Journal of Geometry and Symmetry in Physics

Note on Reversion, Rotation and Exponentiation in Dimensions Five and Six

Abstract

The explicit matrix realizations of reversion and spin groups depend on the set of matrices chosen to represent a basis of one-vectors for a Clifford algebra. On the other hand, there are iterative procedures to obtain bases of one-vectors for higher dimensional Clifford algebras, starting from those for lower dimensional ones. For a basis of one-vectors for $\mathrm{Cl}(0,5)$ , obtained by applying such procedures to the Pauli basis for $\mathrm{Cl} (3,0)$ the matrix form of reversion involves neither of the two standard matrices representing the symplectic form. However, by making use of the relation between $4\times 4$ real matrices and the quaternion tensor product ($\mathbb{H}\otimes \mathbb{H}$), the matrix form of reversion for this basis of one-vectors is identified. The corresponding version of the Lie algebra of the spin group, $\mathfrak{spin}(5)$, has useful matrix properties which are explored. Next, the form of reversion for a basis of one-vectors for $\mathrm{Cl}(0,6)$ obtained iteratively from $\mathrm{Cl} (0,0)$ is obtained. This is then applied to computing exponentials of $5\times 5$ and $6\times 6$ real antisymmetric matrices in closed form, by reduction to the simpler task of computing exponentials of certain $4\times 4$ matrices. For the latter purpose closed form expressions for the minimal polynomials of these $4\times 4$ matrices are obtained, without availing of their eigenstructure. Among the byproducts of this work are natural interpretations for members of an orthogonal basis for $M(4,\mathbb{R})$ provided by the isomorphism with $\mathbb{H}\otimes \mathbb{H}$, and a first principles approach to the spin groups in dimensions five and six.

Article information

Source
J. Geom. Symmetry Phys., Volume 35 (2014), 61-101.

Dates
First available in Project Euclid: 27 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.jgsp/1495850574

Digital Object Identifier
doi:10.7546/jgsp-35-2014-61-101

Mathematical Reviews number (MathSciNet)
MR3363623

Zentralblatt MATH identifier
1335.15034

Citation

Herzig, Emily; Ramakrishna, Viswanath; Dabkowski, Mieczyslaw K. Note on Reversion, Rotation and Exponentiation in Dimensions Five and Six. J. Geom. Symmetry Phys. 35 (2014), 61--101. doi:10.7546/jgsp-35-2014-61-101. https://projecteuclid.org/euclid.jgsp/1495850574