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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.
Contents
- Formal definition
- Adjoint representations
- Infinitesimal Lie group representations
- Basic concepts
- Basic constructions
- Enveloping algebras
- Induced representation
- Representations of a semisimple Lie algebra
- gK module
- Finite dimensional representations of semisimple Lie algebras
- Representation on an algebra
- References
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 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
from
Explicitly, this means that
for all x,y in
The representation
One can equivalently define a
for all x,y in
Adjoint representations
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra
Indeed, by virtue of the Jacobi identity,
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
determines a Lie algebra homomorphism
from
For example, let
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
Let V be a
- 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
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
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
We define
for any A in L, ω in V∗ and X in V. This defines
Let
Enveloping algebras
To each Lie algebra
Since
Induced representation
Let
Furthermore,
The induction is transitive:
Representations of a semisimple Lie algebra
Let
The category of modules over
(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
Finite-dimensional representations of semisimple Lie algebras
Similarly to how semisimple Lie algebras can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified. This is a beautiful, classical theory, described in several textbooks, including (Fulton & Harris 1992), (Hall 2015), and (Humphreys 1972).
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. The last point is the most difficult one; construction of the representations may be given by using Verma modules. This classification generalizes the more elementary representation theory of sl(2;C), where the irreducible representations are classified by the largest eigenvalue of the diagonal element H.
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] = 0Now, 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.