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

Ławrynowicz, Julian ; Rembieliński, Jakub ; Succi, Francesco

Banach Center Publications, Tome 37 (1996), p. 223-240 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1996-01-01

Zbl 0874.46012

EUDML-ID : urn:eudml:doc:208601

Bibliographie

[000] [1] J. Adem, Construction of some normed maps, Bol. Soc. Mat. Mexicana (2) 20 (1975), 59-75. | Zbl 0498.17014

[001] [2] J. Adem, On maximal sets of anticommuting matrices, ibid. (2) 23 (1978), 61-67. | Zbl 0447.15008

[002] [3] J. Adem, On the Hurwitz problem over an arbitrary field I-II, ibid. (2) 26 (1981), 29-41. | Zbl 0514.10016

[003] [4] J. Adem, On Yuzwinsky's theorem concerning admissible triples over an arbitrary field, ibid. (2) 29 (1984), 65-69.

[004] [5] J. Adem, Classification of low-dimensional orthogonal pairings, ibid. (2) 31 (1986), 1-27. | Zbl 0688.10018

[005] [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. | Zbl 0885.15019

[006] [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.

[007] [8] I. Bengtsson and I. Cederwall, Particles, twistors and the division algebras, Nucl. Phys. B 302 (1988), 81-103.

[008] [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. | Zbl 0851.58005

[009] [10] N. Bourbaki, Élements de mathématique. II. Algèbre. Ch. 6-7., 2 éd., Actualités scientifiques et industrielles 1179, Herman, Paris 1964.

[010] [11] T. Brzeziński, H. Dąbrowski, and J. Rembieliński, On the quantum differential calculus and the quantum holomorphicity, J. Math. Phys. 33 (1992), 19-24. | Zbl 0753.17017

[011] [12] J. Cnops, Hurwitz pairs and Clifford valued inner products, this volume, 195-208. | Zbl 0873.15019

[012] [13] A. Crumeyrolle, Clifford Algebras and Spinor Structures, Mathematics and Its Applications 57, Kluwer Academic, Dordrecht 1989. | Zbl 0701.53003

[013] [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. | Zbl 0841.11018

[014] [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.

[015] [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.

[016] [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. | Zbl 0613.35066

[017] [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. | Zbl 0668.30038

[018] [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.

[019] [20] S. Kanemaki and O. Suzuki, Hermitian pre-Hurwitz pairs and the Minkowski space, ibid., 225-232. | Zbl 0669.15012

[020] [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.

[021] [22] T. Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, Reading, MA, 1983.

[022] [23] T. Y. Lam and T. L. Smith, On Yuzwinsky's monomial pairings, Quart. J. Math. Oxford (2) 44 (1993). | Zbl 0802.11013

[023] [24] J. Ławrynowicz, The normed maps $ℝ^11 × ℝ^11 → ℝ^26$ 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.

[024] [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. | Zbl 0794.15014

[025] [26] J. Ławrynowicz, Quantized complex and Clifford structures, ibid. 55 (1993), 113-120.

[026] [27] J. Ławrynowicz, K. Kędzia and O. Suzuki, Supercomplex structures, surface soliton equations, and quasiconformal mappings, Ann. Polon. Math. 55 (1991), 245-268. | Zbl 0761.35091

[027] [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. | Zbl 0920.15010

[028] [29] J. Ławrynowicz in cooperation with J. Krzyż, Quasiconformal Mappings in the Plane. Parametrical Methods, Lecture Notes in Math. 978, Springer, Berlin 1983.

[029] [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. | Zbl 0843.15014

[030] [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. | Zbl 0827.15039

[031] [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. | Zbl 0846.58005

[032] [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. | Zbl 0876.15023

[033] [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.

[034] [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. | Zbl 0605.30050

[035] [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.

[036] [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. | Zbl 0627.15012

[037] [38] J. Ławrynowicz and J. Rembieliński, Pseudo-euclidean Hurwitz pairs and the Kałuża-Klein theories, J. Phys. A: Math. Gen. 20 (1987), 5831-5848. | Zbl 0654.15021

[038] [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]. | Zbl 0701.15025

[039] [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. | Zbl 0828.60081

[040] [41] S. Majid, Transmutation theory and rank for quantum braided groups, Math. Proc. Cambridge Philos. Soc. 113 (1993), 45-70. | Zbl 0781.17006

[041] [42] Yu. I. Manin, Quantum Groups and Non-Commutative Geometry, Centre de Recherches Mathématiques, Montréal 1988. | Zbl 0724.17006

[042] [43] S. Marchiafava and J. Rembieliński, Quantum quaternions, J. Math. Phys. 33 (1992), 171-173.

[043] [44] L. S. Randriamihamison, Paires de Hurwitz pseudo-euclidiennes en signature quelconque, J. Phys. A: Math. Gen. 23 (1990), 2729-2749. | Zbl 0716.15018

[044] [45] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der math. Wissenschaften 270, Springer, Berlin 1985.

[045] [46] D. B. Shapiro, Products of sums of squares, Expo. Math. 2 (1984), 235-261. | Zbl 0541.10025

[046] [47] D. B. Shapiro, Compositions of Quadratic Forms, forthcoming.

[047] [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.

[048] [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). | Zbl 0830.11018

[049] [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.

[050] [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. | Zbl 0957.46514

[051] [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.