Lie Algebra Representation - Formal Definition

Formal Definition

A representation of a Lie algebra is a Lie algebra homomorphism

from to the Lie algebra of endomorphisms on a vector space V (with the commutator as the Lie bracket), sending an element x of to an element ρ_x of .

Explicitly, this means that

for all x,y in . The vector space V, together with the representation ρ, is called a -module. (Many authors abuse terminology and refer to V itself as the representation).

One can equivalently define a -module as a vector space V together with a bilinear map such that

for all x,y in and v in V. This is related to the previous definition by setting x ⋅ v = ρ_x (v).