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 $\mathcal{C}l\left(H_V\right)$ 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.

1. Introduction

It is known that in General Relativity, the gravitational field is understood as an element or facet of a geometrical structure of the spacetime manifold, actually, we have that each gravitational field generated by a given matter distribution is modeling by a pentuple $\left(M,g,D,{\tau }_g,\uparrow \right),$ where M is a 4-dimensional manifold, $g\in \sec T_2^{0}M$ is a Lorentzian metric on $M,D$ is the Levi–Civita connection of $g,$ ${\tau }_g\in \sec {\bigwedge }^4{T}^{*}M$ is the volume element that defines the spatial orientability of M and ↑ means that M is time oriented, here the particles are described by triples $⟨\left(m,q\right),S,\sigma ⟩$ where m is the particle’s mass, q the electric charge, $\sigma$ is the world line and S is the particle, which is characterized in the Clifford bundle formalism, for details see, for e.g., [1,2].

Clifford bundle formalism is a theory that has proven to be a convincing alternative for the study of the gravitational field, for example, in [3] Einstein’s gravitational theory is established with this formalism, this suggests that the gravitational field is in Minkowski spacetime, in [4] by using the Clifford bundle formalism, a Lagrangian theory of the Yang–Mills type for the gravitational field in Minkowski spacetime is developed; here, it is shown how two simple hypotheses permit the interpretation of the formalism in terms of effective Lorentzian or teleparallel geometries, in the case of a Lorentzian geometry interpretation of the theory, the field equations are shown to be equivalent to Einstein’s equations. In [5], using this same formalism, a theory of the gravitational field is presented. This field is represented by an (1, 1)-extensor field h that describes a plastic distortion of the Lorentz vacuum due to the presence of matter. This theory allows the introduction of different types of parallelism rules in the world manifold, which can be interpreted as distortions of the parallelism structure of Minkowski spacetime.

For the study of the theory of gravitation and other aspects of theoretical physics, the Clifford bundle formalism can be generalized over a Clifford algebra associated with a hyperbolic space, where the elements of algebra now consist of both multiforms and multivectors. The simultaneous treatment of multiforms and multivectors in a single algebra is physically motivated by the need to provide a unified description of fields and their dual counterparts, as required for the study of superfields and for formulations of gravitational theories that go beyond the standard Clifford bundle approach. This general algebraic structure is called the Hyperbolic Clifford algebra and the aim of this work is to introduce a structure into this theory, which is fundamental for the study of various topics in Theoretical Physics.

The Hyperbolic Clifford algebra of a real n-dimensional vector space V, oriented to the study of superfields in theoretical physics, was first introduced in [6], where an incipient development on the hyperbolic structure described these superfields was presented, and later in [7] the same authors present a detailed development of the hyperbolic space and the Hyperbolic Clifford algebra, oriented to the study of superfields. In this last work, the authors recall the construction of a hyperbolic space $H_V=\left(V\oplus V^{*},⟨,⟩\right)$ endowed with a non-degenerate bilinear form $⟨,⟩$ and they show, among other things, that for a symmetric bilinear form b of arbitrary signature, there exists a relation of $H_V$ and the exterior direct sums of the hyperbolic spaces associated with the pair $(V,b)$ and $(V,-b)$, they define the Clifford algebra $\mathcal{C}l\left(H_V\right)$ of multivecfors associated with the hyperbolic space $H_V$ and present several formulas that are similar to those of a Clifford algebra of multivectors on V, which are fundamental for the development of the theory. Later in [8], we presented an extensive review of $\mathcal{C}l\left(H_V\right)$ and a review of the differential structure of the Hyperbolic Clifford algebra, where we exposed the properties of the duality product of multivectors and multiforms and the theory of k multivector and l multiform variables multivector extensors over $V.$ Also in that work, we studied the theory of parallelism structures on an arbitrary smooth manifold M, and the concepts of covariant derivatives, deformed derivatives and relative covariant derivatives of a multivector, multiform field and extensor fields. All these theories and concepts are fundamental to the study of various physical theories, see, e.g., [5]. However, none of the previous works have carried out a formal development of the structure of the Hyperbolic Clifford algebra, which is necessary to characterize the properties of the particles, as well as the topics developed in the context of the Clifford algebra of multivectors or multiforms. The present paper fills this gap.

Unlike standard Clifford algebras, which deal exclusively with multivectors, the Hyperbolic Clifford algebra $\mathcal{C}\mathsf{\ell }\left(H_V\right)$ provides a unified framework that simultaneously accommodates both multivectors (elements of $\bigwedge V$) and multiforms (elements of $\bigwedge V^{*}$) in a single algebraic structure. This unified treatment is essential for the study of superfields in theoretical physics and for developing gravitational theories that require symmetric treatment of fields and their dual counterparts, it can be shown, see, e.g., [7], that $\bigwedge V^{*}$ is a minimal left ideal of the Hyperbolic Clifford algebra, i.e., $\bigwedge V^{*}$ = $\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right){\theta }^{*}$ and the elements of $\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right){\theta }^{*}$ are Witten superfields [9].

The primary contributions of this paper are as follows:

(a)

Formal development of structures for Hyperbolic Clifford algebras, extending results from standard Clifford algebra theory to this generalized hyperbolic setting, see Table 1.

(b)

Definition and analysis of hyperbolic orthogonal groups, Clifford–Lipschitz groups, Pin and Spin groups associated with $\mathcal{C}\mathsf{\ell }\left(H_V\right)$, with explicit surjective homomorphisms to the respective orthogonal transformation groups.

(c)

Construction of hyperbolic Clifford bundles on Lorentzian manifolds, establishing the associated vector bundle structure and reduction in the structure group.

(d)

Unified formalism for studying gravitational fields and particles in theories that require simultaneous treatment of fields and their dual counterparts.

Table 1. Comparison between the standard Clifford algebra $\mathcal{C}\mathsf{\ell }(V,g)$ and the Hyperbolic Clifford algebra $\mathcal{C}\mathsf{\ell }\left(H_V\right)$.

Property	$\mathcal{C}\mathsf{\ell }(\mathit{V},\mathit{g})$	$\mathcal{C}\mathsf{\ell }\left({\mathit{H}}_{\mathit{V}}\right)$
Underlying space	$(V,g)$, $\dim V=n$	$H_V=V\oplus V^{*}$, $\dim H_V=2n$
Bilinear form	Non-degenerate, signature $(p,q)$	Neutral, signature $(n,n)$
Elements	Multivectors	Multivecfors ($v\oplus \omega$, $v\in V$, $\omega \in V^{*}$)
Algebra	$\mathcal{C}\mathsf{\ell }(V,g)$	$\mathcal{C}\mathsf{\ell }\left(H_V\right)\simeq \mathrm{End}(\bigwedge V)$
Relation	Subalgebra of $\mathcal{C}\mathsf{\ell }\left(H_V\right)$	“Mother algebra”: contains $\mathcal{C}\mathsf{\ell }(V,b)$ and $\mathcal{C}\mathsf{\ell }(V,-b)$
Isotropic subspaces	Depend on signature of g	V and $V^{*}$ are maximal totally isotropic
Superfield applications	Limited	Natural framework for superfields

Table 1. Comparison between the standard Clifford algebra $\mathcal{C}\mathsf{\ell }(V,g)$ and the Hyperbolic Clifford algebra $\mathcal{C}\mathsf{\ell }\left(H_V\right)$.

Property	$\mathcal{C}\mathsf{\ell }(\mathit{V},\mathit{g})$	$\mathcal{C}\mathsf{\ell }\left({\mathit{H}}_{\mathit{V}}\right)$
Underlying space	$(V,g)$, $\dim V=n$	$H_V=V\oplus V^{*}$, $\dim H_V=2n$
Bilinear form	Non-degenerate, signature $(p,q)$	Neutral, signature $(n,n)$
Elements	Multivectors	Multivecfors ($v\oplus \omega$, $v\in V$, $\omega \in V^{*}$)
Algebra	$\mathcal{C}\mathsf{\ell }(V,g)$	$\mathcal{C}\mathsf{\ell }\left(H_V\right)\simeq \mathrm{End}(\bigwedge V)$
Relation	Subalgebra of $\mathcal{C}\mathsf{\ell }\left(H_V\right)$	“Mother algebra”: contains $\mathcal{C}\mathsf{\ell }(V,b)$ and $\mathcal{C}\mathsf{\ell }(V,-b)$
Isotropic subspaces	Depend on signature of g	V and $V^{*}$ are maximal totally isotropic
Superfield applications	Limited	Natural framework for superfields

In this paper, we present the construction of the frame bundle associated with the Hyperbolic Clifford algebra and the Hyperbolic Clifford bundle of a real n-dimensional vector space V. We show that this is a vector bundle associated with the principal bundle of orthonormal frames. In Section 2, we present a brief review of the Hyperbolic space $H_V$ and the Hyperbolic Clifford algebra $\mathcal{C}l\left(H_V\right)$ associated with this space, as well as some identities necessary for the development of the theory. In Section 3 we study the groups associated with $\mathcal{C}l\left(H_V\right)$, namely the orthogonal groups and symmetries in the $H_V$ space as well as the Clifford–Lipschitz groups and the Pin and groups associated with $\mathcal{C}l\left(H_V\right).$ Finally, in Section 4, we present the construction of the frame bundle associated with $\mathcal{C}l\left(H_V\right)$ and Hyperbolic Clifford bundle, then we derive the structure for $\mathcal{C}l\left(H_V\right).$

2. Hyperbolic Spaces

2.1. Definition and Basic Properties

In this section, we recall some definitions and basic properties of Hyperbolic space. Let $V,V^{*}$ be a pair of dual n-dimensional vector spaces over the real field $\mathbb{R}$ and $V\oplus V^{*}$ the exterior direct sum of the vector spaces $V$ and $V^{*}$, for more details see [7,8]. We remember that a hyperbolic structure over V is the pair

$$H_V=\left(V\oplus V^{*},⟨,⟩\right)$$

where $⟨,⟩$ is the non-degenerate symmetric bilinear form of index n defined by (This non degenerated symmetric bilinear form was first introduced in [10]).

$$⟨\mathbf{x},\mathbf{y}⟩=x^{*}\left(y_{*}\right)+y^{*}\left(x_{*}\right)$$

(1)

for all $\mathbf{x}=x_{*}\oplus x^{*},\mathbf{y}=y_{*}\oplus y^{*}\in H_V$.

