On the Geometric Structure of Hyperbolic Clifford Bundles and Associated Spin Groups

Abstract

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real $n$-dimensional vector space $V$, which is denoted by $\mathcal{C}l\left(H_{V}\right)$. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra $\mathcal{C}l(H_V)$ simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the ``mother algebra''—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space $H_{V}$ are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with $\mathcal{C}l\left(H_{V}\right)$ are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived.