Generalized Hurwitz maps of the type S × V → W, anti-involutions, and quantum braided Clifford algebras

Julian Ławrynowicz; Jakub Rembieliński; Francesco Succi

Banach Center Publications (1996)

  • Volume: 37, Issue: 1, page 223-240
  • ISSN: 0137-6934

Abstract

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The notion of a J 3 -triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then the interrelation with quantum groups and related Clifford-type structures is indicated via anti-involutions which also play a central role in the theory of symmetric complex manifolds. Finally, the theory is linked with a natural generalization of general linear inhomogeneous groups as quantum braided groups. This generalization is in the spirit of the theory initiated and developed by S. Majid, however, our construction differs in the interrelation between the homogeneous and inhomogeneous parts of the group. In order to study the quantum braided orthogonal groups, we consider a kind of quantum geometry in the covector space. This enables us to investigate a quantum braided Clifford algebra structure related to the spinor representation of that group.

How to cite

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Ławrynowicz, Julian, Rembieliński, Jakub, and Succi, Francesco. "Generalized Hurwitz maps of the type S × V → W, anti-involutions, and quantum braided Clifford algebras." Banach Center Publications 37.1 (1996): 223-240. <http://eudml.org/doc/208601>.

@article{Ławrynowicz1996,
abstract = {The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then the interrelation with quantum groups and related Clifford-type structures is indicated via anti-involutions which also play a central role in the theory of symmetric complex manifolds. Finally, the theory is linked with a natural generalization of general linear inhomogeneous groups as quantum braided groups. This generalization is in the spirit of the theory initiated and developed by S. Majid, however, our construction differs in the interrelation between the homogeneous and inhomogeneous parts of the group. In order to study the quantum braided orthogonal groups, we consider a kind of quantum geometry in the covector space. This enables us to investigate a quantum braided Clifford algebra structure related to the spinor representation of that group.},
author = {Ławrynowicz, Julian, Rembieliński, Jakub, Succi, Francesco},
journal = {Banach Center Publications},
keywords = {-triple; geometrical approach; generalized Hurwitz problem; quadratic or bilinear forms; symmetry; antisymmetry; choices of the metric tensors of scalar products of the basis elements; quantum groups; related Clifford-type structures; symmetric complex manifolds; quantum braided groups; quantum geometry in the covector space; quantum braided Clifford algebra structure; spinor representation},
language = {eng},
number = {1},
pages = {223-240},
title = {Generalized Hurwitz maps of the type S × V → W, anti-involutions, and quantum braided Clifford algebras},
url = {http://eudml.org/doc/208601},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Ławrynowicz, Julian
AU - Rembieliński, Jakub
AU - Succi, Francesco
TI - Generalized Hurwitz maps of the type S × V → W, anti-involutions, and quantum braided Clifford algebras
JO - Banach Center Publications
PY - 1996
VL - 37
IS - 1
SP - 223
EP - 240
AB - The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then the interrelation with quantum groups and related Clifford-type structures is indicated via anti-involutions which also play a central role in the theory of symmetric complex manifolds. Finally, the theory is linked with a natural generalization of general linear inhomogeneous groups as quantum braided groups. This generalization is in the spirit of the theory initiated and developed by S. Majid, however, our construction differs in the interrelation between the homogeneous and inhomogeneous parts of the group. In order to study the quantum braided orthogonal groups, we consider a kind of quantum geometry in the covector space. This enables us to investigate a quantum braided Clifford algebra structure related to the spinor representation of that group.
LA - eng
KW - -triple; geometrical approach; generalized Hurwitz problem; quadratic or bilinear forms; symmetry; antisymmetry; choices of the metric tensors of scalar products of the basis elements; quantum groups; related Clifford-type structures; symmetric complex manifolds; quantum braided groups; quantum geometry in the covector space; quantum braided Clifford algebra structure; spinor representation
UR - http://eudml.org/doc/208601
ER -

