Embedding theorems as a bridge between supertraces and supergeometry

TL;DR

This paper explores how embedding theorems connect supertraces and supergeometry, focusing on conditions for embedding supertrace algebras into tensor products with supercommutative algebras, and relating their smoothness properties.

Contribution

It introduces conditions for embedding supertrace algebras into tensor products with supercommutative algebras, linking supertrace identities with supergeometry.

Findings

Established criteria for embedding supertrace algebras into $A$-universal supermaps.

Connected formal smoothness of supertrace algebras with their embeddings.

Provided a framework relating supertrace identities to supergeometry.

Abstract

Any algebra herein is intended over a field of characteristic 0. Let $E$ denote the infinite dimensional Grassman algebra. Given a power associative finite dimensional {$\mathbb{Z}_2$-graded-central-simple} $A$ and a supertrace algebra $B$, so that $B$ belongs to the same variety of $A\otimes E$, we study conditions on $B$ so that it can be embedded into $A\otimes\Xi$, where $\Xi$ is a supercommutative algebra, called $A$-universal supermap of $B$, provided $B$ satisfies all the supertrace identities of $A\otimes E$. We use this result in order to relate the formal smoothness of $B$ with that of its $A$-universal supermap.