algebras, described above. Specifically, the finite-dimensional irreducible representations are constructed as quotients of Verma modules, and Verma modules are constructed as quotients of the universal enveloping algebra. The construction of is as follows. Let T be the tensor algebra of the vector space . Thus, by definition, and the multiplication on it is given by . Let be the quotient ring of T by the ideal generated by elements of the form . There is a natural linear map from into obtained by restricting the quotient map of to degree one piece. The PBW theorem implies that the canonical map is actually injective. Thus, every Lie algebra can be embedded into an associative algebra in such a way that the bracket on is given by in . If is abelian, then is the symmetric algebra of the vector space . Since is a module over itself via adjoint representation, the enveloping algebra becomes a -module by extending the adjoint representation. But one can also use the left and right regular representation to make the enveloping algebra a -module; namely, with the notation , the mapping defines a representation of on . The right regular representation is defined similarly. Induced representation Let be a finite-dimensional Lie algebra over a field of characteristic zero and a subalgebra. acts on from the right and thus, for any -module W, one can form the left -module . It is a -module denoted by and called the -module induced by W. It satisfies (and is in fact characterized by) the universal property: for any -module E . Furthermore, is an exact functor from the category of -modules to the category of -modules. These uses the fact that is a free right module over . In particular, if is simple (resp. absolutely simple), then W is simple (resp. absolutely simple). Here, a -module V is absolutely simple if is simple for any field extension . The induction is transitive: for any Lie subalgebra and any Lie subalgebra . The induction commutes with restriction: let be subalgebra and an ideal of that is contained in . Set and . Then . Infinite-dimensional representations and "category O" Let be a finite-dimensional semisimple Lie algebra over a field of characteristic zero. (in the solvable or nilpotent case, one studies primitive ideals of the enveloping algebra; cf. Dixmier for the definitive account.) The category of (possibly infinite-dimensional) modules over turns