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On the transposition anti-involution in real Clifford algebras. I: The transposition map. (English) Zbl 1393.15029

Summary: A particular orthogonal map on a finite-dimensional real quadratic vector space \((V, Q)\) with a non-degenerate quadratic form \(Q\) of any signature \((p, q)\) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra \(C\ell (V^*, Q)\) of linear functionals (multiforms) acting on the universal Clifford algebra \(C\ell (V, Q)\). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of \(C\ell (V, Q)\). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of \(C\ell (V, Q)\). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [ibid. 59, No. 12, 1359-1381 (2011; Zbl 1393.15030)].

MSC:

15A66 Clifford algebras, spinors
11E88 Quadratic spaces; Clifford algebras
68W30 Symbolic computation and algebraic computation

Software:

CLIFFORD; GfG
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Full Text: DOI arXiv

References:

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