Complexity of modules over classical Lie superalgebras

Abstract

The complexity of the simple and the Kac modules over the general linear Lie superalgebra gl(m|n) of type A was computed by Boe, Kujawa, and Nakano in [2]. A natural continuation to their work is computing the complexity of the same family of modules over the ortho-symplectic Lie superalgebra osp(2|2n) of type C. The two Lie superalgebras are both of Type I which will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. We also compute a categorical invariant, z-complexity, introduced in [2], and we interpret this invariant geometrically in terms of a specific detecting subsuperalgebra. In addition, we compute the complexity and the z-complexity of the simple modules over the Type II Lie superalgebras osp(3|2), D(2, 1; α), G(3), and F (4).