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

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 sl(m, n, A). We calculate H-2 (sl(m, n, A)) and H-2 (sl(m, n, A)). Finally, we introduce a new method using the nonabelian ensor product of Lie superalgebras to find the connection between H-2 (sl(m, n, A)) and the cyclic homology of associative superalgebras for m + n >= 3.