Universal central extensions of the matrix Lie superalgebras \(\text{sl}(m,n,A)\). (Q2704363)

From the text: Universal central extensions play a significant role in group theory and in algebraic \(K\)-theory. In a like manner universal central extensions came into wide use in the theory of Lie algebras. In particular, C. Kassel and J.-L. Loday, developing the paper of \textit{S. Bloch} [Algebraic \(K\)-theory, Evanston 1980, Lect. Notes Math. 854 (1981; Zbl 0469.14009)], described the structure of the universal central extension of \(\text{sl}_n(A)\). In [J. Pure Appl. Algebra 34, 265--275 (1984; Zbl 0549.17009)] \textit{C. Kassel} has solved this problem for all simple Lie algebras over a commutative algebra. The case of the Steinberg Lie algebra \(\text{st}_2(R)\) over an algebra \(R\) was considered by \textit{Y. Gao} in [Commun. Algebra 21, No. 10, 3691--3706 (1993; Zbl 0822.17018)]. NEWLINENEWLINEIn this note we describe the universal central extensions of matrix Lie superalgebras using, as in [\textit{C. Kassel} and \textit{J.-L. Loday}, Ann. Inst. Fourier 32, No. 4, 119--142 (1982; Zbl 0485.17006)], the advantages of matrix realization.NEWLINENEWLINEThe following is proved:NEWLINENEWLINETheorem 1. If \(m+n\geq 3\), then \((\text{st}(m,n,A),\varphi)\) is a central extension of the Lie superalgebra \(\text{sl}(m,n,A)\).NEWLINENEWLINETheorem 2. Let \((W,\psi)\) be a central extension of the Lie superalgebra \(\text{sl}(m,n,A)\), and \(m+n\geq 5\). Then there exists a unique homomorphism \(\eta:\text{st}(m,n,A)\to W\) such that \(\psi\eta=\varphi\).NEWLINENEWLINETheorem 4. The kernel \(T(m,n)\) of the universal central extension \((\text{st}(m,n,A),\varphi)\) \((m+n\geq 5)\) of the Lie superalgebra \(\text{sl}(m,n,A)\) is isomorphic to \(HC_2(A)\).NEWLINENEWLINEFor the entire collection see [Zbl 0953.00031].