Remark 1. 

In the Witt basis $\{e_1,\dots ,e_n,{\theta }^1,\dots ,{\theta }^n\}$ of $H_V$, the matrix representation of the bilinear form $⟨·,·⟩$ is

$$\left(\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right),$$

where $I_n$ is the $n\times n$ identity matrix. This makes it explicit that the form is neutral (i.e., of signature $(n,n)$), and that the subspaces V and $V^{*}$ are totally isotropic, since the diagonal blocks are zero.

The elements of $H_V$ are referred to as vecfors. A vecfor $\mathbf{x}=x_{*}\oplus x^{*}\in H_V$ is positive if $x^{*}\left(x_{*}\right)>0,$ null if $x^{*}\left(x_{*}\right)=0$ and negative if $x^{*}\left(x_{*}\right)<0.$ If $x^{*}\left(x_{*}\right)=1$ we say that $\mathbf{x}=x_{*}\oplus x^{*}$ is a unit vecfor.

Note that the spaces V and $V^{*}$ are maximal totally isotropic subspace of $H_V.$ In reality these spaces can be identified in $H_V$ under the image of the inclusions $i_{*}:V\to V\oplus V^{*},x_{*}\mapsto i_{*}{x}_{*}=x_{*}\oplus 0\equiv x_{*}$ and $i^{*}:V^{*}\to V\oplus V^{*},x^{*}\mapsto i^{*}{x}^{*}=0\oplus x^{*}\equiv x^{*},$ then from (1) we have

$$⟨i_{*}{x}_{*},i_{*}{y}_{*}⟩=⟨x_{*},y_{*}⟩=0,⟨i^{*}{x}^{*},i^{*}{y}^{*}⟩=⟨x^{*},y^{*}⟩=0,⟨i^{*}{x}^{*},i_{*}{y}_{*}⟩=x^{*}\left(y_{*}\right)$$

for all $x_{*},y_{*}\in V\subset V\oplus V^{*}$ and $x^{*},y^{*}\in V^{*}\subset V\oplus V^{*}.$

More generally, to each subspace $S\subset V$ (or analogously, $S^{\prime }\subset V^{*})$ we can associated a maximal totally isotropic subspace as follows, define the null subspace $S^{\prime }\subset V^{*}$ of $S\subset V$ by

$$S^{\prime }=\left\{x^{*}\in V^{*}:x^{*}\left(y_{*}\right)=0,\forall y_{*}\in V\right\}.$$

Note that $S^{\prime }$ is a subspace of $V^{*}.$ Furthermore, it is easy to prove that if $\dim V=n,$ then $\dim \left(S^{\prime }\right)=n-\dim S,$ then the n-dimensional vector subspace $S\oplus S^{\prime }$ of $H_V,$ is a maximal totally isotropic subspace of $H_V.$

On the other hand, if we consider the basis $\left\{e_1,\cdots ,e_n\right\}$ of V and $\left\{{\theta }^1,\cdots ,{\theta }^n\right\}$ of $V^{*},$ then in $H_V$ we have that

$$⟨e_i,e_j⟩=0,⟨{\theta }^i,{\theta }^j⟩=0\mathrm{and}⟨{\theta }^i,e_j⟩={\delta }_j^i.$$

To obtain an orthogonal basis $\left\{{\mathsf{œ}}_1,\cdots ,{\mathsf{œ}}_{2n}\right\}$ to $H_V$, we consider a Witt basis $\left\{e_1,\cdots ,e_n,{\theta }^1,\cdots ,{\theta }^n\right\}$ and we define the elements of this base as follows,

$${\mathsf{œ}}_k={\displaystyle \frac{1}{\sqrt{2}}}\left(e_k\oplus {\theta }^k\right)\mathrm{and}{\mathsf{œ}}_{n+k}={\displaystyle \frac{1}{\sqrt{2}}}\left({\overline{e}}_k\oplus {\theta }^k\right)$$

(2)

where ${\overline{e}}_k=-e_k$, $k=1,\dots ,n$. Thus, to $k,l=1,\dots ,n$ we have

$$⟨{\mathsf{œ}}_k,{\mathsf{œ}}_l⟩={\delta }_{kl},⟨{\mathsf{œ}}_{n+k},{\mathsf{œ}}_{n+l}⟩=-{\delta }_{kl}\mathrm{and}⟨{\mathsf{œ}}_k,{\mathsf{œ}}_{n+l}⟩=0.$$

It is important to note here that, base vectors ${\mathsf{œ}}_{n+l}$ are obtained from the ${\mathsf{œ}}_k$ through the involution [$\mathbf{x}=x_{*}\oplus x^{*}\mapsto \overline{\mathbf{x}}=(-x_{*})\oplus x^{*}$], where $\overline{\mathbf{x}}$ is the (hyperbolic) conjugate of the vecfor $\mathbf{x}.$

A very important result concerning the theory of hyperbolic space is the following,

Proposition 1. 

Given an arbitrary non-degenerate symmetric bilinear form b on $V,$ there is an isomorphism

$$H_V\approx \left(V,b\right)\oplus \left(V,-b\right)=H_{bV}.$$

This result can be found in [7,11]. The explicit isomorphism involved in this proposition is given by

$${\chi }_b:H_V⟶H_{bV},{\chi }_b\left(x_{*}+x^{*}\right)=x_{+}+x_{-}$$

(3)

where

$$x_{\pm }={\displaystyle \frac{1}{\sqrt{2}}}\left(b^{*}{x}^{*}\pm x_{*}\right)$$

with $b^{*}{x}^{*}\equiv b^{*}\left(x^{*},·\right)$ for all $x^{*}\in V^{*},$ where $b^{*}$ is a bilinear form in $V^{*},$ reciprocal of $V,$ i.e., $b^{ik}{b}_{kj}={\delta }_j^i$, $b_{ij}\left(e_i,e_j\right)$, $b^{ik}=b^{*}\left({\theta }^i,{\theta }^k\right).$ For details on how this isomorphism operates on a basis of $H_V$, see [7]. In particular, we can take the vector space $V={\mathbb{R}}^{p,q}$ with the bilinear form b of signature $\left(p,q\right).$

2.2. Exterior Algebra and Contractions of a Hyperbolic Space

The Grassman algebra $\bigwedge H_V$ of the hyperbolic structure $H_V$ is the pair

$$\bigwedge H_V=\left(\bigwedge \left(V\oplus V^{*}\right),⟨,⟩\right)$$

where

$$\bigwedge \left(V\oplus V^{*}\right)=\sum _{r=0}^{2n}{\bigwedge }^r\left(V\oplus V^{*}\right)$$

is the exterior algebra of $V\oplus V^{*}$ and $⟨,⟩$ is the canonical bilinear form on $\bigwedge \left(V\oplus V^{*}\right)$ induced by the bilinear form $⟨,⟩$ of $H_V$, and it is extended by linearity and orthogonality to all of the algebra $\bigwedge H_V.$ Of course, due to the isomorphisms $H_V^{*}\simeq H_{V^{*}}\simeq H_V,$ we have

$$\bigwedge H_V^{*}\simeq \bigwedge H_{V^{*}}\simeq \bigwedge H_V,$$

and it follows that

$$\bigwedge H_V\simeq {\left(\bigwedge H_V\right)}^{*},$$

i.e., $\bigwedge H_V$ is itself an auto-dual space. The elements of $\bigwedge H_V$ are called multivecfors.

Grade involution, reversion and conjugation in the algebra $\bigwedge H_V$ are defined as usual, see, e.g., [1,7]. For homogeneous multivecfors $\mathbf{u}\in {\bigwedge }^r{H}_V,$

$$\widehat{\mathbf{u}}={\left(-1\right)}^r\mathbf{u},\tilde{\mathbf{u}}={\left(-1\right)}^{{\displaystyle \frac{1}{2}}r\left(r-1\right)}\mathbf{u},\overline{\mathbf{u}}=\widehat{\tilde{\mathbf{u}}}={\left(-1\right)}^{{\displaystyle \frac{1}{2}}r\left(r+1\right)}\mathbf{u}$$

and we call that every element $\mathbf{u}\in \bigwedge H_V$ is uniquely decomposed into a sum of the even and odd parts of $\mathbf{u}$, i.e., $\mathbf{u}={\mathbf{u}}_{+}+{\mathbf{u}}_{-}$, where

$${\mathbf{u}}_{+}={\displaystyle \frac{1}{2}}\left(\mathbf{u}+\widehat{\mathbf{u}}\right)\mathrm{and}{\mathbf{u}}_{-}={\displaystyle \frac{1}{2}}\left(\mathbf{u}-\widehat{\mathbf{u}}\right).$$

Remark 2. 

