Lie algebra representation

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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 ρ_x 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 x ⋅ v = ρ_x (v).

Examples

Adjoint representations

Main article: Adjoint representation of a Lie algebra

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: G → H 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.

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 $d_e\operatorname{Ad} = \operatorname{ad}.$

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.^[1]

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)

V is a direct sum of simple modules.
V is the sum of its simple submodules.
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).^[2] 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 V₁ ⊗ V₂ 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 × V→ V 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.^[3] 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.^[4]

(g,K)-module

Main articles: (g,K)-module and Harish-Chandra 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. The irreducible representations, in turn, may be classified by the "theorem of the highest weight." The theorem states that every irreducible representation has a dominant integral highest weight, two irreducible representations with the same highest weight are isomorphic, and that every dominant integral element occurs as the highest weight of some irreducible representation.^[5] The last point is the most difficult one; construction of the representations may be given by using Verma modules.^[6]

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 super Jacobi 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.

Notes

↑ Hall 2015 Theorem 5.6
↑ Dixmier 1977, Theorem 1.6.3
↑ Jacobson 1962
↑ http://mathoverflow.net/questions/64931/why-the-bgg-category-o
↑ Hall 2015 Theorems 9.4 and 9.5
↑ Hall 2015 Sections 9.5-9.7

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
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics 222 (2nd ed.), Springer
Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, D-modules, perverse sheaves, and representation theory; translated by Kiyoshi Takeuch
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.
N. Jacobson, Lie algebras, Courier Dover Publications, 1979.
Garrett Birkhoff; Philip M. Whitman (1949). "Representation of Jordan and Lie Algebras" (PDF). Trans. Amer. Math. Soc. 65: 116–136.