$\textrm{U}(n)$-structures and their induced minimal left ideals

TL;DR

This paper explores the relationship between U(n)-structures and minimal left ideals in Clifford algebras, extending previous work on special holonomy groups and their algebraic representations.

Contribution

It identifies U(n)-structures with minimal left ideals in Clifford algebras using the induced Kahler polynomial, linking geometric structures to algebraic classifications.

Findings

Established correspondence between U(n)-structures and Clifford algebra ideals

Extended previous classifications to include U(n)-structures

Connected geometric stabilizers with algebraic minimal ideals

Abstract

In previous work, we associated to $\textrm{SU(3)}$, $\mathrm{G}_2$, and $\textrm{Spin(7)}$-structures minimal left ideals for the Clifford algebras $\mathbb{R}_{0,6},\mathbb{R}_{0,7}$, and $\mathbb{R}_{0,8}$, respectively. In this paper, we continue to analyze the link between Berger's classification theorem and the structure theorem of minimal left ideals for Clifford algebras of signature $(p,q)$ by identifying $\mathrm{U}(n)$-structures with minimal left ideals for Clifford algebras of various signatures via the induced Kahler polynomial $P(\omega_{0})$ associated with the symplectic form $\omega_{0}$ that defines the $\mathrm{U}(n)$-structure as a stabilizer subgroup of $\mathrm{O}(n)$.