Universal central extensions of [formula omitted] over [formula omitted]-graded algebras

Abstract

We study central extensions of the Lie superalgebra slm|n(A), where A is a Z/2Z-graded superalgebra over a commutative ring K. The Steinberg Lie superalgebra stm|n(A) plays a crucial role. We show that stm|n(A) is a central extension of slm|n(A) for m+n≥3. We use a Z/2Z-graded version of cyclic homology to show that the center of the extension is isomorphic to HC1(A) as K-modules. For m+n≥5, we prove that stm|n(A) is the universal central extension of slm|n(A). For m+n=3,4, we prove that st2|1(A) and st3|1(A) are both centrally closed. The universal central extension of st2|2(A) is constructed explicitly.