Abłamowicz, Rafał; Fauser, Bertfried On the transposition anti-involution in real Clifford algebras. I: The transposition map. (English) Zbl 1393.15029 Linear Multilinear Algebra 59, No. 12, 1331-1358 (2011). 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)]. Cited in 2 ReviewsCited in 16 Documents MSC: 15A66 Clifford algebras, spinors 11E88 Quadratic spaces; Clifford algebras 68W30 Symbolic computation and algebraic computation Keywords:conjugation; contraction; correlation; dual space; exterior algebra; grade involution; graded tensor product; spinor modules; indecomposable module; involution; left regular representation; minimal left ideal; monomial order; primitive idempotent; quadratic form; reversion; simple algebra; transpose of linear mapping Citations:Zbl 1393.15030; Zbl 1388.15021 Software:CLIFFORD; GfG PDFBibTeX XMLCite \textit{R. Abłamowicz} and \textit{B. Fauser}, Linear Multilinear Algebra 59, No. 12, 1331--1358 (2011; Zbl 1393.15029) Full Text: DOI arXiv References: [1] R. Abłamowicz,Computations with Clifford and Grassmann algebras, Adv. Applied Clifford Algebras 19 (2009), pp. 499-545 · Zbl 1182.65068 · doi:10.1007/s00006-009-0182-3 [2] R. Abłamowicz,Computation of non-commutative Gröbner bases in Grassmann and Clifford algebras, Adv. Applied Clifford Algebras 20 (2010), pp. 447-476 · Zbl 1200.13043 · doi:10.1007/s00006-010-0205-0 [3] DOI: 10.1007/978-1-4612-1368-0_13 · doi:10.1007/978-1-4612-1368-0_13 [4] Abłamowicz R, CLIFFORD for Maple (2009) [5] Abłamowicz R, GfG – Groebner for Grassmann – A Maple 12 Package for Groebner Bases in Grassmann Algebras (2010) [6] R. Abłamowicz and B. Fauser,On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear · Zbl 1393.15030 [7] Chisholm JSR, Clifford Algebras and their Applications in Mathematical Physics pp 27– (1992) · doi:10.1007/978-94-015-8090-8_3 [8] DOI: 10.1088/0305-4470/32/10/010 · Zbl 1097.81533 · doi:10.1088/0305-4470/32/10/010 [9] Fauser, B.On the relation of Clifford-Lipschitz groups to q-symmetric groups, XXII International Colloquium on Group Theoretical Methods in Physics, Hobart, Tasmania, 13–19 July 1998 [10] DOI: 10.1088/0305-4470/34/1/308 · Zbl 1033.81060 · doi:10.1088/0305-4470/34/1/308 [11] DOI: 10.1002/mma.247 · Zbl 0990.15017 · doi:10.1002/mma.247 [12] Fauser B, Habilitation Thesis (2002) [13] Fauser B, Clifford Algebras – Applications to Mathematics, Physics, and Engineering pp 279– (2004) [14] DOI: 10.1007/s10773-006-9111-6 · Zbl 1119.18007 · doi:10.1007/s10773-006-9111-6 [15] DOI: 10.1007/978-1-4612-1368-0_18 · doi:10.1007/978-1-4612-1368-0_18 [16] Fauser, B and Stumpf, H.Positronium as an example of algebraic composite calculations, International Conference on the Theory of the Electron, Cuautitlan, Mexico, 27–29 September 1995 · Zbl 1221.81167 [17] Greub W, Multilinear Algebra,, 2. ed. (1978) · doi:10.1007/978-1-4613-9425-9 [18] Hahn AJ, Undergraduate Texts in Mathematics (1994) [19] Helmstetter J, Algébres de Clifford et algébres de Weyl 25 (1982) [20] DOI: 10.1016/0021-8693(87)90241-9 · Zbl 0667.15024 · doi:10.1016/0021-8693(87)90241-9 [21] Helmstetter J, Clifford Algebras and their Applications in Mathematical Physics pp 33– (1992) · doi:10.1007/978-94-015-8090-8_4 [22] Helmstetter J, Quadratic Mappings and Clifford Algebra (2008) [23] Kadison L, University Lecture Series 14 (1999) [24] Lam TY, The Algebraic Theory of Quadratic Forms (1973) [25] Lang S, Algebra (1993) [26] DOI: 10.1007/PL00001276 · Zbl 0958.05128 · doi:10.1007/PL00001276 [27] Lipschutz S, Schaum’s Outline Series (1968) [28] DOI: 10.1017/CBO9780511526022 · doi:10.1017/CBO9780511526022 [29] DOI: 10.1017/CBO9780511470912 · doi:10.1017/CBO9780511470912 [30] DOI: 10.1073/pnas.91.26.13057 · Zbl 0831.16025 · doi:10.1073/pnas.91.26.13057 [31] DOI: 10.1063/1.524893 · Zbl 0459.15017 · doi:10.1063/1.524893 [32] Salingaros N, J. Math. Phys. 23 (1982) [33] DOI: 10.1063/1.526260 · Zbl 0552.20008 · doi:10.1063/1.526260 [34] Schmeikal B, Clifford Algebras with Symbolic and Numeric Computations pp 83– (1996) · doi:10.1007/978-1-4615-8157-4_5 [35] DOI: 10.4067/S0719-06462010000200018 · Zbl 1217.15035 · doi:10.4067/S0719-06462010000200018 [36] DOI: 10.1007/s00006-004-0006-4 · Zbl 1100.20502 · doi:10.1007/s00006-004-0006-4 [37] Vicary J, Categorical Formulation of Quantum Algebras (2008) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.