On the Goldie quotient ring of the enveloping algebra of a classical simple Lie superalgebra

If g is a classical simple Lie superalgebra (g not equal P(n)), the envelop ing algebra U(g) is a prime ring and hence has a simple artinian ring of qu otients a(U(g)) by Goldie's Theorem. We show that if g has Type I then Q(U( g)) is a matrix ring over g(U(g(0))). On the other hand, if g = osp(1, 2r) then by extending the center of U(g) we obtain a prime ring whose Goldie qu otient ring is a matrix ring over the quotient division ring of a Weyl alge bra. This is an analog of a result of Gelfand and Kirillov. (C) 2001 Academ ic Press.