# 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

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topŁawrynowicz, 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 -

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