The involution degree of $\mathbf{x}\in H_V,$ denoted by $\widehat{\mathbf{x}},$ is also denoted by the map $\alpha :H_V\to H_V,$ with $\alpha \left(\mathbf{x}\right)=\widehat{\mathbf{x}}=\left(-x_{*}\right)\oplus x^{*}$. In general, every Clifford algebra admits a unique canonical automorphism or involutionary automorphism $\alpha :\mathcal{C}\mathsf{\ell }\left(\mathbb{V}\right)\to \mathcal{C}\mathsf{\ell }\left(\mathbb{V}\right)$ that maps every odd element $x\in \mathcal{C}{\mathsf{\ell }}^1\left(\mathbb{V}\right)$ to $-x$, and which satisfies the following properties,

$$\alpha \circ \alpha =id;\alpha \left(xy\right)=\alpha \left(x\right)\alpha \left(y\right);\alpha \left(v\right)=-v,\forall v\in \mathbb{V}.$$

The spaces $\bigwedge V$ and $\bigwedge V^{*}$ are identified with their images in $\bigwedge H_V$ under the homomorphisms $i^w:\bigwedge V\to \bigwedge H_V$ defined by

$$i^w\left(x_{1*}\wedge \dots \wedge x_{r*}\right)=\left(x_{1*}\oplus 0\right)\wedge \dots \wedge \left(x_{r*}\oplus 0\right)\equiv x_{1*}\wedge \dots \wedge x_{r*}$$

and $i_w:\bigwedge V^{*}\to \bigwedge H_V$ defined by

$$i_w\left(x_1^{*}\wedge \dots \wedge x_r^{*}\right)=\left(0\oplus x_1^{*}\right)\wedge \dots \wedge \left(0\oplus x_r^{*}\right)\equiv x_1^{*}\wedge \dots \wedge x_r^{*}.$$

Then, for $u_{*},v_{*}\in \bigwedge V\subset \bigwedge H_V$ and $u^{*},v^{*}\in \bigwedge V^{*}\subset \bigwedge H_V,$ we have

$$⟨u_{*},v_{*}⟩=0,⟨u^{*},v^{*}⟩=0\mathrm{and}⟨u^{*},v_{*}⟩=u^{*}\left(v_{*}\right).$$

(4)

Thus, $\bigwedge V$ and $\bigwedge V^{*}$ are a totally isotropic subspace of $\bigwedge H_V.$ But they are no longer maximal, $⟨,⟩$ being neutral, the dimension of a maximal totally isotropic subspace is $2^{2n-1},$ whereas $\dim \bigwedge V=\dim \bigwedge V^{*}=2^n.$ Indeed, the bilinear form $⟨·,·⟩$ on $\bigwedge H_V$ is neutral, i.e., it has signature $(2^{2n-1},2^{2n-1})$. For a neutral form on a $2^{2n}$-dimensional space, the dimension of a maximal totally isotropic subspace equals exactly half the total dimension, i.e., $2^{2n-1}$. Since $\dim \bigwedge V=2^n<2^{2n-1}$ for $n\ge 2$, the subspaces $\bigwedge V$ and $\bigwedge V^{*}$ are totally isotropic but not maximal. For elements $\mathbf{u}=u_{*}\wedge u^{*},\mathbf{v}=v_{*}\wedge v^{*}\in \bigwedge H_V$ with $u_{*}\in {\bigwedge }^{r}V,u^{*}\in {\bigwedge }^s{V}^{*},v_{*}\in {\bigwedge }^{s}V$ and $v^{*}\in {\bigwedge }^r{V}^{*}$ it holds

$$⟨\mathbf{u},\mathbf{v}⟩={\left(-1\right)}^{rs}{u}^{*}\left(v_{*}\right)v^{*}\left(u_{*}\right).$$

It is also very important to remember that the volume element $\mathsf{œ}={\mathsf{œ}}_1\wedge \dots \wedge {\mathsf{œ}}_{2n}\in {\bigwedge }^{2n}{H}_V$ with ${\mathsf{œ}}_x\ne 0$ $\forall x\in M,$ provides an orientation to Hyperbolic space $H_V$, here $\left\{{\mathsf{œ}}_1,\cdots ,{\mathsf{œ}}_{2n}\right\}$ is the orthonormal basis of $H_V$ naturally associated with the dual basis $\left\{e_1,\cdots ,e_n\right\}$ of V and $\left\{{\theta }^1,\cdots ,{\theta }^n\right\}$ of $V^{*}.$

On the other hand, a left contraction $⌟:\bigwedge H_V\times \bigwedge H_V\to \bigwedge H_V$ and a right contraction $⌞:\bigwedge H_V\times \bigwedge H_V\to \bigwedge H_V$ are introduced in the algebra $\bigwedge H_V$ in the usual way (see, e.g., [1,7]), i.e., by

$$⟨\mathbf{u}⌟\mathbf{v},\mathbf{w}⟩=⟨\mathbf{v},\tilde{\mathbf{u}}\wedge \mathbf{w}⟩\mathrm{and}⟨\mathbf{v}⌞\mathbf{u},\mathbf{w}⟩=⟨\mathbf{v},\mathbf{w}\wedge \tilde{\mathbf{u}}⟩,$$

for all $\mathbf{u},\mathbf{v},\mathbf{w}\in \bigwedge H_V.$ The general properties, for all $\mathbf{x},\mathbf{y}\in H_V$, for all $\mathbf{u},\mathbf{v},\mathbf{w}\in \bigwedge H_V$ and $\mathsf{œ}\in {\bigwedge }^{2n}{H}_V,$ can be seen in [7,8]. Moreover, from Equation (4) we obtain

$$u_{*}⌟v_{*}=u_{*}⌞v_{*}=0,u^{*}⌟v^{*}=u^{*}⌞v^{*}=0$$

(5)

for all $u_{*},v_{*}\in \bigwedge V$ and $u^{*},v^{*}\in \bigwedge V^{*},$ so that for elements of the form $\mathbf{u}=u_{*}\wedge u^{*}$ and $\mathbf{x}=x_{*}\oplus x^{*},$ it holds

$$\begin{array}{c}\mathbf{x}⌟\mathbf{u}=\left(x^{*}⌟u_{*}\right)\wedge u^{*}+{\widehat{u}}_{*}\wedge \left(x_{*}⌟u^{*}\right)\\ \mathbf{u}⌞\mathbf{x}=u_{*}\wedge \left(u^{*}⌞x_{*}\right)+\left(x^{*}⌟u_{*}\right)\wedge {\widehat{u}}^{*}\end{array}$$

For more details about the properties of the left and right contractions, see, e.g., [1,12].

2.3. Clifford Algebra of a Hyperbolic Space

Introduce in ${\displaystyle \bigwedge }{H}_V$ the Clifford product of a vecfor $\mathbf{x}\in H_V$ by an element $\mathbf{u}\in \bigwedge H_V$ by

$$\mathbf{xu}=\mathbf{x}⌟\mathbf{u}+\mathbf{x}\wedge \mathbf{u}$$

and extend this product by linearity and associativity to all of the space $\bigwedge H_V$, the general properties of the Clifford product can be seen, e.g., in [1]. The resulting algebra is isomorphic to the Clifford algebra $\mathcal{C}\mathsf{\ell }\left(H_V\right)$ of the hyperbolic structure $H_V$ and will thereby be identified with it and is called the mother algebra (or the Hyperbolic Clifford algebra) of the vector space V. The even and odd subspaces of $\mathcal{C}\mathsf{\ell }\left(H_V\right)$ will be denoted, respectively, by $\mathcal{C}{\mathsf{\ell }}^{\left(0\right)}\left(H_V\right)$ and $\mathcal{C}{\mathsf{\ell }}^{\left(1\right)}\left(H_V\right)$, so that

$$\mathcal{C}\mathsf{\ell }\left(H_V\right)=\mathcal{C}{\mathsf{\ell }}^{\left(0\right)}\left(H_V\right)\oplus \mathcal{C}{\mathsf{\ell }}^{\left(1\right)}\left(H_V\right)$$

and the same notation of the exterior algebra is used for grade involution, reversion, and conjugation in $\mathcal{C}\mathsf{\ell }\left(H_V\right)$, which, obviously, satisfy

$${\left(\mathbf{uv}\right)}^{^}=\widehat{\mathbf{u}}\widehat{\mathbf{v}},{\left(\mathbf{uv}\right)}^{˜}=\tilde{\mathbf{v}}\tilde{\mathbf{u}},{\left(\mathbf{uv}\right)}^{¯}=\overline{\mathbf{v}}\overline{\mathbf{u}}.$$

For vecfors $\mathbf{x},\mathbf{y}\in H_V$, we have the relation

$$\mathbf{xy}+\mathbf{yx}=2<\mathbf{x},\mathbf{y}>.$$

(6)

In particular, just as with multivectors, in Clifford algebra the norm application $N:\mathcal{C}\mathsf{\ell }\left(H_V\right)\to \mathcal{C}\mathsf{\ell }\left(H_V\right)$ can be defined as

$$N\left(\mathbf{x}\right)=\mathbf{x}\overline{\mathbf{x}},$$

(7)

the square of the volume $2n$-vecfor $\mathsf{œ}$ satisfy

$${\mathsf{œ}}^2=1,$$

and some properties of the Clifford product that will be useful in this work are as follows: for a homogeneous multivecfor $\mathbf{u}\in {\bigwedge }^r{H}_V,$ the grade involution is $\widehat{\mathbf{u}}={\left(-1\right)}^r\mathbf{u},$ in particular, $\widehat{\mathsf{œ}}={\left(-1\right)}^{2n}\mathsf{œ}=\mathsf{œ}.$ On the other hand,

$$\mathbf{u}⌟\mathsf{œ}=\mathbf{u}\mathsf{œ},\mathsf{œ}⌞\mathbf{u}=\mathsf{œ}\mathbf{u}$$

(8)

and if $\mathbf{x}\in H_V,$

$$\mathbf{x}⌟\mathbf{u}={\displaystyle \frac{1}{2}}\left(\mathbf{xu}-\widehat{\mathbf{u}}\mathbf{x}\right),\mathbf{u}⌟\mathbf{x}={\displaystyle \frac{1}{2}}\left(\mathbf{ux}-\mathbf{x}\widehat{\mathbf{u}}\right)$$

$$\mathbf{x}\wedge \mathbf{u}={\displaystyle \frac{1}{2}}\left(\mathbf{xu}+\widehat{\mathbf{u}}\mathbf{x}\right),\mathbf{u}\wedge \mathbf{x}={\displaystyle \frac{1}{2}}\left(\mathbf{ux}+\mathbf{x}\widehat{\mathbf{u}}\right).$$

Also note that,

$$\mathbf{x}\mathsf{œ}=-\widehat{\mathsf{œ}}\mathbf{x}=-\mathsf{œ}\mathbf{x}.$$

(9)

The following results on isomorphisms can be found in [7].

Proposition 2. 

There is the following natural isomorphism

$$\mathcal{C}\mathsf{\ell }\left(H_V\right)\simeq \mathrm{End}\left({\displaystyle \bigwedge }V\right).$$

In addition, being b a non-degenerate symmetric bilinear form on V, it holds also

$$\mathcal{C}\mathsf{\ell }\left(H_V\right)\simeq \mathcal{C}\mathsf{\ell }\left(H_{bV}\right)\simeq \mathcal{C}\mathsf{\ell }(V,b)\widehat{\otimes }\mathcal{C}\mathsf{\ell }(V,-b).$$

where $\widehat{\otimes }$ denotes the graded tensor product.

