Superalgebra in Characteristic 2

TL;DR

This paper explores the structure of supervector spaces in characteristic 2 through the category sVec_2, introducing d-algebras with twisted commutativity, generalizing algebraic results, and establishing foundational Lie algebra theory in this setting.

Contribution

It introduces and studies d-algebras in sVec_2, generalizes classical algebraic results, and develops Lie algebra theory including a PBW theorem in characteristic 2.

Findings

Artinian d-algebras decompose into local d-algebras

No noncommutative d-algebras of dimension ≤7 exist

Exactly one d-algebra of dimension 7 up to isomorphism

Abstract

Following the work of Siddharth Venkatesh, we study the category $\textbf{sVec}_2$. This category is a proposed candidate for the category of supervector spaces over fields of characteristic $2$ (as the ordinary notion of a supervector space does not make sense in charcacteristic $2$). In particular, we study commutative algebras in $\textbf{sVec}_2$, known as $d$-algebras, which are ordinary associative algebras $A$ together with a linear derivation $d:A \to A$ satisfying the twisted commutativity rule: $ab = ba + d(b)d(a)$. In this paper, we generalize many results from standard commutative algebra to the setting of $d$-algebras; most notably, we give two proofs of the statement that Artinian $d$-algebras may be decomposed as a direct product of local $d$-algebras. In addition, we show that there exists no noncommutative $d$-algebras of dimension $\leq 7$, and that up to isomorphism there exists exactly one $d$-algebra of dimension $7$. Finally, we give the notion of a Lie algebra in the category $\textbf{sVec}_2$, and we state and prove the Poincare-Birkhoff-Witt theorem for this category.