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 adX:V→V: Y→[X,Y], then ad:g→End(V): X→adX is a representation of g called the adjoint representation.

Direct sum and tensor product
Given two representations ρi:g→End(Vi) 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 addional relative minus sign between the two summands.

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.