Lie superalgebras in characteristic 2 and mixed characteristic

TL;DR

This paper introduces a unified framework for Lie superalgebras in characteristic 2, proves the PBW theorem for them, and explores their deformation theory and liftings to mixed characteristic settings.

Contribution

It defines a new unified notion of Lie superalgebras in characteristic 2, extending existing concepts, and establishes foundational theorems and deformation theory for these structures.

Findings

Unified definition of Lie superalgebras in characteristic 2.

Proved PBW theorem for the new notion.

Developed deformation theory and lifting to mixed characteristic.

Abstract

We define the notion of a Lie superalgebra over a field $k$ of characteristic $2$ which unifies the two pre-existing ones - $\mathbb{Z}/2$-graded Lie algebras with a squaring map and Lie algebras in the Verlinde category ${\rm Ver}_4^+(k)$, and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect $k$), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero - the notion of a mixed Lie superalgebra over a ramified quadratic extension $R$ of the ring of Witt vectors $W(k)$.