Universal central extensions of $\mathfrak{sl}(m,n,A)$ of small rank over associative superalgebras

Abstract

We complete the solution of the problem of finding the universal central extension of the matrix superalgebras $\mathfrak{sl}(m,n,A)$ where $A$ is an associative superalgebra and computing $H_2(\mathfrak{sl}(m,n,A))$. The Steinberg Lie superalgebra $\mathfrak{st}(m,n,A)$ has a very important role and we will also find out $H_2(\mathfrak{st}(m,n,A))$. In Chen and Sun (arXiv:1311.7079, 2013) it is solved the problem where $m+n\ge 5$ and in Chen and Guay (Algebr. Represent. Theory, 2013) it is solved when $n=0$, so here we work out the three remaining cases $\mathfrak{sl}(2,1,A),\mathfrak{sl}(3,1,A)$ and $\mathfrak{sl}(2,2,A)$.

16 pages