When Is a Bogolyubov Automorphism Inner?

TL;DR

This work characterizes when Bogolyubov automorphisms of the Clifford algebra $\\mathrm{Cl}(V)$ are inner for infinite-dimensional $V$ over an algebraically closed field $\\mathbb{F}$ with $\\operatorname{char}(\\mathbb{F})\\neq 2$. It combines finitary orthogonal transformations, finite-codimensional invariant subspaces, and centralizer analysis to derive a precise criterion. The main result states that $[\\varphi]$ is inner iff $\\varphi$ is finitary and either $\\varphi=\\mathrm{Id}$ with $\\det(\\varphi|_{V/V(1)})=1$, or $-\\varphi$ is finitary with $\\dim(V/V(-1))=k\\ge 1$ and $(-1)^k\\det(\\varphi|_{V/V(-1)})=-1$, with a concrete countable-dimensional example giving innerness by conjugation with $v_1$. The findings yield a complete algebraic criterion for inner Bogolyubov automorphisms in this setting and connect to the structure of locally matrix algebras via a decomposition $\\mathrm{Cl}(V) \\cong \\mathrm{Cl}(V')\\otimes Z_{\\mathrm{Cl}(V)}(\\mathrm{Cl}(V'))$.

Abstract

Let $V$ be an infinite-dimensional vector space over a field of characteristic not equal to $2$. Given a nondegenerate quadratic form $f$ on $V$, we consider the Clifford algebra $\mathrm{Cl}(V,f)$. Any orthogonal linear transformation of $V$ extends to a Bogolyubov automorphism of $\mathrm{Cl}(V,f)$. We obtain necessary and sufficient conditions for a Bogolyubov automorphism to be inner.

Theorems & Definitions (9)

• Theorem 1
• Example 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem $\ref{['Th_1']}$