$\textrm{U}(n)$-structures and their induced minimal left ideals
Ricardo Suárez
TL;DR
This work builds a bridge between Berger's holonomy-based G-structures and Clifford algebra spinor theory by encoding ${\mathrm U}(n)$-structures through the rational Kahler polynomial $P^{\mathbb{Q}}(\omega_{0})$ and identifying induced minimal left ideals in Clifford algebras ${\mathbb R}_{p,q}$ with these structures. For ${\mathbb R}^{2n}$, a ${\mathrm U}(n)$-structure induces a minimal left ideal in ${\mathbb R}_{n,n}$ and, by projection, embedded ideals in ${\mathbb R}_{n,n+1}$ and ${\mathbb R}_{n,n+2}$, with the primitive idempotent given by $f_{\mathrm{U}(n)}=q^{*}(P^{\mathbb{Q}}(\omega_{0}))$; the type of the spinor module is real, complex, or quaternionic accordingly. The paper provides explicit ${\mathrm U}(3)$-structure examples in ${\mathbb R}_{3,3}$, ${\mathbb R}_{3,4}$, and ${\mathbb R}_{3,5}$, showing how complex and quaternionic structures arise as extensions of the base real spinor space. It also discusses recoverability by projection and lays out a general scheme for recovering ${\mathrm U}(m)$-structures from Clifford algebras of various signatures, including SU and holonomy implications. The results hint at a unified spinorial viewpoint for holonomy-related geometries and motivate future work on spinor bundles and holonomy in broader settings.
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 $\mathrm{O}(n)$.