Atypical modules of the lie superalgebra gl (m/n)

Abstract

Let G --GoSG ~ be the genera/l/near Lie superalgebra gl(rn/n) [2] consisting of complex A B matrices (v D) of size (rn -b n) 2. The even subspace G O of G consists of the matrices A0 (o D) and the odd subspace G i consists of the matrices (o ° s ) . The bracket between homogeneous elements is defined by [a,b] = a b (-1)=Zba for a e G=,b e G~ (a,~ e (0, i} ---~2). Thus the even subalgebra is isomorphic to gl(rn) ~ gl(n). G admits a consistent ~-grading G = G-1 ~B Go (9 G+I where Go = Go, G+x is the space of matrices of the form (0°0 s ) and G-1 is the space of matrices of the form (o0) . The special linear Lie superalgebra sl(m/n) is the subalgebra of gl(m/n) consisting of matrices with vanishing supertrace. In what follows we put G = gl(m/n), but all of the results can be reformulated for ~l(m/n) as well. The Cartan subalgebra H of G consists of the subspace of diagonal matrices. The roo$ or weight space H* is the dual space of H and is spanned by the forms ei (i 1 , . . . , m ) and 6j (3 -1 , . . . , n ) . The inner product on the weight space H* is given by [5] (ei[(i) = 61j, (ei[6i) : 0, (6~[6y) = -6ii, where 61i is the usual Kronecker symbol. In this e6-basis the even roots of G are of the form e l ei or ~ i 6i, and the odd roots are of the form q-(el 6j). Let A denote the set of all roots, Ao the set of even roots and Ax the set of odd roots. As a system of simple roots one takes the so-called distinguished set [3] ez-e~, e 2 e s , . . . , e=-61, 61-62, . . . , Q 1 Q . Then the set A + of posi~ive roots consists of the elements e, ej (i < j) , 61 ~ (i < j) and e, ~j. Now the notations A+ and A + are obvious; in particular :