Central extensions of generalized orthosymplectic Lie superalgebras

Abstract

The key ingredient of this paper is the universal central extension of the generalized orthosymplectic Lie superalgebra ${\mathfrak{osp}}_{m|2n}(R,{}^{-})$ coordinatized by a unital associative superalgebra $(R,{}^{-})$ with superinvolution. Such a universal central extension will be constructed via a Steinberg orthosymplectic Lie superalgebra coordinated by $(R,{}^{-})$. The research on the universal central extension of ${\mathfrak{osp}}_{m|2n}(R,{}^{-})$ will yield an identification between the second homology group of the generalized orthosymplectic Lie superalgebra ${\mathfrak{osp}}_{m|2n}(R,{}^{-})$ and the first $\mathbb{Z}/2\mathbb{Z}$-graded skew-dihedral homology group of $(R,{}^{-})$ for $(m,n)\ne (2,1),(1,1)$. The universal central extensions of ${\mathfrak{osp}}_{2|2}(R,{}^{-})$ and ${\mathfrak{osp}}_{1|2}(R,{}^{-})$ will also be treated separately.

The decomposition of ${\mathfrak{osp}}_{m|2n}(R,{}^{-})$ given after the proof of Proposition 3.2 has been revised. Accordingly, the decomposition of ${\mathfrak{sto}}_{m|2n}(R,{}^{-})$ in Proposition 3.6 and 4.1 has been revised. A few typos have been fixed

References in corpus (1)

• Universal central extensions of $s{l}_{m|n}$ over $Z/2Z$-graded algebras