## 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