algebra representation of a lie superalgebra

If we have a Lie superalgebra L, then, a (not necessarily associative) Z₂ graded algebra 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, Hab=(Ha)b+(-1)^aHa(Hb) Also, if A is unital, H1=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 super Jacobi identity! This is a special case of an algebra representation of a Hopf algebra. See also unitary representation of a Lie superalgebra.

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