The Embedding Problem in Algebras with Involution

TL;DR

Addresses the embedding problem for algebras with involution over an algebraically closed field $K$ of characteristic not equal to $2$. Uses the framework of $*$-polynomial identities, the free associative algebra with involution $K\langle X, *\rangle$, and the decomposition $A = A^+ \oplus A^-$ to study embeddings that preserve involutions. Relies on the classification of finite-dimensional $*$-simple algebras as $(M_k(K), t)$, $(M_k(K), s)$ for even $k$, and $(M_k(K)\oplus M_k(K)^{op}, ex)$, together with Amitsur–Levitzki style results on standard $*$-identities $St_d$ to compare identity sets and infer embeddability. Provides positive embedding results in several cases, notably when $A$ is central and $B$ has orthogonal or symplectic involution, and discusses how the identities control possible target algebras such as $(M_n(K), *)$ or $(M_{2n}(K), s)$. However, the scarcity of explicit $*$-identities for higher matrix algebras limits the general applicability of the approach.

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.

Theorems & Definitions (13)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3: StandardPolynomialMatrices, Lemma 4.1
• Theorem 4: slin1979special, Proposition 2
• Theorem 5: rowen1982simple, Theorem 3
• Proposition 1: becher2018involutions, Proposition 4.4
• Corollary 1
• Proposition 2: rowen1980polynomial, Corollary 2.5.12
• Proposition 3