Central extensions of generalized orthosymplectic Lie superalgebras

The key ingredient of this paper is the universal central extension of the generalized orthosymplectic Lie superalgebra ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{m\mathrm{\mid }2n}(R,{}^{-})$ 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 ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{m\mathrm{\mid }2n}(R,{}^{-})$ will yield an identification between the second homology group of the generalized orthosymplectic Lie superalgebra ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{m\mathrm{\mid }2n}(R,{}^{-})$ and the first $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-graded skew-dihedral homology group of $(R,{}^{-})$ for $(m,n)\mathrm{\ne }(2,1),(1,1)$. The universal central extensions of ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{2\mathrm{\mid }2}(R,{}^{-})$ and ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{1\mathrm{\mid }2}(R,{}^{-})$ will also be treated separately.