Open Access
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.


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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

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