Complexity of Modules over Lie Superalgebras

Abstract

In this dissertation, a fair amount of work is dedicated to computing the complexity of modules over a classical Lie superalgebra $\fg=\fg_{\0}\oplus \fg_{\1}$ over the complex numbers $\C$. We will consider the category $\F$ of finite dimensional $\fg$-supermodules which are completely reducible as $\fg_{\0}$-modules. Every module $M\in \F$ admits a minimal projective resolution whose terms have dimensions which increase at a polynomial rate of growth. This rate of growth is called the \emph{complexity} of $M$. In \cite{BKN1} the authors compute the complexity of the simple and the Kac modules over the general linear Lie superalgebra $\gl(m|n)$ of type $A$. 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 \emph{Type I}, thus the Kac modules in the two cases are constructed by the same induction functor. This similarity will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. The complexity is not a categorical invariant. However, we compute a categorical invariant called the $z$-complexity, introduced in \cite{BKN1}, 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 \emph{Type II} Lie superalgebras $\osp(3|2)$, $\dd$, $G(3)$, and $F(4)$.