Lie Algebra Representation - Representation On An Algebra

Representation On An Algebra

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

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

H = (H)y + (−1)xHx(H)

Also, if A is unital, then

H = 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.

A Lie (super)algebra is an algebra and it has an adjoint representation of itself. This is a representation on an algebra: the (anti)derivation property is the superJacobi identity.

If a vector space is both an associative algebra and a Lie algebra and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra.