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Note on reversion, rotation and exponentiation in dimensions five and six. (English) Zbl 1335.15034

Authors’ abstract: The explicit matrix realizations of the reversion anti-automorphism and the spin group depend on the set of matrices chosen to represent a basis of 1-vectors for a given Clifford algebra. On the other hand, there are iterative procedures to obtain bases of 1-vectors for higher-dimensional Clifford algebras, starting from those for lower-dimensional ones. For a basis of 1-vectors for \(\mathrm{Cl} (0, 5)\), obtained by applying such procedures to the Pauli basis of 1-vectors for \(\mathrm{Cl}(3,0)\), we find that the matrix form of reversion involves neither of the two standard representations of the symplectic bilinear form. However, by making use of the relation between \(4 \times 4\) real matrices and the tensor product of the quaternions with themselves, the matrix form of reversion for this basis of 1-vectors is identified. The corresponding version of the Lie algebra of the spin group has useful matrix properties which are explored. Next, the form of reversion for a basis of 1-vectors for \(\mathrm{Cl}(0,6)\) obtained iteratively from \(\mathrm{Cl}(0,0)\) is obtained. This is then applied to the task of computing exponentials of \(5 \times 5\) and \(6 \times 6\) real skew-symmetric matrices in closed form by reducing this 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 having to compute their eigenstructure. Finally, a novel representation of \(\mathrm{Sp}(4)\) is provided which may be of independent interest. Among the byproducts of this work are natural interpretations for some members of an orthogonal basis for \(M(4, R)\) provided by the isomorphism with the quaternion tensor product, and a first principles approach to the spin groups in dimensions five and six.

MSC:

15A66 Clifford algebras, spinors
22E70 Applications of Lie groups to the sciences; explicit representations
15A16 Matrix exponential and similar functions of matrices
15A69 Multilinear algebra, tensor calculus
11E88 Quadratic spaces; Clifford algebras
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15B57 Hermitian, skew-Hermitian, and related matrices
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