Universal central extensions of $s{l}_{m|n}$ over $Z/2Z$-graded algebras

Abstract

We study central extensions of the Lie superalgebra $s{l}_{m|n}(A)$, where $A$ is a $Z/2Z$-graded superalgebra over a commutative ring $K$. The Steinberg Lie superalgebra $s{t}_{m|n}(A)$ plays a crucial role. We show that $s{t}_{m|n}(A)$ is a central extension of $s{l}_{m|n}(A)$ for $m+n\ge 3$. We use a $Z/2Z$-graded version of cyclic homology to show that the center of the extension is isomorphic to $H{C}_1(A)$ as $K$-modules. For $m+n\ge 5$, we prove that $s{t}_{m|n}(A)$ is the universal central extension of $s{l}_{m|n}(A)$. For $m+n=3,4$, we prove that $s{t}_{2|1}(A)$ and $s{t}_{3|1}(A)$ are both centrally closed. The universal central extension of $s{t}_{2|2}(A)$ is constructed explicitly.

18 pages; section 7 added; reference [KT] added; the authors thank the referee for comments on the previous version

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