Classification of primitive ideals of U(o(\infty)) and U(sp(\infty))
- The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras U(o(\infty)) and U(sp(\infty)). Let g(\infty) denote any of the Lie algebras o(\infty) or sp(\infty). Then g(\infty)=\Bigcap_{n\geq 2}g(2n) for g(2n) = o(2n) or g(2n) = sp(2n), respectively. We show that each primitive ideal I of U(g(1)) is weakly bounded, i.e., I \ U(g(2n)) equals the intersection of annihilators of bounded weight g(2n)-modules. To every primitive ideal I of g(\infty) we attach a unique irreducible coherent local system of bounded ideals, which is an analog of a coherent local system of finite-dimensional modules, as introduced earlier by A. Zhilinskii. As a result, primitive ideals of U(g(\infty)) are parametrized by triples (x;y;Z) where x is a nonnegative integer, y is a nonnegative integer or half-integer, and Z is a Young diagram. In the case of o(1), each primitive ideal is integrable, and our classification reduces to a classification of integrable ideals going back to A. Zhilinskii, A. Penkov and I. Petukhov. In the case of sp(\infty), only 'half' of the primitive ideals are integrable, and nonintegrable primitive ideals correspond to triples (x;y;Z) where y is a half-integer.
