Second homology of generalized periplectic Lie superalgebras

Let $(R,{}^{-})$ be an arbitrary unital associative superalgebra with superinvolution over a commutative ring $\mathbb{k}$ with $2$ invertible. The second homology of the generalized periplectic Lie superalgebra ${\mathfrak{p}}_m(R,{}^{-})$ for $m⩾3$ has been completely determined via an explicit construction of its universal central extension. In particular, this second homology could be identified with the first $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-graded dihedral homology of $R$ with certain superinvolution whenever $m⩾5$.