Central extensions of generalized orthosymplectic Lie superalgebras

TL;DR

This paper constructs the universal central extension of generalized orthosymplectic Lie superalgebras over superalgebras with involution, linking it to graded skew-dihedral homology, and explores special cases separately.

Contribution

It introduces a method to explicitly construct the universal central extension of generalized orthosymplectic Lie superalgebras using Steinberg superalgebras, connecting it to homology groups.

Findings

Universal central extension constructed via Steinberg orthosymplectic Lie superalgebra.

Identification of the second homology group with graded skew-dihedral homology.

Separate treatment of special cases $oldsymbol{ ext{osp}_{2|2}}$ and $oldsymbol{ ext{osp}_{1|2}}$.

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 $(m,n)\neq(2,1),(1,1)$. The universal central extensions of $\mathfrak{osp}_{2|2}(R,{}^-)$ and $\mathfrak{osp}_{1|2}(R,{}^-)$ will also be treated separately.