Second homology of generalized periplectic Lie superalgebras

TL;DR

This paper determines the second homology of generalized periplectic Lie superalgebras over superalgebras with superinvolution, linking it to graded dihedral homology, and provides explicit universal central extensions.

Contribution

It explicitly computes the second homology of these superalgebras and connects it to graded dihedral homology, extending previous understanding.

Findings

Second homology fully determined for m ≥ 3.

Identifies second homology with graded dihedral homology for m ≥ 5.

Provides explicit universal central extensions.

Abstract

Let $(R,{}^-)$ be an arbitrary unital associative superalgebra with superinvolution over a commutative ring $\Bbbk$ with $2$ invertible. The second homology of the generalized periplectic Lie superalgebra $\mathfrak{p}_m(R,{}^-)$ for $m\geqslant3$ has been completely determined via an explicit construction of its universal central extension. In particular, this second homology could be identified with the first $\mathbb{Z}/2\mathbb{Z}$-graded dihedral homology of $R$ with certain superinvolution whenever $m\geqslant 5$.