Representations of nilpotent Lie algebras and superalgebras.

摘要

This thesis studies the representation theory of complex finite-dimensional nilpotent Lie algebras and superalgebras.;Let G be a simply connected, nilpotent Lie group with Lie algebra g . The group G acts on the dual space g* by the coadjoint action. By the orbit method of Kirillov, the simple unitary representations of G are in bijective correspondence with the coadjoint orbits in g* , which in turn are in bijective correspondence with the primitive ideals of the universal enveloping algebra of g . The number of simple g -modules which have a common eigenvector for a subalgebra of g and are annihilated by a primitive ideal I is shown by Benoist to depend on geometric properties of a certain subvariety of the coadjoint orbit corresponding to I. We determine the exact number of such modules when the coadjoint orbit is two-dimensional.;A complete description of the coadjoint orbits for A+n-1 , the Lie algebra of n x n strictly upper triangular matrices, has not yet been obtained, though there has been steady progress on it ever since the orbit method was devised. We apply methods developed by Andre to find defining equations for the elementary coadjoint orbits for the maximal nilpotent Lie subalgebras of the orthogonal Lie algebras, and we also determine all the possible dimensions of coadjoint orbits in the case of A+n-1 .;Bell and Musson showed that the algebras obtained by factoring the universal enveloping superalgebra of a Lie superalgebra by graded-primitive ideals are isomorphic to tensor products of Weyl algebras and Clifford algebras. We describe certain cases where the factors are purely Weyl algebras and determine how the sizes of these Weyl algebras depend on the graded-primitive ideals.;We apply results due to Kac to determine the dimensions of the finite-dimensional simple modules of the nilradical of a fixed Borel subsuperalgebra of some families of classical Lie superalgebras. For the superalgebra A m-1,0, we compute the dimension of the simple modules for the nilradical of any Borel subsuperalgebra. These calculations indicate that the (strong) Bruhat order plays a significant role.