Duals of simple two-sided vector spaces

TL;DR

This paper explores the duals of simple two-sided vector spaces over a perfect field extension, linking their structure to arithmetic data and establishing conditions for their symmetric algebra constructions.

Contribution

It characterizes the arithmetic data corresponding to simple two-sided vector spaces with equal left and right dimensions and describes their duals.

Findings

Identifies arithmetic data for dual vector spaces.

Establishes conditions for symmetric algebra existence.

Connects vector space duals with underlying arithmetic data.

Abstract

Let $K$ be a perfect field and let $k \subset K$ be a subfield. In previous work of the second author and C. Pappacena, left finite dimensional simple two-sided $k$-central vector spaces over $K$ were classified by arithmetic data associated to the extension $K/k$. In this paper, we continue to study the relationship between simple two-sided vector spaces and their associated arithmetic data. In particular, we determine which arithmetic data corresponds to simple two-sided vector spaces with the same left and right dimension, and we determine the arithmetic data associated to the left and right dual of a simple two-sided vector space. As an immediate application, we prove the existence of the non-commutative symmetric algebra of any $k$-central two-sided vector space over $K$ which has the same left and right dimension.