Translator Disclaimer
2014 Note on Reversion, Rotation and Exponentiation in Dimensions Five and Six
Emily Herzig, Viswanath Ramakrishna, Mieczyslaw K. Dabkowski
J. Geom. Symmetry Phys. 35: 61-101 (2014). DOI: 10.7546/jgsp-35-2014-61-101


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.


Download Citation

Emily Herzig. Viswanath Ramakrishna. Mieczyslaw K. Dabkowski. "Note on Reversion, Rotation and Exponentiation in Dimensions Five and Six." J. Geom. Symmetry Phys. 35 61 - 101, 2014.


Published: 2014
First available in Project Euclid: 27 May 2017

zbMATH: 1335.15034
MathSciNet: MR3363623
Digital Object Identifier: 10.7546/jgsp-35-2014-61-101

Rights: Copyright © 2014 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences


Back to Top