Corollary 1. 

The even and odd subspaces of the Hyperbolic Clifford algebra are

$$\mathcal{C}{\mathsf{\ell }}^{\left(0\right)}\left(H_V\right)\simeq \mathrm{End}\left({\displaystyle {\bigwedge }^{\left(0\right)}}V\right)\oplus \mathrm{End}\left({\displaystyle {\bigwedge }^{\left(1\right)}}V\right)$$

and

$$\mathcal{C}{\mathsf{\ell }}^{\left(1\right)}\left(H_V\right)\simeq L({\displaystyle {\bigwedge }^{\left(0\right)}}V,{\displaystyle {\bigwedge }^{\left(1\right)}}V)\oplus L({\displaystyle {\bigwedge }^{\left(1\right)}}{\displaystyle \bigwedge },{\displaystyle {\bigwedge }^{\left(0\right)}}V)$$

where $L(V,W)$ denotes the space of the linear mappings from V to W and ${\bigwedge }^{\left(0\right)}V$ and ${\bigwedge }^{\left(1\right)}V$ denote, respectively, the spaces of the even and of odd elements of $\bigwedge V.$

3. Groups Associated with $\mathcal{C}\mathsf{\ell }\left({\mathit{H}}_{\mathit{V}}\right)$

Next, we will describe the groups and transformations involved in the structure of the Hyperbolic Clifford algebra, just as in the Clifford algebra of multivectors or multiforms, the associated groups are the orthogonal group, the Clifford–Lipschitz group and the Pin and groups, which in the case of $\mathcal{C}\mathsf{\ell }\left(H_V\right)$ are appropriately defined.

3.1. Orthogonal Transformations and Reflection in $H_V$

Let g be a symmetric bilinear form defined on a vector space V. A map $T:V\to V$ is an isometry or an orthogonal transformation if

$$g\left(T\left(v\right),T\left(u\right)\right)=g\left(v,u\right),\forall v,u\in V,$$

it immediately follows that ${\left(\det T\right)}^2=1,$ i.e., $\det T=\pm 1.$

Orthogonal transformations where $\det T=1$ are called rotations and when $\det T=-1$ are called reflections. The orthogonal group on the vector space $H_V=V\oplus V^{*}$ will be denoted by

$$O_{H_V}=\left\{\mathrm{Isometries}/T:H_V\to H_V\right\}.$$

The subgroup of $O_{H_V}$, when $\det T=1$ is called a special orthogonal group and will be denoted by ${SO}_{H_V}.$

It is very important for the development of a theory in $\mathcal{C}\mathsf{\ell }\left(H_V\right)$ to specify the reflections, we know that the spaces V and $V^{*}$ are totally maximal isotropic subspaces of $H_V$. Then these spaces can be identified in $H_V$ under the image of the inclusions $i_{*}:V\to H_V,x_{*}\mapsto i_{*}{x}_{*}=x_{*}\oplus 0\equiv x_{*}$ and $i^{*}:V^{*}\to H_V,x^{*}\mapsto i^{*}{x}^{*}=0\oplus x^{*}\equiv x^{*},$ so given a linear map $\varphi \in \mathrm{End}V$ (or similarly ${\varphi }^{*}\in \mathrm{End}{V}^{*}$), it can be seen in $\mathrm{End}{H}_V$, under the inclusion $i_{*}$ (or under the inclusion $i^{*}$, respectively). Thus, every linear map $\varphi \in \mathrm{End}V$ (or, equivalently ${\varphi }^{*}\in \mathrm{End}{V}^{*}$) induces a linear map $I\left(\varphi \right)\in \mathrm{End}{H}_V$, called the isotropic extension of $\varphi$ (see [7]), defined by

$$I\left(\varphi \right)=\varphi \oplus {\varphi }^{*}.$$

(10)

In this sense, every vecfor $\mathbf{x}\in H_V$ can be associated with a reflection $S_{\mathbf{x}}:V\to V$ and a reflection $S^{\mathbf{x}}:V^{*}\to V^{*}$ (here $V,V^{*}\hookrightarrow H_V$ under the inclusions $i_{*}$ and $i^{*}$, respectively) defined, respectively, by

$$S_{\mathbf{x}}{y}_{*}=y_{*}-2{\displaystyle \frac{x^{*}\left(y_{*}\right)}{x^{*}\left(x_{*}\right)}}{x}_{*}\mathrm{and}{S}^{\mathbf{x}}{y}^{*}=y^{*}-2{\displaystyle \frac{y^{*}\left(x_{*}\right)}{x^{*}\left(x_{*}\right)}}{x}^{*}.$$

(11)

So, we can define the isotropic extension for the space $H_V,$$I\left(S_{\mathbf{x}}\right)={\mathbf{S}}_{\mathbf{x}}:H_V\to H_V$ given by

$${\mathbf{S}}_{\mathbf{x}}=S_{\mathbf{x}}\oplus S^{\mathbf{x}},$$

or explicitly for $\mathbf{y}\in H_V,$ we have

$${\mathbf{S}}_{\mathbf{x}}\mathbf{y}=\mathbf{y}-2{\displaystyle \frac{⟨\mathbf{x},\mathbf{y}⟩}{⟨\mathbf{x},\mathbf{x}⟩}}\mathbf{x}.$$

(12)

We immediately noticed that

$${\mathbf{S}}_{\mathbf{x}}\mathbf{x}=\mathbf{x}-2{\displaystyle \frac{⟨\mathbf{x},\mathbf{x}⟩}{⟨\mathbf{x},\mathbf{x}⟩}}\mathbf{x}=-\mathbf{x}.$$

(13)

We also easily observe that ${\mathbf{S}}_{\mathbf{x}}$ is an orthogonal application with respect to the bilinear form $⟨,⟩$ of $H_V,$ i.e.,

$$⟨{\mathbf{S}}_{\mathbf{x}}\mathbf{y},{\mathbf{S}}_{\mathbf{x}}\mathbf{z}⟩=⟨\mathbf{y},\mathbf{z}⟩,$$

for all $\mathbf{y},\mathbf{z}\in H_V.$

Furthermore, we can show that $\det {\mathbf{S}}_{\mathbf{x}}=-1$, in fact we already know that ${\mathbf{S}}_{\mathbf{x}}\mathbf{x}=-\mathbf{x},$ then consider the volume element $\mathsf{œ}={\mathsf{œ}}_1\wedge \dots \wedge {\mathsf{œ}}_{2n}$ where $\left\{{\mathsf{œ}}_1,\dots ,{\mathsf{œ}}_{2n}\right\}$ is an orthonormal basis of $H_V$ and define the action of ${\mathbf{S}}_x$ on an element $\mathbf{z}=z_1\wedge \cdots \wedge z_p\in {\bigwedge }^p{H}_V$ by ${\mathbf{S}}_x\left(\mathbf{z}\right)={\mathbf{S}}_x\left(z_1\right){\mathbf{S}}_x\left(z_2\right)\cdots {\mathbf{S}}_x\left(z_p\right).$ On the other hand, we have the relation (for example, see [13]) ${\mathbf{S}}_{\mathbf{x}}\left(\mathsf{œ}\right)=\left(\det {\mathbf{S}}_{\mathbf{x}}\right)\mathsf{œ}$, and using the properties (8) and (9), we have

$$\begin{array}{ccc}\left(\det {\mathbf{S}}_{\mathbf{x}}\right)\mathsf{œ}\hfill & =\hfill & {\mathbf{S}}_{\mathbf{x}}\left(\mathsf{œ}\right)={\mathbf{S}}_{\mathbf{x}}\left({\mathsf{œ}}_1{\mathsf{œ}}_2\dots {\mathsf{œ}}_{2n}\right)\hfill \\ & =\hfill & {\mathbf{S}}_{\mathbf{x}}\left({\mathsf{œ}}_1\right){\mathbf{S}}_{\mathbf{x}}\left({\mathsf{œ}}_2\right)\cdots {\mathbf{S}}_{\mathbf{x}}\left({\mathsf{œ}}_{2n}\right)\hfill \\ & =\hfill & {\left(-1\right)}^{2n}\mathbf{x}\mathsf{œ}{\mathbf{x}}^{-1}=-{\left(-1\right)}^{2n}\mathsf{œ}{\mathbf{xx}}^{-1}=-\mathsf{œ},\hfill \end{array}$$

where we obtain $\det {\mathbf{S}}_{\mathbf{x}}=-1,$ from which we conclude that ${\mathbf{S}}_{\mathbf{x}}$ is a reflection.

