Editing Lie algebra representation (section)

==Representation on an algebra==
If we have a Lie superalgebra ''L'', then a representation of ''L'' on an algebra is a (not necessarily [[associative]]) [[graded algebra|'''Z'''<sub>2</sub> graded]] [[algebra over a field|algebra]] ''A'' which is a representation of ''L'' as a '''Z'''<sub>2</sub> [[graded vector space]] and in addition, the elements of ''L'' acts as [[Derivation (abstract algebra)|derivation]]s/[[antiderivation]]s on ''A''.

More specifically, if ''H'' is a [[pure element]] of ''L'' and ''x'' and ''y'' are [[pure element]]s of ''A'',

:''H''[''xy''] = (''H''[''x''])''y'' + (&minus;1)<sup>''xH''</sup>''x''(''H''[''y''])

Also, if ''A'' is [[unital algebra|unital]], then

:''H''[1] = 0

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

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