$\textrm{U}(n)$-structures and their induced minimal left 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 $P(ω_{0})$ associated with the symplectic form $ω_{0}$ that defines the $\mathrm{U}(n)$-structure as a stabilizer subgroup of $\mathrm{O}(n)$.