References

top
  1. [1] J. Adem, Construction of some normed maps, Bol. Soc. Mat. Mexicana (2) 20 (1975), 59-75. Zbl0498.17014
  2. [2] J. Adem, On maximal sets of anticommuting matrices, ibid. (2) 23 (1978), 61-67. Zbl0447.15008
  3. [3] J. Adem, On the Hurwitz problem over an arbitrary field I-II, ibid. (2) 26 (1981), 29-41. Zbl0514.10016
  4. [4] J. Adem, On Yuzwinsky's theorem concerning admissible triples over an arbitrary field, ibid. (2) 29 (1984), 65-69. 
  5. [5] J. Adem, Classification of low-dimensional orthogonal pairings, ibid. (2) 31 (1986), 1-27. Zbl0688.10018
  6. [6] J. Adem, J. Ławrynowicz and J. Rembieliński, Generalized Hurwitz maps of the type S × V → W, (a) Dep. de Mat. Centro de Investiación y de Estudios Avanzados México Preprint no. 80 (1990), ii + 20 pp., (b) Rep. Math. Phys. 37 (1996), 325-336. Zbl0885.15019
  7. [7] N. L. Alling and N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math. 219, Springer, Berlin-Heidelberg-New York 1971. 
  8. [8] I. Bengtsson and I. Cederwall, Particles, twistors and the division algebras, Nucl. Phys. B 302 (1988), 81-103. 
  9. [9] J. Bingener and T. Lehmkuhl, Infinitedimensional supergeometry, in: Complex Analysis and Geometry, V. Ancona, E. Ballico and A. Silva (eds.), Lecture Notes in Pure and Applied Math. 178, Dekker, New York 1995, 27-93. Zbl0851.58005
  10. [10] N. Bourbaki, Élements de mathématique. II. Algèbre. Ch. 6-7., 2 éd., Actualités scientifiques et industrielles 1179, Herman, Paris 1964. 
  11. [11] T. Brzeziński, H. Dąbrowski, and J. Rembieliński, On the quantum differential calculus and the quantum holomorphicity, J. Math. Phys. 33 (1992), 19-24. Zbl0753.17017
  12. [12] J. Cnops, Hurwitz pairs and Clifford valued inner products, this volume, 195-208. Zbl0873.15019
  13. [13] A. Crumeyrolle, Clifford Algebras and Spinor Structures, Mathematics and Its Applications 57, Kluwer Academic, Dordrecht 1989. Zbl0701.53003
  14. [14] I. Furuoya, S. Kanemaki, J. Ławrynowicz and O. Suzuki, Hermitian Hurwitz pairs, in: Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics, J. Ławrynowicz (ed.), Kluwer Academic, Dordrecht 1994, 135-154. Zbl0841.11018
  15. [15] A. Hurwitz, Über die Komposition der quadratischen Formen von beliebig vielen Variablen, Nachrichten von der königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-phys. Kl. 1898, 309-316; reprinted in: A. Hurwitz, Mathematische Werke II, Birkhäuser, Basel 1933, 565-571. 
  16. [16] A. Hurwitz, Über die Komposition der quadratischen Formen, Math. Ann. 88 (1923), 1-25; reprinted in: A. Hurwitz, Mathematische Werke II, Birkhäuser, Basel 1933, 641-666. 
  17. [17] J. Kalina, J. Ławrynowicz, and O. Suzuki, A field equation defined by a Hurwitz pair, Suppl. Rend. Circ. Mat. Palermo (2) 9 (1985), 117-128. Zbl0613.35066
  18. [18] J. Kalina, J. Ławrynowicz, and O. Suzuki, Partial differential equations connected with some Clifford structures and the related quasiconformal mappings, Rend. Sem. Fac. Sci. Univ. Cagliari 57 (1987), 131-142. Zbl0668.30038
  19. [19] S. Kanemaki, Hurwitz pairs and octonions, in: Deformations of Mathematical Structures. Complex Analysis with Physical Applications, J. Ławrynowicz (ed.), Kluwer Academic, Dordrecht 1989, 215-223. 
  20. [20] S. Kanemaki and O. Suzuki, Hermitian pre-Hurwitz pairs and the Minkowski space, ibid., 225-232. Zbl0669.15012
