Journal of Geometry and Symmetry in Physics

On Matrix Representations of Geometric (Clifford) Algebras

Ramon G. Calvet

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Abstract

Representations of geometric (Clifford) algebras with real square matrices are reviewed by providing the general theorem as well as examples of lowest dimensions. New definitions for isometry and norm are proposed. Direct and indirect isometries are identified respectively with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its matrix representation of order $n$. It is deduced in which geometric algebras direct isometries are inner automorphisms (similarity transformations of matrices). Indirect isometries need reversion too. Finally, the most common isometries are reviewed in order to write them in this way.

Article information

Source
J. Geom. Symmetry Phys., Volume 43 (2017), 1-36.

Dates
First available in Project Euclid: 12 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.jgsp/1494605778

Digital Object Identifier
doi:10.7546/jgsp-43-2017-1-36

Mathematical Reviews number (MathSciNet)
MR3644812

Zentralblatt MATH identifier
1377.15011

Subjects
Primary: 15A66: Clifford algebras, spinors
Secondary: 16G30: Representations of orders, lattices, algebras over commutative rings [See also 16Hxx]

Keywords
Clifford algebra geometric algebra isometries matrix representation

Citation

Calvet, Ramon G. On Matrix Representations of Geometric (Clifford) Algebras. J. Geom. Symmetry Phys. 43 (2017), 1--36. doi:10.7546/jgsp-43-2017-1-36. https://projecteuclid.org/euclid.jgsp/1494605778


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