[Paper Review] Duality Products of Multivectors and Multiforms, and Extensors

V. V. Fernández, A. M. Moya+2 more|ArXiv.org|Mar 18, 2007

Algebraic and Geometric Analysis5 references4 citations

TL;DR

This paper develops a comprehensive algebraic framework for duality products between multivectors and multiforms, introducing k-multivector and l-multiform variable extensors over a finite-dimensional real vector space V. It establishes key operations—such as exterior products, adjoint and extension operators—on spaces denoted $\text{ext}_k^l(V)$ and $\text{ext}_k^{l*}(V)$, with detailed properties of generalized operators like $\underset{\smile}{\gamma}$, enabling a simplified and powerful formalism for differential geometry on manifolds with arbitrary connections.

ABSTRACT

In this paper we study in details the properties of the duality product of multivectors and multiforms (used in the definition of the hyperbolic Clifford algebra of multivefors) and introduce the theory of the k multivector and l multiform variables multivector (or multiform) extensors over V studying their properties with considerable detail.

Motivation & Objective

• To formalize the duality product between multivectors in $\bigwedge V$ and multiforms in $\bigwedge V^*$, essential for defining the hyperbolic Clifford algebra of multivefors.
• To develop a systematic theory of $k$-multivector and $l$-multiform variable extensors over $V$, defining the spaces $\text{ext}_k^l(V)$ and $\text{ext}_k^{l*}(V)$.
• To introduce and study fundamental operations on extensors, including exterior product, adjoint, extension, and generalized operator procedures.
• To lay the algebraic groundwork for a simplified and improved presentation of differential geometry on arbitrary manifolds with arbitrary connections using the Clifford bundle formalism.

Proposed method

• The paper defines the duality product $\langle \cdot, \cdot \rangle$ between elements of $\bigwedge V$ and $\bigwedge V^*$, establishing its role in the hyperbolic Clifford algebra $\mathcal{C}\ell(V \oplus V^*, \langle \cdot, \cdot \rangle)$.
• It introduces the left and right contracted products between multivectors and multiforms, providing a notation that clarifies their algebraic behavior.
• The spaces $\text{ext}_k^l(V)$ and $\text{ext}_k^{l*}(V)$ are defined as spaces of multivector or multiform extensors with $k$ multivector and $l$ multiform variables, respectively.
• The exterior product of extensors is defined, enabling composition of extensor fields in a graded algebraic structure.
• Generalized operators such as $\underset{\smile}{\gamma}$ are introduced, with properties like $\underset{\smile}{\gamma}(\tau \wedge \sigma) = (\underset{\smile}{\gamma}\tau) \wedge \sigma + \tau \wedge (\underset{\smile}{\gamma}\sigma)$, ensuring compatibility with the algebraic structure.
• The adjoint and extension operators are defined and studied in detail, with explicit transformation rules under duality and contraction.

Experimental results

Research questions

• RQ1 How can the duality product between multivectors and multiforms be systematically formalized to support the construction of the hyperbolic Clifford algebra of multivefors?
• RQ2 What are the algebraic properties of the left and right contracted products between elements of $\bigwedge V$ and $\bigwedge V^*$, and how can they be used to define generalized operators?
• RQ3 How can extensor fields with $k$ multivector and $l$ multiform variables be constructed and manipulated algebraically using operations like exterior product and adjoint?
• RQ4 What are the transformation rules for generalized operators such as $\underset{\smile}{\gamma}$ acting on extensors, and how do they preserve algebraic consistency?
• RQ5 How does the proposed extensor calculus improve the formalism for differential geometry on manifolds with arbitrary connections compared to prior approaches?

Key findings

• The duality product $\langle \cdot, \cdot \rangle$ between multivectors and multiforms is rigorously defined and shown to underlie the structure of the hyperbolic Clifford algebra $\mathcal{C}\ell(V \oplus V^*, \langle \cdot, \cdot \rangle)$, enabling a unified treatment of geometric objects.
• The generalized operator $\underset{\smile}{\gamma}$ satisfies the Leibniz rule over the exterior product: $\underset{\smile}{\gamma}(\tau \wedge \sigma) = (\underset{\smile}{\gamma}\tau) \wedge \sigma + \tau \wedge (\underset{\smile}{\gamma}\sigma)$, ensuring compatibility with extensor algebra.
• The adjoint and extension operators are defined on $\text{ext}_k^l(V)$ and $\text{ext}_k^{l*}(V)$, with explicit transformation rules under duality and contraction, enabling consistent manipulation of extensor fields.
• The action of $\underset{\smile}{\gamma}$ on inner products satisfies $\underset{\smile}{ olimits\gamma}\langle \tau, \sigma \rangle = \langle \underset{\smile}{\gamma}\tau, \sigma \rangle - \langle \tau, \underset{\smile}{\gamma}^{\bigtriangleup}\sigma \rangle$, preserving the duality structure.
• The theory of extensors provides a powerful and simplifying framework for differential geometry on arbitrary manifolds, as demonstrated by the consistent transformation laws and algebraic closure of operations.
• The formalism presented generalizes and improves upon earlier approaches using only multivector and extensor calculus, offering a more systematic and computationally tractable tool for geometric analysis.