  21. [21] T. Kimura and I. Oda, Superparticles and division algebras. Six dimensions and quaternions, Progr. Theoret. Phys. 80, no. 1: Progress Letters (1988), 1-6. 
  22. [22] T. Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, Reading, MA, 1983. 
  23. [23] T. Y. Lam and T. L. Smith, On Yuzwinsky's monomial pairings, Quart. J. Math. Oxford (2) 44 (1993). Zbl0802.11013
  24. [24] J. Ławrynowicz, The normed maps ℝ 1 1 × ℝ 1 1 → ℝ 2 6 in hypercomplex analysis and in physics, in: Clifford Algebras and Their Applications in Mathematical Physics, A. Micali, P. Boudet, and J. Helmstetter (eds.), Kluwer Academic, Dordrecht 1992, 447-461. 
  25. [25] J. Ławrynowicz, Clifford analysis and the five-dimensional analogues of the quanternionic structure of the Kałuża-Klein and Penrose types, Ber. Univ. Jyväskylä Math. Inst. 55 (1993), 97-112. Zbl0794.15014
  26. [26] J. Ławrynowicz, Quantized complex and Clifford structures, ibid. 55 (1993), 113-120. 
  27. [27] J. Ławrynowicz, K. Kędzia and O. Suzuki, Supercomplex structures, surface soliton equations, and quasiconformal mappings, Ann. Polon. Math. 55 (1991), 245-268. Zbl0761.35091
  28. [28] J. Ławrynowicz, S. Koshi and O. Suzuki, Dualities generated by the generalized Hurwitz problem and variation of the Yang-Mills field: in: Analysis, Geometry and Groups: A Riemann Legacy Volume, H. M. Srivastava and Th. M. Rassias (eds.), Hadronic Press, Palm Harbor, FL, 1993, 275-288. Zbl0920.15010
  29. [29] J. Ławrynowicz in cooperation with J. Krzyż, Quasiconformal Mappings in the Plane. Parametrical Methods, Lecture Notes in Math. 978, Springer, Berlin 1983. 
  30. [30] J. Ławrynowicz, L. C. Papaloucas and J. Rembieliński, Quantum braided Clifford algebras, in: Clifford Algebras and Spinor Structures. A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919-1992), R. Abłamowicz and P. Lounesto (eds.), Mathematics and Its Applications 321, Kluwer Academic, Dordrecht 1995, 387-395. Zbl0843.15014
  31. [31] J. Ławrynowicz, R. M. Porter, E. Ramírez de Arellano, and J. Rembieliński, On dualities generated by the generalized Hurwitz problem, in: Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics, J. Ławrynowicz (ed.), Kluwer Academic, Dordrecht 1994, 189-208. Zbl0827.15039
  32. [32] J. Ławrynowicz and E. Ramírez de Arellano, Anti-involutions, symmetric complex manifolds, and quantum spaces, in: Complex Analysis and Geometry, V. Ancona, E. Ballico and A. Silva (eds.), Lecture Notes in Pure and Applied Math. 178, Dekker, New York 1995, 309-317. Zbl0846.58005
  33. [33] J. Ławrynowicz, E. Ramírez de Arellano and J. Rembieliński, The correspondence between type-reversing transformations of pseudo-euclidean Hurwitz pair and Clifford algebras I-II, Bull. Soc. Sci. Lettres Łódź 40 Sér. Rech. Déform. 8 (1990), 61-97 and 99-129. Zbl0876.15023
  34. [34] J. Ławrynowicz and J. Rembieliński, Hurwitz pairs equipped with complex structures, in: Seminar on Deformations, Proceedings, Łódź-Warsaw 1982/84, J. Ławrynowicz (ed.), Lecture Notes in Math. 1165, Springer, Berlin 1985, 185-195. 
  35. [35] J. Ławrynowicz and J. Rembieliński, Supercomplex vector spaces and spontaneous symmetry breaking, in: Seminari di Geometria 1984, S. Coen (ed.), Università di Bologna, Bologna 1985, 131-154. Zbl0605.30050
