Central extensions of generalized orthosymplectic Lie superalgebras

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.