}\to {\mathfrak {h}}}$ 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

${\displaystyle \phi :G\to \mathrm {GL} (V)\,}$

determines a Lie algebra homomorphism

${\displaystyle d\phi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)}$

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

For example, let ${\displaystyle c_{g}(x)=gxg^{-1}}$. Then the differential of ${\displaystyle c_{g}:G\to G}$ at the identity is an element of ${\displaystyle \mathrm {GL} ({\mathfrak {g}})}$. Denoting it by ${\displaystyle \operatorname {Ad} (g)}$ one obtains a representation ${\displaystyle \operatorname {Ad} }$ of G on the vector space ${\displaystyle {\mathfrak {g}}}$. Applying the preceding, one gets the Lie algebra representation ${\displaystyle d\operatorname {Ad} }$. It can be shown that ${\displaystyle d_{e}\operatorname {Ad} =\operatorname {ad} .}$

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 ${\displaystyle {\mathfrak {g}}}$ be a Lie algebra. Let V, W be ${\displaystyle {\mathfrak {g}}}$-modules. Then a linear map ${\displaystyle f:V\to W}$ is a homomorphism of ${\displaystyle {\mathfrak {g}}}$-modules if it is ${\displaystyle {\mathfrak {g}}}$-equivariant; i.e., ${\displaystyle f(xv)=xf(v)}$ for any ${\displaystyle x\in {\mathfrak {g}},v\in V}$. If f is bijective, ${\displaystyle V,W}$ are said to be equivalent. Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.

Let V be a ${\displaystyle {\mathfrak {g}}}$-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 ${\displaystyle {\mathfrak {g}}}$ 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).[1] 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 ${\displaystyle {\mathfrak {g}}}$-invariant if ${\displaystyle xv=0}$ for all ${\displaystyle x\in {\mathfrak {g}}}$. The set of all invariant elements is denoted by ${\displaystyle V^{\mathfrak {g}}}$. ${\displaystyle V\mapsto V^{\mathfrak {g}}}$ 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

${\displaystyle x[v_{1}\otimes v_{2}]=x[v_{1}]\otimes v_{2}+v_{1}\otimes x[v_{2}].}$

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 ${\displaystyle (z\omega )[X]={\bar {z}}\omega [X]}$ 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].

We define ${\displaystyle {\bar {\rho }}}$ as follows:

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

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

Let ${\displaystyle V,W}$ be ${\displaystyle {\mathfrak {g}}}$-modules, ${\displaystyle {\mathfrak {g}}}$ a Lie algebra. Then ${\displaystyle \operatorname {Hom} (V,W)}$ becomes a ${\displaystyle {\mathfrak {g}}}$-module by setting ${\displaystyle (x\cdot f)(v)=xf(v)-f(xv)}$. In particular, ${\displaystyle \operatorname {Hom} _{\mathfrak {g}}(V,W)=\operatorname {Hom} (V,W)^{\mathfrak {g}}}$. Since any field becomes a ${\displaystyle {\mathfrak {g}}}$-module with a trivial action, taking W to be the base field, the dual vector space ${\displaystyle V^{*}}$ becomes a ${\displaystyle {\mathfrak {g}}}$-module.

## Enveloping algebras

To each Lie algebra ${\displaystyle {\mathfrak {g}}}$ over a field k, one can associate a certain ring called the universal enveloping algebra of ${\displaystyle {\mathfrak {g}}}$. The construction is universal and consequently (along with the PBW theorem) representations of ${\displaystyle {\mathfrak {g}}}$ corresponds in one-to-one with algebra representations of universal enveloping algebra of ${\displaystyle {\mathfrak {g}}}$. The construction is as follows.[2] Let T be the tensor algebra of the vector space ${\displaystyle {\mathfrak {g}}}$. Thus, by definition, ${\displaystyle T=\oplus _{n=0}^{\infty }\otimes _{1}^{n}{\mathfrak {g}}}$ and the multiplication on it is given by ${\displaystyle \otimes }$. Let ${\displaystyle U({\mathfrak {g}})}$ be the quotient ring of T by the ideal generated by elements ${\displaystyle [x,y]-x\otimes y+y\otimes x}$. Since ${\displaystyle U({\mathfrak {g}})}$ is an associative algebra over the field k, it can be turned into a Lie algebra via the commutator ${\displaystyle [x,y]=xy-yx}$ (omitting ${\displaystyle \otimes }$ from the notation). There is a canonical morphism of Lie algebras ${\displaystyle {\mathfrak {g}}\to U({\mathfrak {g}})}$ obtained by restricting ${\displaystyle T\to U({\mathfrak {g}})}$ to degree one piece. The PBW theorem implies that the canonical map is actually injective. Note if ${\displaystyle {\mathfrak {g}}}$ is abelian, then ${\displaystyle U({\mathfrak {g}})}$ is the symmetric algebra of the vector space ${\displaystyle {\mathfrak {g}}}$.