  36. [36] J. Ławrynowicz and J. Rembieliński, Pseudo-euclidean Hurwitz pair and generalized Fueter equations, in: Clifford Algebras and Their Applications in Mathematical Physics, Proceedings, Canterbury 1985, J. S. R. Chisholm and A. K. Common (eds.), NATO-ASI Series C: Mathematical and Physical Sciences 183, Reidel, Dordrecht 1986, 39-48. 
  37. [37] J. Ławrynowicz and J. Rembieliński, Complete classification for pseudoeuclidean Hurwitz pairs including the symmetry operations, Bull. Soc. Sci. Lettres Łódź 36, no 29 Sér. Rech. Déform. 4, no. 39 (1986), 15. Zbl0627.15012
  38. [38] J. Ławrynowicz and J. Rembieliński, Pseudo-euclidean Hurwitz pairs and the Kałuża-Klein theories, J. Phys. A: Math. Gen. 20 (1987), 5831-5848. Zbl0654.15021
  39. [39] J. Ławrynowicz and J. Rembieliński, On the composition of nondegenerate quadratic forms with an arbitrary index, Ann. Fac. Sci. Toulouse Math (5) 10 (1989), 141-168 [due to a printing error in vol. 10 the whole article was reprinted in vol. 11 (1990), no. 1, of the same journal, 141-168]. Zbl0701.15025
  40. [40] J. Ławrynowicz and L. Wojtczak in cooperation with S. Koshi and O. Suzuki, Stochastical mechanics of particle systems in Clifford-analytical formulation related to Hurwitz pairs of dimension (8,5), in: Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics, J. Ławrynowicz (ed.), Kluwer Academic, Dordrecht 1994, 213-262. Zbl0828.60081
  41. [41] S. Majid, Transmutation theory and rank for quantum braided groups, Math. Proc. Cambridge Philos. Soc. 113 (1993), 45-70. Zbl0781.17006
  42. [42] Yu. I. Manin, Quantum Groups and Non-Commutative Geometry, Centre de Recherches Mathématiques, Montréal 1988. Zbl0724.17006
  43. [43] S. Marchiafava and J. Rembieliński, Quantum quaternions, J. Math. Phys. 33 (1992), 171-173. 
  44. [44] L. S. Randriamihamison, Paires de Hurwitz pseudo-euclidiennes en signature quelconque, J. Phys. A: Math. Gen. 23 (1990), 2729-2749. Zbl0716.15018
  45. [45] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der math. Wissenschaften 270, Springer, Berlin 1985. 
  46. [46] D. B. Shapiro, Products of sums of squares, Expo. Math. 2 (1984), 235-261. Zbl0541.10025
  47. [47] D. B. Shapiro, Compositions of Quadratic Forms, forthcoming. 
  48. [48] F. Succi, Symmetric complex manifolds, dianalytic structures, and Klein manifolds, in: Ergebnisse der Tagung Komplexe Analysis und komplexe Differentialgeometrie, A. Duma (ed.), Hagen 1986, 9-16. 
  49. [49] T. L. Smith and P. Yiu, Constructions of sums of squares formulae with integer coefficients, Bol. Soc. Mat. Mexicana (2) 37 (1992), 479-495 (1994). Zbl0830.11018
  50. [50] O. Suzuki, J. Ławrynowicz and J. Kalina, A geometric approach to the Kadomtsev-Petviashvili system II, Proc. Inst. Nat. Sci. College Hum. Sci. Nihon Univ. 26 (1991), 51-68. 
  51. [51] J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane, in: Recent Advances in Field Theory, Proceedings, Annecy-le-Vieux 1990, Nuclear Phys. B, Proc. Suppl. 18B (1990), 302-312. Zbl0957.46514
  52. [52] P. Yiu, Composition of sums of squares with integer coefficients, in: Deformations of Mathematical Structures II. Hurwitz-Type Structures and Applications to Surface Physics, J. Ławrynowicz (ed.), Kluwer Academic, Dordrecht 1994, 7-100. 

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