The Embedding Problem in Algebras with Involution

TL;DR

This paper investigates the conditions under which a simple algebra with involution can be embedded into another, focusing on polynomial identities with involution over an algebraically closed field of characteristic zero.

Contribution

It provides new insights into the embedding problem for algebras with involution, especially relating to polynomial identities and involution-preserving embeddings.

Findings

Established criteria for embedding based on polynomial identities with involution.

Characterized when an algebra with involution can be embedded into another.

Extended understanding of involution-preserving algebra embeddings in characteristic zero.

Abstract

Let $K$ be an algebraically closed field of characteristic zero, and let $A$ and $B$ be two simple algebras with involution over $K$. In this note we study the embedding problem for algebras with involution. More specifically, if the algebra $A$ satisfies the polynomial identities with involution of the algebra $B$, we investigate whether there exists an embedding of $A$ into $B$ that preserves the involutions.