out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory category O is a better place for the representation theory in the semisimple case in zero characteristic. For instance, the category O turned out to be of a right size to formulate the celebrated BGG reciprocity. (g,K)-module One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups. The application is based on the idea that if is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification and the connected maximal compact subgroup K. The -module structure of allows algebraic especially homological methods to be applied and -module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups. 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[xy] = (H[x])y + (−1)xHx(H[y]) 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. 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. See also Representation of a Lie group Weight (representation theory) Weyl's theorem on complete reducibility Root system Weyl character formula Representation theory of a connected compact Lie group Whitehead's lemma (Lie algebras) Kazhdan–Lusztig conjectures Quillen's lemma - analog of Schur's lemma Notes References Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of Representations that are generated by