Spinors in braided geometry

Mićo Đurđević; Zbigniew Oziewicz

Banach Center Publications (1996)

  • Volume: 37, Issue: 1, page 315-325
  • ISSN: 0137-6934

Abstract

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Let V be a ℂ-space, σ ∈ E n d ( V ⊗ 2 ) be a pre-braid operator and let F ∈ l i n ( V ⊗ 2 , ℂ ) . This paper offers a sufficient condition on (σ,F) that there exists a Clifford algebra Cl(V,σ,F) as the Chevalley F-dependent deformation of an exterior algebra C l ( V , σ , 0 ) ≡ V ∧ ( σ ) . If σ ≠ σ - 1 and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal category. A spinor representation as a left Cl(V,σ,F)-module is identified with an exterior algebra over F-isotropic ℂ-subspace of V. We give a sufficient condition on braid σ that the spinor representation is faithful and irreducible.

How to cite

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Đurđević, Mićo, and Oziewicz, Zbigniew. "Spinors in braided geometry." Banach Center Publications 37.1 (1996): 315-325. <http://eudml.org/doc/208608>.

@article{Đurđević1996,
abstract = {Let V be a ℂ-space, $σ ∈ End(V^\{⊗2\})$ be a pre-braid operator and let $F ∈ lin(V^\{⊗2\},ℂ).$ This paper offers a sufficient condition on (σ,F) that there exists a Clifford algebra Cl(V,σ,F) as the Chevalley F-dependent deformation of an exterior algebra $Cl(V,σ,0) ≡ V^\{∧\}(σ)$. If $σ ≠ σ^\{-1\}$ and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal category. A spinor representation as a left Cl(V,σ,F)-module is identified with an exterior algebra over F-isotropic ℂ-subspace of V. We give a sufficient condition on braid σ that the spinor representation is faithful and irreducible.},
author = {Đurđević, Mićo, Oziewicz, Zbigniew},
journal = {Banach Center Publications},
keywords = {Clifford algebras; spinor representations; braided geometry; braided exterior algebra; Hopf algebra; braided derivation; braided Chevalley deformation; quantum Yang Baxter equation; Grassmann algebra},
language = {eng},
number = {1},
pages = {315-325},
title = {Spinors in braided geometry},
url = {http://eudml.org/doc/208608},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Đurđević, Mićo
AU - Oziewicz, Zbigniew
TI - Spinors in braided geometry
JO - Banach Center Publications
PY - 1996
VL - 37
IS - 1
SP - 315
EP - 325
AB - Let V be a ℂ-space, $σ ∈ End(V^{⊗2})$ be a pre-braid operator and let $F ∈ lin(V^{⊗2},ℂ).$ This paper offers a sufficient condition on (σ,F) that there exists a Clifford algebra Cl(V,σ,F) as the Chevalley F-dependent deformation of an exterior algebra $Cl(V,σ,0) ≡ V^{∧}(σ)$. If $σ ≠ σ^{-1}$ and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal category. A spinor representation as a left Cl(V,σ,F)-module is identified with an exterior algebra over F-isotropic ℂ-subspace of V. We give a sufficient condition on braid σ that the spinor representation is faithful and irreducible.
LA - eng
KW - Clifford algebras; spinor representations; braided geometry; braided exterior algebra; Hopf algebra; braided derivation; braided Chevalley deformation; quantum Yang Baxter equation; Grassmann algebra
UR - http://eudml.org/doc/208608
ER -

References

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  12. [12] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press 1995. Zbl0857.17009
  13. [13] Z. Oziewicz, E. Paal and J. Różański, Derivations in braided geometry, Acta Physica Polonica B 26 (7) (1995), 1253-1273. Zbl0966.81517
  14. [14] Z. Oziewicz, Clifford algebra for Hecke braid, in: Clifford Algebras and Spinor Structures, R. Ablamowicz and P. Lounesto (ed.), Mathematics and Its Applications vol. 321, Kluwer Academic Publishers, Dordrecht 1995, pp. 397-411. Zbl0845.15011
  15. [15] S. Shnider and S. Sternberg, Quantum Groups, from coalgebras to Drinfeld algebras a guided tour, International Press Incorporated, Boston, 1993. Zbl0845.17015
  16. [16] M. E. Sweedler, Hopf Algebras, Benjamin, Inc., New York, 1969. 
  17. [17] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122 (1989), 125-170. Zbl0751.58042

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