Lie algebra/representation

In mathematics, more specific algebra, a Representation of a Lie algebra is a simplification of the abstract Lie algebra to a simpler matrix Lie algebra. A representation of a Lie algebra g is a homomorphism from the Lie algebra to the endomorphisms (linear maps) of some vector space.

Examples

• Given a matrix Lie algebra g⊆End(V) of some vector space V, then i:g⊆End(V) is a representation. If the dimension of V is minimal, this is called the fundamental representation of the matrix Lie algebra.

Note that for so(3) this is the representation via Pauli matrices, i.e. on C² as su(2).

Adjoint representation

Let g be a Lie algebra. Consider the vector space V=g and construct the linear maps ad_X:V→V: Y→[X,Y], then ad:g→End(V): X→ad_X is a representation of g called the adjoint representation.

Direct sum and tensor product

Given two representations ρ_i:g→End(V_i) we can consider its direct sum ρ₁⊕ρ₂:g→End(V₁⊕V₂): X→ρ₁(X)⊕ρ₂(X).

Conversely, given a Lie algebra representation ρ we can ask whether it can non-trivially be written as direct sum of two representations. If it cannot and is not the 0-representation, then ρ is called irreducible. The goal of classification of representations is thus to decompose representations into irreducible ones. Given the root system of a Lie algebra it is easy to write down all its irreducible representations.

Given two representations of the same Lie algebra g as above, we can also construct a new representation as

$\rho _{1}\otimes \rho _{2}\colon \mathbf {g} \to \operatorname {End} (V_{1}\otimes V_{2}):X\mapsto \rho _{1}(X)\otimes {\boldsymbol {1}}+{\boldsymbol {1}}\otimes \rho _{2}(X)$.

The meaning of fundamental representation of a Lie algebra g is that all irreducible representations of g occur as irreducible factors of tensor products of the fundamental representation.

Other methods to construct new representations is by taking exterior powers of a representation. This is analog to the tensor product with an additional relative minus sign between the two summands.

Character of a representation

Given a finite dimensional representation ρ: g→End(V) we can consider the trace to construct a characterizing map, i.e. χ:g→k : X → tr ρ(X). The character is a linear map, but unfortunately it vanishes on commutators, i.e. χ[X,Y] = tr (ρ(X)ρ(Y)-ρ(Y)ρ(X)) = 0. Therefore characters are more useful in the study of representations of solvable Lie algebras.

Universal enveloping algebra

Given a Lie algebra g we can ask for homomorphisms to associative algebras (endowed with the commutator bracket). The universal enveloping algebra U(g) is now defined as an associative algebra together with an embedding of g fulfilling the following universal property:

Every homomorphism φ from the Lie algebra g to an associative algebra A extends uniquely to a homomorphism φ' from the universal enveloping algebra U(g) to A.

The existence of universal envelopping algebras follows from the quotient of the tensor algebra by the ideal generated by

$\{X\otimes Y-Y\otimes X-[X,Y]:X,Y\in \mathbf {g} \}$.

Unfortunately the tensor algebra as well as the quotient are infinite dimensional (except in the trivial case g=0) even if g is finite dimensional.

Ado's theorem

Given an abstract Lie algebra g we can ask whether we can write it as a subalgebra of a matrix algebra. In terms of representations we are asking for a faithful representation. For arbitrary fields the answer is no.

Ado's theorem states however that every finite dimensional Lie algebra over an algebraically closed field has a faithful finite dimensional representation.

References

1. V.S. Varadarajan: Lie groups, Lie algebras, and their representations, Springer (1984), ISBN 0-387-90969-9.