Universal central extensions of sl(m,n,A)sl(m,n,A) over associative superalgebras

Abstract

We find the universal central extension of the matrix superalgebras sl(m, n, A), where A is an associative superalgebra and m+n = 3, 4, and its relation with the Steinberg superalgebra st(m, n, A). We calculate H2 ( sl(m, n, A) ) and H2 ( st(m, n, A) ) . Finally, we introduce a new method using the nonabelian tensor product of Lie superalgebras to find the connection between H2 ( sl(m, n, A) ) and the cyclic homology of associative superalgebras for m + n ≥ 3.