Abstract
Woronowicz in 1989 introduced a graded exterior algebra for arbitrary braid operator. In the present paper we give necessary and sufficient conditions between scalar product and braid operator that exists a Clifford algebra and that exists the Chevalley deformation of Woronowicz’s exterior algebra. In particular a quantum Clifford and Weyl algebras for a Hecke braid are isomorphic to Chevalley’s deformations of an algebras of quantum fermions and of quantum bosons respectively. We discuss deformation versus quantization, braided monoidal category, inner product for arbitrary braid and the Crumeyrolle algebra isomorphisms.
This paper is in final form and no version of it will be submitted for publication elsewhere.
Research partially supported by State Committee of Scientific Research, Poland, KBN grant # 2 P302 023 07.
The paper was written in Centro de Investigaciones Teoricas, Facultad de Estudios Superiores Cuautitlán, UNAM, Apartado Postal # 25, CP 54700 Cuautitlán Izcalli, Estado de México, (oziewicz@servidor.unam.mx).
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References
Ablamowicz Rafal and Pertti Lounesto (1993), Private letter to Garret Sobczyk and Zbigniew Oziewicz, June 3, 5 pages
Bourbaki Nicolás (1959), Algèbre, chap. 9: formes sesquilinéaires et formes quadratiques, Paris, Hermann
Chevalley Claude (1954), The Algebraic Theory of Spinors, Columbia University Press, New York
Crumeyrolle Albert (1990), Orthogonal and Symplectic Clifford Algebras, Spinor Structures, Kluwer Academic Publishers, Dordrecht
Grassmann Hermann (1877), Der ort der Hamilton’schen quaternionen in der Ausdehnungslehre, Math. Ann., 12 375
Joyal A. and R. Street (1985), Braided Monoidal Categories, Macquarie Mathematics Reports 850067 (1985) and 860081 (1986)
Kähler Erich (1960), Innerer und äusserer Differentialkalkül, Abh. Dt. Akad. Wiss. Berlin, Kl. für Mathematik, Physik und Technologie, Jahrg. 1960, Nr. 4
Kähler Erich (1962), Der innere Differentialkalkül, Rendiconti di Matematica e delle sue Applicazioni, Roma, 21 (3–4) 425–523
Lounesto Pertti (1993), What is a bivector?, in ‘Spinors, Twistors, Clifford Algebras and Quantum Deformations’, in series ‘Fundamental Theories of Physics’, edited by Z. Oziewicz, B. Jancewicz and A. Borowiec, Kluwer Academic Publishers, Dordrecht/Boston/London
Mac Lane Saunders (1971), Categories for Working Mathematician, Graduate Texts in Mathematics # 5, Springer-Verlag
Woronowicz Stanislaw Lech (1989), Differential calculus on compact matrix pseud ogroups (quantum groups), Commun. Math. Phys. 122 125–170
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Oziewicz, Z. (1995). Clifford Algebra for Hecke Braid. In: Ablamowicz, R., Lounesto, P. (eds) Clifford Algebras and Spinor Structures. Mathematics and Its Applications, vol 321. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8422-7_26
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DOI: https://doi.org/10.1007/978-94-015-8422-7_26
Publisher Name: Springer, Dordrecht
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