# Lie algebra representation

(Redirected from Representation of a Lie algebra)

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

A representation of a Lie algebra $\mathfrak g$ is a Lie algebra homomorphism

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

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

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.

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

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

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

## Examples

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 $d \phi: \mathfrak g \to \mathfrak h$ 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.

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

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 $\mathfrak{g}$ be a Lie algebra. Let V, W be $\mathfrak{g}$-modules. Then a linear map $f: V \to W$ is a homomorphism of $\mathfrak{g}$-modules if it is $\mathfrak{g}$-equivariant; i.e., $f(xv) = xf(v)$ for any $x \in \mathfrak{g}, v \in V$. If f is bijective, $V, W$ 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 $\mathfrak{g}$-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).[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 $\mathfrak{g}$-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].

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

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

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

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

## Enveloping algebras

To each Lie algebra $\mathfrak{g}$ over a field k, one can associate a certain ring called the universal enveloping algebra of $\mathfrak{g}$. The construction is universal and consequently (along with the PBW theorem) representations of $\mathfrak{g}$ corresponds in one-to-one with algebra representations of universal enveloping algebra of $\mathfrak{g}$. The construction is as follows.[2] Let T be the tensor algebra of the vector space $\mathfrak{g}$. Thus, by definition, $T = \oplus_{n=0}^\infty \otimes_1^n \mathfrak{g}$ and the multiplication on it is given by $\otimes$. Let $U(\mathfrak{g})$ be the quotient ring of T by the ideal generated by elements $[x, y] - x \otimes y + y \otimes x$. Since $U(\mathfrak{g})$ is an associative algebra over the field k, it can be turned into a Lie algebra via the commutator $[x, y] = x y - yx$ (omitting $\otimes$ from the notation). There is a canonical morphism of Lie algebras $\mathfrak{g} \to U(\mathfrak{g})$ obtained by restricting $T \to U(\mathfrak{g})$ to degree one piece. The PBW theorem implies that the canonical map is actually injective. Note if $\mathfrak{g}$ is abelian, then $U(\mathfrak{g})$ is the symmetric algebra of the vector space $\mathfrak{g}$.

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

## Induced representation

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

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

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

The induction is transitive: $\operatorname{Ind}_\mathfrak{h}^\mathfrak{g} \simeq \operatorname{Ind}_\mathfrak{h'}^\mathfrak{g} \circ \operatorname{Ind}_\mathfrak{h}^\mathfrak{h'}$ for any Lie subalgebra $\mathfrak{h'} \subset \mathfrak{g}$ and any Lie subalgebra $\mathfrak{h} \subset \mathfrak{h}'$. The induction commutes with restriction: let $\mathfrak{h} \subset \mathfrak{g}$ be subalgebra and $\mathfrak{n}$ an ideal of $\mathfrak{g}$ that is contained in $\mathfrak{h}$. Set $\mathfrak{g}_1 = \mathfrak{g}/\mathfrak{n}$ and $\mathfrak{h}_1 = \mathfrak{h}/\mathfrak{n}$. Then $\operatorname{Ind}^\mathfrak{g}_\mathfrak{h} \circ \operatorname{Res}_\mathfrak{h} \simeq \operatorname{Res}_\mathfrak{g} \circ \operatorname{Ind}^\mathfrak{g_1}_\mathfrak{h_1}$.

## 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.[3]

## (g,K)-module

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

For more details on this topic, see Weight (representation theory).

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 (Fulton & Harris 1992).

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)
• Dixmier, J. (1977), Enveloping Algebras, Amsterdam, New York, Oxford: North-Holland, ISBN 0-444-11077-1.
• 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.
• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
• 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.