×

zbMATH — the first resource for mathematics

Quantum deformations of \(\text{GL}_ n\). (English) Zbl 0753.17015
The authors construct a family of deformations depending on \(1+{n \choose 2}\) parameters for the ring of functions on \(\text{GL}_ n\). First of all, they construct a family of Hopf algebras using the method of Yu. I. Manin, introduced in [Quantum groups and non-commutative geometry (Univ. Montreal, 1988; Zbl 0724.17006)]. Then they show that the algebras in the family are twists of the one-parameter family \({\mathcal O}(\text{GL}_ n(q))\) of deformations of the ring of functions on \(\text{GL}_ n\) by 2-cocycles.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
20G42 Quantum groups (quantized function algebras) and their representations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Artin, Adv. Math. 66 pp 171– (1987)
[2] , and , Some algebras associated to automorphisms of elliptic curves, The Grothendiech Festschrift Vol. 1, pp. 33–84, Birkhäuser, Boston-Basel-Berlin, 1990.
[3] Bergman, Adv. Math. 29 pp 178– (1978)
[4] and , Quantum GLn, preprint.
[5] , and , Two parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras, J. London Math. Soc., to appear.
[6] and , Quantum groups as deformations of Hopf algebras, preprint.
[7] Maltsiniotis, C.R. Acad. Sci. Paris 311 pp 831– (1990)
[8] Quantum Groups and Non-Commutative Geometry, publ. du CRM, 1988, Université de Montréal.
[9] Manin, Comm. Math. Phys. 123 pp 163– (1989)
[10] and , Noncommutative Noetherian Rings, Wiley & Sons Ltd., 1987.
[11] Parshall, Memoirs AMS No. 439 (1991)
[12] Reshetikhin, Lett. Math. Phys. 20 pp 331– (1990)
[13] Quantum groups: An introduction and survey for ring theorists, preprint. · Zbl 0744.16023
[14] Consistent multiparameter quantization of GL(n), preprint.
[15] Takeuchi, Proc. Japan Acad. 66 pp 112– (1990)
[16] Matrix bialgebras and quantum groups, Israel J. Math., to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.