Universal central extensions of slm|n over Z/2Z-graded algebras

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© 2015 Elsevier B.V. We study central extensions of the Lie superalgebra sl m|n (A), where A is a Z/2Z-graded superalgebra over a commutative ring K. The Steinberg Lie superalgebra st m|n (A) plays a crucial role. We show that st m|n (A) is a central extension of sl m|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 HC 1 (A) as K-modules. For m+n≥5, we prove that st m|n (A) is the universal central extension of sl m|n (A). For m+n=3, 4, we prove that st 2|1 (A) and st 3|1 (A) are both centrally closed. The universal central extension of st 2|2 (A) is constructed explicitly.

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Journal of Pure and Applied Algebra