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

Abstract

In previous work, we associated to $\text{SU(3)}$, ${\mathrm{G}}_2$, and $\text{Spin(7)}$-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 $P(\ddot{I}{‰}_0)$ associated with the symplectic form $\ddot{I}{‰}_0$ that defines the $\mathrm{U}(n)$-structure as a stabilizer subgroup of $\mathrm{O}(n)$.