On the other hand, in terms of $\mathcal{C}\mathsf{\ell }\left(H_V\right),$ from (6), we have for vecfors $\mathbf{x},\mathbf{y}\in H_V$ and the relationship

$$\mathbf{xy}+\mathbf{yx}=2<\mathbf{x},\mathbf{y}>,$$

(14)

from which ${\mathbf{x}}^2=<\mathbf{x},\mathbf{x}>,$ so the vecfor $\mathbf{x}/<\mathbf{x},\mathbf{x}>$ can be interpreted as the inverse of $\mathbf{x}$ since

$$\mathbf{x}{\displaystyle \frac{\mathbf{x}}{<\mathbf{x},\mathbf{x}>}}={\displaystyle \frac{\mathbf{x}}{<\mathbf{x},\mathbf{x}>}}\mathbf{x}={\displaystyle \frac{{\mathbf{x}}^2}{<\mathbf{x},\mathbf{x}>}}=\mathbf{1}.$$

(15)

So from (12) and (14) we can write

$$\begin{array}{ccc}{\mathbf{S}}_{\mathbf{x}}\mathbf{y}\hfill & =\hfill & \mathbf{y}-\left[{\displaystyle \frac{\mathbf{xy}+\mathbf{yx}}{<\mathbf{x},\mathbf{x}>}}\right]\mathbf{x}\hfill \\ \hfill & =\hfill & \mathbf{y}-\left[\mathbf{xy}+\mathbf{yx}\right]{\mathbf{x}}^{-1}\hfill \\ \hfill & =\hfill & -{\mathbf{xyx}}^{-1}=\widehat{\mathbf{x}}{\mathbf{yx}}^{-1}.\hfill \end{array}$$

(16)

In conclusion, a reflection of the hyperplane orthogonal to the vecfor $\mathbf{x}$, in terms of $\mathcal{C}\mathsf{\ell }\left(H_V\right)$, satisfies the identity (In [7] a matrix representation of the reflection in the orthonormal basis for a hyperbolic space can be found).

$${\mathbf{S}}_{\mathbf{x}}\mathbf{y}=\widehat{\mathbf{x}}{\mathbf{yx}}^{-1},\forall \mathbf{x},\mathbf{y}\in H_V.$$

(17)

3.2. The Clifford–Lipschitz Group Associated with $\mathcal{C}\mathsf{\ell }\left(H_V\right)$

To define the Clifford–Lipschitz group associated with $\mathcal{C}\mathsf{\ell }\left(H_V\right),$ we will try to preserve the notation as in the case of multivector algebra $\mathcal{C}\mathsf{\ell }\left(V\right).$ Then, taking the vector space $V={\mathbb{R}}^{p,q},$ that is, $H_V=H_{{\mathbb{R}}^{p,q}}={\mathbb{R}}^{p,q}\oplus {\left({\mathbb{R}}^{p,q}\right)}^{*}$ and to simplify the notation, instead of writing $H_{{\mathbb{R}}^{p,q}}$ we will write $H_{p,q}$ and denote $\mathcal{C}\mathsf{\ell }\left(H_{{\mathbb{R}}^{p,q}}\right)$ as $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}.$ Following this same reasoning, the orthogonal group and special orthogonal group will be denoted by $O_{H_{p,q}}$ and ${SO}_{H_{p,q}}$, respectively, for $V={\mathbb{R}}^{p,q}.$

As in the multivector algebra, the most natural group of $\mathcal{C}\mathsf{\ell }\left(H_V\right)$ is the group of invertible elements of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}},$ denoted by

$$\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{*}=\left\{\mathbf{x}\in \mathcal{C}{\mathsf{\ell }}_{H_{p,q}}/\exists {\mathbf{x}}^{-1}\right\}.$$

This group acts naturally on $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$ as an algebra of homomorphisms via its adjoint representation $Ad$ or twisted adjoint representation $\widehat{A}d$, respectively.

$$Ad:\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{*}\to \mathrm{Aut}\left(\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}\right);\mathbf{x}⟼A{d}_{\mathbf{x}},withA{d}_{\mathbf{x}}\left(\mathbf{u}\right)={\mathbf{xux}}^{-1}$$

$$\widehat{A}d:\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{*}\to \mathrm{Aut}\left(\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}\right);x⟼\widehat{A}{d}_{\mathbf{x}},with\widehat{A}{d}_{\mathbf{x}}\left(\mathbf{u}\right)=\widehat{\mathbf{x}}{\mathbf{ux}}^{-1}.$$

To define a cover of the group of orthogonal transformations of $H_{p,q},$ one can use its twisted adjoint action on $H_{p,q},$ $\widehat{A}{d}_{\mathbf{x}}\left(u\right)=\widehat{\mathbf{x}}{\mathbf{ux}}^{-1},$ since from (17) it is a reflection of the hyperplane orthogonal to the vecfor $\mathbf{x},$ then automatically preserves the quadratic form in $H_{p,q},$ i.e, $\widehat{A}{d}_{\mathbf{x}}\in O_{H_{p,q}}$. In what follows, we will denote the twisted adjoint action $\widehat{A}{d}_{\mathbf{x}}$ by ${\rho }_{\mathbf{x}}.$ Thus, using ${\rho }_{\mathbf{x}}$ we can define the Clifford–Lipschitz group for $H_{p,q}$, as the subgroup of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{*},$ defined by

$${\mathrm{\Gamma }}_{H_{p,q}}=\left\{\mathbf{x}\in \mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{*}/\alpha \left(\mathbf{x}\right){\mathbf{vx}}^{-1}\in H_{p,q},\forall \mathbf{v}\in H_{p,q}\right\},$$

(18)

where $\alpha \left(\mathbf{x}\right)\equiv \widehat{\mathbf{x}}$ is the graduated involution, a notation that we will also adopt in what follows.

Furthermore, the twisted adjoint action $\rho$ of the group ${\mathrm{\Gamma }}_{H_{p,q}}$ on the hyperbolic vector space $H_{p,q}$ is defined by

$$\rho :{\mathrm{\Gamma }}_{H_{p,q}}\to \mathrm{Aut}\left(H_{p,q}\right),{\rho }_{\mathbf{x}}\mathbf{v}=\alpha \left(\mathbf{x}\right){\mathbf{vx}}^{-1},\forall \mathbf{v}\in H_{p,q}.$$

(19)

It is easy to show that ${\rho }_{\mathbf{x}}$ is a linear action on $H_{p,q}$ and that ${\mathrm{\Gamma }}_{H_{p,q}}$ is a group. Furthermore, note that ${\rho }_{\mathbf{x}}$ is an isomorphism on $H_{p,q}$ for all $\mathbf{x}\in {\mathrm{\Gamma }}_{H_{p,q}}$. Indeed, first note that ${\rho }_{\mathbf{x}}$ is injective, since suppose that ${\rho }_{\mathbf{x}}\left(u\right)=0,$ so we have that $\alpha \left(\mathbf{x}\right)u{\mathbf{x}}^{-1}=0,$ then, multiplying on the right by $\mathbf{x}$ and on the left by $\alpha \left({\mathbf{x}}^{-1}\right)$ we obtain that $u=0$, that is $\ker {\rho }_{\mathbf{x}}=\left\{0\right\}$ and since $\dim H_{p,q}$ is finite, then ${\rho }_{\mathbf{x}}:H_{p,q}\to H_{p,q}$ is an isomorphism $\forall \mathbf{x}\in {\mathrm{\Gamma }}_{H_{p,q}}$.

Theorem 1. 

The map $\rho :{\mathrm{\Gamma }}_{H_{p,q}}\to O_{H_{p,q}}$ is a surjective homomorphism whose kernel is the multiplicative group ${\mathbb{R}}^{*}\mathbf{1}$ of the nonzero scalar multiples of the unit $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$. The restriction of $N\left(\mathbf{x}\right)$ to ${\mathrm{\Gamma }}_{H_{p,q}}$ is a nonzero scalar.

Proof. 

Let us first note that if $\mathbf{x}\in \ker \rho ,$ by definition ${\rho }_{\mathbf{x}}v=v,\forall \mathbf{x}\in {\mathrm{\Gamma }}_{H_{p,q}},$ this is ${\rho }_{\mathbf{x}}v=\alpha \left(\mathbf{x}\right)v{\mathbf{x}}^{-1}=v$ then $\alpha \left(\mathbf{x}\right)v=v\mathbf{x}$$\forall v\in H_{p,q}.$

Since $\mathbf{x}\in {\mathrm{\Gamma }}_{p,q},$ in particular, $\mathbf{x}\in \mathcal{C}{\mathsf{\ell }}_{H_{p,q}}=\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{\left(0\right)}\oplus \mathcal{C}{\mathsf{\ell }}_{H_{p,q}}^{\left(1\right)},$ then we can separate the even part and the odd part of $\mathbf{x}\in {\mathrm{\Gamma }}_{p,q},$ this is:

$$\mathbf{x}=x_0+x_1,$$

(20)

where $x_0$ is even and $x_1$ is odd, thus $\alpha \left(\mathbf{x}\right)=x_0-x_1.$ Then,

$$\alpha \left(\mathbf{x}\right)v=v\left(x_0+x_1\right)\forall v\in H_{p,q}.$$

Then, setting $x_0=a_0+a_1{\sigma }_1$ and $x_1=b_0{\sigma }_1+b_1,$ where $a_0,b_0$ are even and $a_1,b_1$ are odd and all are elements of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$, independent of ${\sigma }_1$. Substituting into the condition $\alpha \left(\mathbf{x}\right)v=v\mathbf{x}$ for all $v\in H_{p,q}$ and taking $v={\sigma }_1$, one obtains

$$(x_0-x_1){\sigma }_1={\sigma }_1(x_0+x_1).$$

Using ${\sigma }_1^2=⟨{\sigma }_1,{\sigma }_1⟩\in {\mathbb{R}}^{*}$ and the anticommutation relations of the basis vectors, this forces $a_1=0$ and $b_0=0$, i.e., $x_0$ and $x_1$ are independent of ${\sigma }_1$. Following the same reasoning for each basis element ${\sigma }_k$ with $k=1,\dots ,2n$ (see [14,15] for full details), the argument is valid for all elements of the basis $\left\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_{2n}\right\}$ of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$, and therefore, if $\mathbf{x}\in \ker \rho$ does not depend on any element of the basis, then $\mathbf{x}$ commutes with every basis element of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$. Since the center of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$ consists only of real scalar multiples of the identity (for $p+q\ge 1$), $\mathbf{x}$ is a real number multiplied by the unit of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}.$

On the other hand, since $\alpha \left(\overline{u}\right)=u\forall u\in H_V,$ then if $\mathbf{x}\in {\mathrm{\Gamma }}_{p,q}$ we have $\alpha \left(\mathbf{x}\right)v{\mathbf{x}}^{-1}\in H_V,$ which implies that

$$\alpha \left(\mathbf{x}\right)v{\mathbf{x}}^{-1}=\alpha \left(\overline{\alpha \left(\mathbf{x}\right)v{\mathbf{x}}^{-1}}\right)=\alpha \left(\overline{{\mathbf{x}}^{-1}}\overline{v}\overline{\alpha \left(\mathbf{x}\right)}\right)=\alpha \left({\overline{\mathbf{x}}}^{-1}\right)v\overline{\mathbf{x}}.$$

Thus, from the previous equality we have

$$\begin{array}{cc}\alpha \left(N{\left(\mathbf{x}\right)}^{-1}\right)vN\left(\mathbf{x}\right)\hfill & =\alpha \left({\left(\mathbf{x}\overline{\mathbf{x}}\right)}^{-1}\right)v\mathbf{x}\overline{\mathbf{x}}=\alpha \left({\overline{\mathbf{x}}}^{-1}\right)\alpha \left({\mathbf{x}}^{-1}\right)v\mathbf{x}\overline{\mathbf{x}}\hfill \\ & =\alpha \left(\mathbf{x}\right)\alpha \left({\mathbf{x}}^{-1}\right)v{\mathbf{xx}}^{-1}=v,\hfill \end{array}$$

from which for $\mathbf{x}\in {\mathrm{\Gamma }}_{p,q}$, we have that $N\left(\mathbf{x}\right)\in {\mathbb{R}}^{*}\mathbf{1}.$ □

Remark 3. 

