A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra

Abstract

For a Lie superalgebra 9 we denote the even and odd parts of 9 by go and g,, respectively. The simple Lie superalgebra 9 is called classical if f0 is reductive. For 9 classical simple we study primitive ideals in the enveloping algebra U(g). Our main result is that any graded primitive ideal is the annihilator of a graded simple quotient of a Verma module. This is an analogue of the well-known theorem of Duflo [D] on primitive ideals in the enveloping algebra of a semisimple Lie algebra. The proof is based on Duflo’s theorem and some work of E. Letzter [Ll, L2] on primitive ideals in finite ring extensions. The definition of a Verma module depends on the existence of a triangular decomposition in 9. This is dicussed in Section 1. A more precise statement of the main theorem is given Section 2. In Section 3 we discuss some corollaries, for example we show that if JZ # Q(H) then graded prime ideals are prime (Corollary 3.1), and if f # P(n), then any factor ring of U(g) has the same left and right Krull dimension (Corollary 3.3). Classical simple Lie superalgebras which are not Lie algebras have been classified by Kac [Kl, Theorem 2, p. 441 (see also [Sch, Theorem 1, p. 1401). In the notation of Kac these algebras are as follows. Scheunert’s notation, if different is given in parentheses. A(m,n)=sl(m+l,n+l), m#n,m,n~O(spl(m+1,n+1)) A(n, n) = son + 1, n + l)/(Zzn+z>, n>O(spl(n+1,n+1)/@rz,+2)