Algebra representation of a Lie algebra - Definition and Overview

If we have a Lie superalgebra L, then, a (not necessarily associative) Z₂ gradedalgebra A is an algebra representation of L if as a Z₂graded vector space, A is a vector space rep of L and in addition, the elements of L acts as derivations/antiderivations.

More specifically, if H is a pure element of L and x and y are a pure elements of A,

H[ab]=(H[a])b+(-1)^aHa(H[b])

Also, if A is unital,

H[1]=0

Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (-1) to the some power factors.

Now here comes an interesting part. What if have a vector space which happens to be an associative algebra AND a Lie algebra at the same time AND in addition, as an associative algebra, it is a rep of itself as a Lie algebra? Why, we have a Poisson algebra! And what about the corresponding case for an associative superalgebra? We have a Poisson superalgebra!

Let's have more fun. A Lie (super)algebra is an algebra and it also happens to be an adjoint representation of itself. Now what does the (anti)derivation rule say? It's the superJacobi identity!

This is a special case of an algebra representation of a Hopf algebra.

See also unitary representation of a Lie superalgebra.