The surjective homomorphism $\rho :{\mathrm{\Gamma }}_{H_{p,q}}\to O_{H_{p,q}}$ with kernel ${\mathbb{R}}^{*}$ parallels the classical construction for standard Clifford algebras. The fundamental difference lies in the structure: while standard Clifford–Lipschitz groups act on n-dimensional vector spaces, the hyperbolic version acts on $2n$-dimensional hyperbolic structures $H_{p,q}$. This allows the simultaneous representation of both multivectors ($\bigwedge V$) and multiforms ($\bigwedge V^{*}$), providing a unified algebraic framework absent in classical theory.

3.3. The Pin and Groups of $\mathcal{C}{\mathsf{\ell }}_{H_{p,q}}$

The Pin and Spin groups are the most “reduced” subgroups of ${\mathrm{\Gamma }}_{H_{p,q}}$ to describe orthogonal transformations, these groups are defined by restricting the group ${\mathrm{\Gamma }}_{H_{p,q}}$ to those elements that have norm $\pm 1.$ That is, the group $Pi{n}_{H_{p,q}}$ is defined as

$${\mathrm{Pin}}_{H_{p,q}}=\left\{a\in {\mathrm{\Gamma }}_{H_{p,q}}/N\left(a\right)=\pm 1\right\},$$

then, restricting the domain of the twisted adjoint action $\rho$ to the group ${\mathrm{Pin}}_{H_{p,q}},$

$$\rho :{\mathrm{Pin}}_{H_{p,q}}\to O_{H_{p,q}}$$

we obtain that

$${\left.\ker \rho \right|}_{{\mathrm{Pin}}_{H_{p,q}}}=\ker \rho \cap \ker N=\left\{\pm 1\right\}={\mathbb{Z}}_2.$$

The group $H_{p,q},$ is the subgroup of the group ${\mathrm{\Gamma }}_{H_{p,q}}^{+}={\mathrm{\Gamma }}_{H_{p,q}}\cap \mathcal{C}{\mathsf{\ell }}_{p,q}^{+},$ defined as

$$H_{p,q}=\left\{a\in {\mathrm{\Gamma }}_{H_{p,q}}^{+}/N\left(a\right)=\pm 1\right\}.$$

So, the twisted adjoint $\rho$ restricted to group $H_{p,q}$ has its image in ${SO}_{H_{p,q}},$ that is

$${\rho :}_{H_{p,q}}\to {SO}_{H_{p,q}},$$

(21)

and also ${\left.\ker \rho \right|}_{H_{p,q}}=\left\{\pm 1\right\}={\mathbb{Z}}_2.$

The above results can be summarized in the following theorem, similar to the case of multivector Clifford algebra $\mathcal{C}\mathsf{\ell }\left(V\right).$

Theorem 2. 

Let $\rho :{\mathrm{\Gamma }}_{H_{p,q}}\to O_{H_{p,q}}$ be the surjective homomorphism studied previously, then we have the following relations depending on the domain of ρ:

(a) 

${\left.\rho \right|}_{{\mathrm{Pin}}_{H_{p,q}}}:{\mathrm{Pin}}_{H_{p,q}}\to O_{H_{p,q}}$ is surjective with Kernel ${\mathbb{Z}}_2.$

(b) 

${\left.\rho \right|}_{H_{p,q}}:H_{p,q}\to {SO}_{H_{p,q}}$ is surjective with Kernel ${\mathbb{Z}}_2.$

(c) 

${\left.\rho \right|}_{{\stackrel{e}{H}}_{p,q}}:{\stackrel{e}{H}}_{p,q}\to {SO}_{H_{p,q}}^e$ is surjective with Kernel ${\mathbb{Z}}_2.$

(d) 

$$O_{H_{p,q}}={\displaystyle \frac{{\mathrm{Pin}}_{H_{p,q}}}{{\mathbb{Z}}_2}};{SO}_{H_{p,q}}={\displaystyle \frac{H_{p,q}}{{\mathbb{Z}}_2}};{SO}_{H_{p,q}}^e={\displaystyle \frac{{\stackrel{e}{H}}_{p,q}}{{\mathbb{Z}}_2}}.$$

Remark 4. 

The definitions of ${\mathrm{Pin}}_{H_{p,q}}$ and ${\mathrm{Spin}}_{H_{p,q}}$ via restriction to elements of norm $\pm 1$ follow the standard construction. However, the hyperbolic framework introduces a crucial new feature: both ${\mathrm{SO}}_{H_{p,q}}$ (the special orthogonal group) and its oriented version ${\mathrm{SO}}_{H_{p,q}}^e$ admit double covers via the respective groups. This is essential for constructions or representations on hyperbolic Lorentzian manifolds and for developing consistent structures on spacetime with simultaneous treatment of field components and their duals.

4. Bundle Structure of the Hyperbolic Clifford algebra

To introduce the bundle on $\mathcal{C}\mathsf{\ell }\left(H_V\right),$ we must take into account that in the case of hyperbolic space, two vector bundles are involved, the tangent bundle and the cotangent bundle, on the same manifold $M,$ let us say $\left(E_1,M,{\pi }_1,G,V_1\right)$ and $\left(E_2,M,{\pi }_2,G,V_2\right).$ The Whitney sum of $E_1$ and $E_2$ is the vector bundle with total space $E_1\oplus E_2,$ which is given by

$$E_1\oplus E_2=\left\{\left(p_1,p_2\right)\in E_1\times E_2:{\pi }_1\left(p_1\right)={\pi }_2\left(p_2\right)\right\},$$

with projection $\mathrm{\Pi }:E_1\oplus E_2\to M$ given by

$$\mathrm{\Pi }\left(\left(p_1,p_2\right)\right)={\pi }_1\left(p_1\right)={\pi }_2\left(p_2\right).$$

(22)

Then, the fiber corresponding to $x\in M$ of $E_1\oplus E_2$ is the direct sum of the vector spaces ${\pi }_1^{-1}\left(x\right)\oplus {\pi }_2^{-1}\left(x\right).$ To verify the local trivialization condition in $E_1\oplus E_2,$ it suffices to take trivializing functions $\left\{{\phi }_{\alpha }\right\}$ and $\left\{{\psi }_{\alpha }\right\}$ for $E_1$ and $E_2$, respectively, subordinated to the same cover ${\left(U_{\alpha }\right)}_{\alpha \in I},$ of $M,$ whose induced transition functions are $s_{\alpha \beta }={\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}$ and $h_{\alpha \beta }={\psi }_{\alpha }\circ {\psi }_{\beta }^{-1}$, then the trivializations for $E_1\oplus E_2$ are given by

$$\left\{{\varphi }_{\alpha }=\left({\phi }_{\alpha }\times {\psi }_{\alpha }\right)\right\},$$

(23)

to which transition functions correspond

$$g_{\alpha \beta }=s_{\alpha \beta }\oplus h_{\alpha \beta }=\left[\begin{array}{cc}{s}_{\alpha \beta }& 0\\ 0& h_{\alpha \beta }\end{array}\right],$$

(24)

remembering that $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G,$ from (24) we will have to $G=G\left(2n,\mathbb{R}\right).$ For details on Whitney sum, see e.g., [16].

A very useful and fundamental property, for the next topic, of Whitney’s sum, is that if ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are trivial vector bundles with vector spaces of dimension n and m, respectively, then ${\mathcal{F}}_3={\mathcal{F}}_1\oplus {\mathcal{F}}_2$ is a trivial vector bundle with vector space of dimension $n+m$, since there exists an isomorphism

$$\chi :{\mathcal{F}}_1\oplus {\mathcal{F}}_2\to {\mathcal{F}}_3$$

defined by $\chi \left(\left(x,u\right),\left(x,v\right)\right)=\left(x,u+v\right),$ where $u+v\in V,$ with $\dim V=n+m.$ Then, to define a structure hyperbolic on $M,$ we can use a known result due to Geroch [17,18], which states that for a Lorentz manifold $(M,g)$, a structure exists if and only if the principal bundle $P_{{SO}_{1,3}^e}\left(M\right)$ is a trivial bundle. Remembering that a principal bundle is trivial if and only if it admits a global section. Therefore, Geroch’s result says that a spacetime admits a structure if and only if it admits a Lorentz frame.

Remark 5. 

It is natural to ask whether the existence of a hyperbolic structure imposes stricter topological constraints on M than a standard structure. The existence of a hyperbolic structure requires the existence of a global section of the principal bundle ${\mathbf{P}}_{{SO}_{H_{p,q}}^e}\left(M\right)$, i.e., a global hyperbolic orthonormal frame. Since the structure group ${SO}_{H_{p,q}}^e$ has a richer topology than ${SO}_{1,3}^e$, the precise topological obstructions depend on the homotopy type of ${SO}_{H_{p,q}}^e$. A detailed analysis of these constraints lies beyond the scope of the present paper and is left as a direction for future research.