Since ${\displaystyle {\mathfrak {g}}}$ is a module over itself via adjoint representation, the enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ becomes a ${\displaystyle {\mathfrak {g}}}$-module by extending the adjoint representation. But one can also use the left and right regular representation to make the enveloping algebra a ${\displaystyle {\mathfrak {g}}}$-module; namely, with the notation ${\displaystyle l_{x}(y)=xy,x\in {\mathfrak {g}},y\in U({\mathfrak {g}})}$, the mapping ${\displaystyle x\mapsto l_{x}}$ defines a representation of ${\displaystyle {\mathfrak {g}}}$ on ${\displaystyle U({\mathfrak {g}})}$. The right regular representation is defined similarly.

## Induced representation

Let ${\displaystyle {\mathfrak {g}}}$ be a finite-dimensional Lie algebra over a field of characteristic zero and ${\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}$ a subalgebra. ${\displaystyle U({\mathfrak {h}})}$ acts on ${\displaystyle U({\mathfrak {g}})}$ from the right and thus, for any ${\displaystyle {\mathfrak {h}}}$-module W, one can form the left ${\displaystyle U({\mathfrak {g}})}$-module ${\displaystyle U({\mathfrak {g}})\otimes _{U({\mathfrak {h}})}W}$. It is a ${\displaystyle {\mathfrak {g}}}$-module denoted by ${\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W}$ and called the ${\displaystyle {\mathfrak {g}}}$-module induced by W. It satisfies (and is in fact characterized by) the universal property: for any ${\displaystyle {\mathfrak {g}}}$-module E

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

Furthermore, ${\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}}$ is an exact functor from the category of ${\displaystyle {\mathfrak {h}}}$-modules to the category of ${\displaystyle {\mathfrak {g}}}$-modules. These uses the fact that ${\displaystyle U({\mathfrak {g}})}$ is a free right module over ${\displaystyle U({\mathfrak {h}})}$. In particular, if ${\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}W}$ is simple (resp. absolutely simple), then W is simple (resp. absolutely simple). Here, a ${\displaystyle {\mathfrak {g}}}$-module V is absolutely simple if ${\displaystyle V\otimes _{k}F}$ is simple for any field extension ${\displaystyle F/k}$.

## Representations of a semisimple Lie algebra

Let ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ 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.[3]

## (g,K)-module

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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 ${\displaystyle \pi }$ is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification ${\displaystyle {\mathfrak {g}}}$ and the connected maximal compact subgroup K. The ${\displaystyle {\mathfrak {g}}}$-module structure of ${\displaystyle \pi }$ allows algebraic especially homological methods to be applied and ${\displaystyle K}$-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[1] = 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.

## References

• Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of Representations that are generated by vectors of highest weight," Functional. Anal. Appl. 5 (1971)
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• A. Beilinson and J. Bernstein, "Localisation de g-modules," C. R. Acad. Sci. Paris Sér. I Math., vol. 292, iss. 1, pp. 15–18, 1981.
• Template:Fulton-Harris
• D. Gaitsgory, Geometric Representation theory, Math 267y, Fall 2005
• Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, D-modules, perverse sheaves, and representation theory; translated by Kiyoshi Takeuch
• J.Humphreys, Introduction to Lie algebras and representation theory, Birkhäuser, 2000.
• N. Jacobson, Lie algebras, Courier Dover Publications, 1979.
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