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 ρx 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 = ρx (v).
Read more about this topic: Lie Algebra Representation
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