Algebra representation of a Lie superalgebra
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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,
- H[ab] = (H[a])b + (−1)aHa(H[b])
Also, if A is unital, then
- 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.
Given 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? We then have a Poisson algebra. And what about the corresponding case for an associative superalgebra? We have a Poisson superalgebra.
A Lie (super)algebra is an algebra and it has an adjoint representation of itself. Now what does the (anti)derivation rule say? It is the superJacobi identity.
This is a special case of an algebra representation of a Hopf algebra.