4.1. Frame Bundle on $\mathcal{C}\mathsf{\ell }\left(H_V\right)$

Similar to the tangent bundle $TM$, we will say that the hyperbolic tangent bundle $TM\oplus T^{*}M$ to a differentiable n-dimensional manifold M is associated to a principal bundle $F_H\left(M\right)={\cup }_{x\in M}{F}_{x}M\oplus F_x^{*}M,$ called the frame bundle, where $F_{x}M\oplus F_x^{*}M$ is the set of frames in $x\in M.$ The structure group of $F_H\left(M\right)$ is $Gl\left(2n,\mathbb{R}\right),$ (here following the notation of a principal fiber bundle, we have that the bundle $F_H\left(M\right)$ is formally written as $\left(F_{H}M,M,\pi ,Gl\left(2n,\mathbb{R}\right)\right)$). Let $\left\{x_{\alpha }^1,x_{\alpha }^2,\cdots ,x_{\alpha }^n\right\}$ be the coordinates associated with a local chart $\left(U_{\alpha },{\phi }_{\alpha }\right)$ of the maximal atlas of M. Thus, the natural basis to $T_{x}M\oplus T_x^{*}M$ on $U_{\alpha }\subset M,$ is given by

$$\left\{{\displaystyle \frac{\partial }{\partial x^1}}\oplus d{x}^1,\cdots ,{\displaystyle \frac{\partial }{\partial x^n}}\oplus d{x}^n,{\displaystyle \frac{-\partial }{\partial x^1}}\oplus d{x}^1,\cdots ,{\displaystyle \frac{-\partial }{\partial x^n}}\oplus d{x}^n\right\}.$$

(25)

where ${\displaystyle \frac{-\partial }{\partial x^k}}\oplus d{x}^k$ are obtained from ${\displaystyle \frac{\partial }{\partial x^k}}\oplus d{x}^k$ through involution

$${\displaystyle \frac{\partial }{\partial x^k}}\oplus d{x}^k\to {\displaystyle \frac{-\partial }{\partial x^k}}\oplus d{x}^k,$$

see (2).

To simplify the notation relating to the indices, we will place

$${\mathbf{fl}}_k={\displaystyle \frac{\partial }{\partial x^k}}\oplus d{x}^{k}and{\mathbf{fl}}_{n+k}={\displaystyle \frac{-\partial }{\partial x^k}}\oplus d{x}^k$$

(26)

with $k=1,\dots ,n.$ Then, from (25) we can write the natural basis of $T_{x}M\oplus T_x^{*}M$ over $U_{\alpha }\subset M,$ as

$$\left\{{\mathbf{fl}}_1,\cdots ,{\mathbf{fl}}_n,{\mathbf{fl}}_{n+1},\cdots ,{\mathbf{fl}}_{2n}\right\}$$

Definition 1. 

A frame at $T_{x}M\oplus T_x^{*}M$ is a set

$${\mathrm{\Theta }}_x=\left\{{\left.z_1\right|}_x,\cdots ,{\left.z_n\right|}_x,{\left.z_{n+1}\right|}_x,\cdots ,{\left.z_{2n}\right|}_x\right\}$$

of linearly independent vectors such that

$${\left.z_i\right|}_x=F_i^j{\left.{\mathbf{fl}}_j\right|}_x,$$

(27)

where the matrix $\left(F_i^j\right)\in Gl\left(2n,\mathbb{R}\right).$

Remark 6. 

In the standard tangent bundle $TM$, a frame at $T_{x}M$ consists of a basis of the n-dimensional tangent space. Here, we extend this concept to the hyperbolic tangent-cotangent bundle $TM\oplus T^{*}M$, where frames incorporate both contravariant (tangent) and covariant (cotangent) directions. This symmetric treatment of primal and dual spaces is a defining feature of the hyperbolic structure and is essential for theories in general relativity that require simultaneous treatment of fields and their dual counterparts.

A local trivialization ${\varphi }_a:{\mathrm{\Pi }}^{-1}\left(U_{\alpha }\right)\to U_{\alpha }\times Gl\left(2n,\mathbb{R}\right)$ of $F_H\left(M\right)$ is defined by

$${\varphi }_{\alpha }\left(p\right)=\left(\mathrm{\Pi }\left(p\right),{\varkappa }_{\alpha }\left(p\right)\right)=\left(x,{\mathrm{\Theta }}_x\right),$$

(28)

where $\mathrm{\Pi }\left(p\right)=x$ and ${\varkappa }_{\alpha }\left(p\right)={\mathrm{\Theta }}_x.$ Here $p=\left(p_1,p_2\right)$ and $\Pi$ and ${\varphi }_a$ are as in (22) and (23), respectively.

The action of $G=\left(G_i^j\right)\in Gl\left(2n,\mathbb{R}\right)$ on a frame $\mathrm{\Omega }\in F_H\left(U\right)$ is given by $\left(\mathrm{\Omega },G\right)\to \mathrm{\Omega }G,$ where the new frame $\mathrm{\Omega }G\in F_H\left(U\right)$ is defined by ${\varphi }_{\alpha }\left(\mathrm{\Omega }G\right)=\left(\mathrm{\Pi }\left(\mathrm{\Omega }G\right),{\varkappa }_{\alpha }\left(\mathrm{\Omega }G\right)\right)=\left(x,{\mathrm{\Theta }}_x^{\prime }\right),$ with $\mathrm{\Pi }\left(\mathrm{\Omega }G\right)=x$ and

$${\varkappa }_{\alpha }\left(\mathrm{\Omega }G\right)={\mathrm{\Theta }}_x^{\prime }=\left\{{\left.z_1^{\prime }\right|}_x,\cdots ,{\left.z_{2n}^{\prime }\right|}_x\right\}$$

with

$${\left.z_i^{\prime }\right|}_x={\left.z_j\right|}_x{G}_i^j,$$

(29)

where $i,j=1,\dots ,2n.$ Conversely, given the frames ${\mathrm{\Theta }}_x$ and ${\mathrm{\Theta }}_x^{\prime }$ there exists $G=\left(G_i^j\right)\in Gl\left(2n,\mathbb{R}\right)$ such that (29) is satisfied, which means that $Gl\left(2n,\mathbb{R}\right)$ acts on $F_H\left(M\right)$ actively.

Let $\left\{x^{\alpha }\right\}$ and $\left\{x^{\prime \alpha }\right\}$ be the coordinates associated with the local charts $\left\{U_{\alpha },{\phi }_{\alpha }\right\}$ and $\left\{U_{\alpha }^{\prime },{\phi }_{\alpha }^{\prime }\right\}$, respectively, and of the maximal atlas of $M.$ If $x\in U_{\alpha }\cap U_{\beta }$ we have

$${\left.z_i\right|}_x=F_i^j{\left.{\mathbf{fl}}_j\right|}_x=F_i^{\prime j}{\left.{\mathbf{fl}}_j^{\prime }\right|}_x,$$

where $\left(F_i^j\right),\left(F_i^{\prime j}\right)\in Gl\left(2n,\mathbb{R}\right)$ and note that the ${\mathbf{fl}}_l^{\prime }$ are defined as in (26) in $\left\{x^{\prime \alpha }\right\}$ coordinates, i.e.,

$${\mathbf{fl}}_l^{\prime }={\displaystyle \frac{\partial }{\partial x^{\prime l}}}\oplus d{x}^{\prime l}\mathrm{and}{\mathbf{fl}}_{n+l}^{\prime }={\displaystyle \frac{-\partial }{\partial x^{\prime l}}}\oplus d{x}^{\prime l},$$

with $l=1,\dots ,n.$ Since $F_i^j=F_k^{\prime j}{\left.\left({\displaystyle \frac{\partial x^k}{\partial x^{\prime j}}}\right)\right|}_x$ we have that the transition functions are

$$g_i^k\left(x\right)={\left.\left({\displaystyle \frac{\partial x^k}{\partial x^{\prime j}}}\right)\right|}_x\in Gl\left(2n,\mathbb{R}\right).$$

(30)

Remember that transition functions are continuous functions

$$g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G,\mathrm{where}{g}_{\alpha \beta }\left(x\right)={\varphi }_{\alpha ,x}\circ {\varphi }_{\beta ,x}^{-1},$$

(31)

and $g_{\alpha \beta }$ is as in (24), i.e.,

$$\begin{array}{c}{g}_{\alpha \beta }\left(x\right)=s_{\alpha \beta }\left(x\right)\oplus h_{\alpha \beta }\left(x\right)\hfill \\ =\left[\begin{array}{cc}{s}_{\alpha \beta }\left(x\right)& 0\\ 0& h_{\alpha \beta }\left(x\right)\end{array}\right]=\left[\begin{array}{cc}{\phi }_{\alpha ,x}\circ {\phi }_{\beta ,x}^{-1}& 0\\ 0& {\psi }_{\alpha ,x}\circ {\psi }_{\beta ,x}^{-1}\end{array}\right]\hfill \end{array}$$

(32)

Now making use of the metric field defined by $⟨\mathbf{x},\mathbf{y}⟩=x^{*}\left(y_{*}\right)+y^{*}\left(x_{*}\right),$ where $\mathbf{x}=x_{*}\oplus x^{*},\mathbf{y}=y_{*}\oplus y^{*}\in H_V,$ defined in (1), we can introduce the orthonormal frame in each $T_{x}U\oplus T_x^{*}U.$ That is, for each $T_{x}U\oplus T_x^{*}U,$ we can denote an orthonormal frame by ${\mathrm{\Theta }}_x=\left\{{\left.z_1\right|}_x,\cdots ,{\left.z_n\right|}_x,{\left.z_{n+1}\right|}_x,\cdots ,{\left.z_{2n}\right|}_x\right\}$, where

$${\left.z_i\right|}_x=h_i^j{\left.{\gamma }_j\right|}_x,$$

with $\left(h_i^j\right)\in O_{H_{p,q}},$ and where $O_{H_{p,q}}$ is the $2n$-dimensional real orthogonal group. In this case, the frame bundle is said to have been reduced to the hyperbolic orthonormal frame bundle, which will be denoted by ${\mathbf{P}}_{O_{H_{p,q}}}\left(M\right)$ or when $\left(h_i^j\right)\in {SO}_{H_{p,q}}$ the hyperbolic frame bundle will be denoted by ${\mathbf{P}}_{{SO}_{H_{p,q}}}\left(M\right).$

