Algebra representation of a Lie superalgebra
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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.Template:Math-stub