In an analogous way, in a hyperbolic principal bundle of oriented orthogonal frames ${\mathbf{P}}_{{SO}_{H_{p,q}}^e}\left(M\right),$ we can define on a Lorentzian manifold modeling space-time and its covering bundle called hyperbolic bundle ${\mathbf{P}}_{\stackrel{e}{H}p,q}\left(M\right).$ Remember that the isomorphism defined in (3) allows us to choose an arbitrary non-degenerate symmetric bilinear form b. Thus, in the case of a hyperbolic space, a Lorentzian manifold is a pair $\left(M,g\right)$, where $g\in \sec T^{2,0}M$ is a Lorentzian metric of signature $\left(1,3\right)$, i.e., for all $x\in M$, $T_{x}M\oplus T_x^{*}M$ $\simeq {\mathbb{R}}_H^{1,3}$ where ${\mathbb{R}}_H^{1,3}$ is the hyperbolic vector Minkowski space. Then, the Hyperbolic Clifford algebra of the ${\mathbb{R}}_H^{1,3}$ will be denoted by ${\mathbb{R}}_{1,3}^H.$ Most of the properties of hyperbolic algebra ${\mathbb{R}}_{1,3}^H$ are inherited from ${\mathbb{R}}_{1,3}$ algebra and are fundamental to this theory, for example, it is straightforward to show that every automorphism of ${\mathbb{R}}_{1,3}^H$ is inner and if we denote by ${\mathbb{R}}_{1,3}^{*H}$ the group of invertible elements of ${\mathbb{R}}_{1,3}^H$, then through the adjoint representation this group acts on ${\mathbb{R}}_{1,3}^H$ as an algebra of automorphisms. In addition, the group ${SO}_{H_{p,q}}^e$ has a natural extension in ${\mathbb{R}}_{1,3}^H.$

4.2. Hyperbolic Clifford Bundle

Definition 2. 

The Hyperbolic Clifford bundle of the metric manifold M is

$$\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)\equiv \mathcal{C}\mathsf{\ell }(TM\oplus T^{*}M,⟨⟩)={\displaystyle \bigcup _{x\in M}}\mathcal{C}\mathsf{\ell }(T_{x}M\oplus T_x^{*}M,⟨⟩),$$

where $\mathcal{C}\mathsf{\ell }(T_{x}M\oplus T_x^{*}M,⟨⟩)$ is the Hyperbolic Clifford algebra of the hyperbolic structure $\left(T_{x}M\oplus T_x^{*}M,⟨⟩\right)$.

The hyperbolic Clifford bundle $\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)$ is a vector bundle associated with the principal bundle of orthonormal frames ${\mathbf{P}}_{{SO}_{H_{1,3}}^e}\left(M\right),$ associated to a Lorentzian manifold, i.e.,

$$\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)={\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right){\times }_{\rho }\mathcal{C}\mathsf{\ell }(T_{x}M\oplus T_x^{*}M,⟨⟩)\simeq {\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right){\times }_{\rho }{\mathbb{R}}_{1,3}^H$$

Indeed, considering the canonical projection ${\mathrm{\Pi }}_H:\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)\to M$ and taking ${\left(U_{\alpha }\right)}_{\alpha \in I}$ an open covering of $M,$ then we can define the trivialization mappings

$${\phi }_{\alpha }:{\mathrm{\Pi }}_H^{-1}\left(U_{\alpha }\right)\to U_{\alpha }\times {\mathbb{R}}_{1,3}^H,$$

such that, ${\phi }_{\alpha }\left(p\right)=\left({\mathrm{\Pi }}_H\left(p\right),{\phi }_{\alpha ,x}\left(p\right)\right)=\left(x,{\phi }_{\alpha ,x}\left(p\right)\right).$ Thus, if $x\in U_{\alpha }\cap U_{\beta }$ and $p\in {\mathrm{\Pi }}_H^{-1}\left(x\right),$ we have

$${\phi }_{\alpha ,x}\left(p\right)=h_{\alpha \beta }\left(x\right){\phi }_{\beta ,x}\left(p\right),$$

for $h_{\alpha \beta }\left(x\right)\in \mathrm{Aut}\left({\mathbb{R}}_{1,3}^H\right),$ where $h_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{Aut}\left({\mathbb{R}}_{1,3}^H\right)$ are the transition mapping of $\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right).$ Then, as every automorphism of ${\mathbb{R}}_{1,3}^H$ is inner, we have

$$h_{\alpha \beta }\left(x\right){\phi }_{\beta ,x}\left(p\right)=g_{\alpha \beta }\left(x\right){\phi }_{\alpha ,x}\left(p\right)g_{\alpha \beta }{\left(x\right)}^{-1},$$

for some $g_{\alpha \beta }\left(x\right)\in {\mathbb{R}}_{1,3}^{*H}.$

On the other hand, taking into account the isomorphism (21), we can deduce that the structure group of the hyperbolic Clifford bundle $\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)$ is reducible from $\mathrm{Aut}\left({\mathbb{R}}_{1,3}^H\right)$ to ${SO}_{H_{1.3}}^e$ and the transition maps for $\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)$ can be taking from ${\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right).$ Then, the hyperbolic Clifford bundle is an associated vector bundle to the principal bundle ${\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right),$ i.e.,

$$\mathcal{C}\mathsf{\ell }\left(H\left(M\right)\right)={\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right){\times }_{\rho }{\mathbb{R}}_{1,3}^H.$$

Details on the construction of a vector bundle in the field of theoretical physics can be found, e.g., [1,19]. Finally, we can define a hyperbolic structure on M, similarly to the case when the structure is a frame bundle, see, e.g., [1], from which we adopt the notation.

Definition 3. 

A hyperbolic structure on M consists of a hyperbolic principal fiber bundle, with group ${Spin}_{H_{1,3}}^e$,

$${\mathrm{\Pi }}_{\mathrm{{\rm Y}}}:{\mathbf{P}}_{{Spin}_{H_{1,3}}^e}\left(M\right)\to M$$

and a map

$$\mathrm{{\rm Y}}:{\mathbf{P}}_{{Spin}_{H_{1,3}}^e}\left(M\right)\to {\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right),$$

satisfying the following conditions,

(a) 

$\mathrm{\Pi }\left(\mathrm{{\rm Y}}\left(p\right)\right)={\mathrm{\Pi }}_{\mathrm{{\rm Y}}}\left(p\right),$$\forall p\in {\mathbf{P}}_{{Spin}_{H_{1,3}}^e}\left(M\right),$ where $\mathrm{\Pi }$ is the projection map of the hyperbolic bundle ${\mathbf{P}}_{{SO}_{H_{1.3}}^e}\left(M\right).$

(b) 

$\mathrm{{\rm Y}}\left(pu\right)=\mathrm{{\rm Y}}\left(p\right){\rho }_u,$$\forall p\in {\mathbf{P}}_{{Spin}_{H_{1,3}}^e}\left(M\right)$ and $\rho :{Spin}_{1,3}^e\to \mathrm{Aut}\left({\mathbb{R}}_{1,3}^H\right)$, ${\rho }_u:{\mathbb{R}}_{1,3}^H\to {\mathbb{R}}_{1,3}^H$ such that ${\rho }_{u}x=ux{u}^{-1}.$

Remark 7. 

The existence of a hyperbolic structure on a spacetime manifold M requires (by analogy with Geroch’s classical theorem) the existence of a global section of the principal bundle $P_{{\mathrm{SO}}_{H_{p,q}}^e}\left(M\right)$. Since the structure group ${\mathrm{SO}}_{H_{p,q}}^e$ is strictly larger than the standard ${\mathrm{SO}}_{1,3}^e$, one might expect different topological constraints. However, a detailed analysis of these constraints and comparison with classical obstruction theory is beyond the scope of this paper and is left for future investigation.

Remark 8. 

Remember that a local section of the fiber bundle $\left(E,M,\pi ,G,F\right)$ on an open set $U\subset M,$ is a mapping $S:U\to E$ such that $\mathrm{\Pi }\circ S=I$ and if $U=M$ the section S is said to be global. Then, any section of the hyperbolic principal fiber bundle ${\mathbf{P}}_{{Spin}_{H_{1,3}}^e}\left(M\right)$ will be called hyperbolic frame field.

5. Conclusions and Future Directions

This paper has presented a comprehensive development of structures associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V.

5.1. Summary of Main Results

The main results are as follows: (i) construction of the frame bundle $F_H\left(M\right)={\cup }_{x\in M}{F}_{x}M\oplus F_x^{*}M$ associated to the hyperbolic tangent-cotangent bundle structure; (ii) derivation of orthogonal transformations and reflections in hyperbolic space, with explicit characterization of the Clifford–Lipschitz group ${\mathrm{\Gamma }}_{H_{p,q}}$ as a surjective cover of $O_{H_{p,q}}$ with kernel ${\mathbb{R}}^{*}$; (iii) definition and analysis of Pin groups ${\mathrm{Pin}}_{H_{p,q}}$ and groups ${\mathrm{Spin}}_{H_{p,q}}$ as double covers of orthogonal transformation groups; (iv) establishment of the hyperbolic Clifford bundle as an associated vector bundle with a reduction in the structure group from $\mathrm{Aut}\left({\mathbb{R}}_{1,3}^H\right)$ to ${\mathrm{SO}}_{H_{1,3}}^e$.

5.2. Physical Significance

These results provide a rigorous mathematical foundation for the study of gravitational fields and particles in theoretical physics, particularly in formulations that require symmetric treatment of fields and their dual counterparts. The Clifford bundle formalism offers an elegant alternative to standard approaches and has proven useful in Yang-Mills type theories of gravitation in Minkowski spacetime.

5.3. Future Research Directions

Several open problems merit investigation:

1.

Topological obstructions: A detailed analysis of the topological constraints for the existence of hyperbolic structures on general spacetime manifolds, comparing with classical Geroch-type results.

2.

Specific gravitational theories: Application to particular formulations of gravitation and comparison of predictions with standard approaches.

3.

Supersymmetric extensions: Generalization to supersymmetric theories and integration with superfield formalisms.

4.

Computational approaches: Development of algorithmic methods for studying Clifford algebras of monomial rings and their automorphism groups.

5.

Quantization: Investigation of quantization procedures in the hyperbolic Clifford bundle framework and their relationship to standard canonical quantization.

We hope this work will inspire further development of the Hyperbolic Clifford algebra formalism and its applications in theoretical physics and algebra.