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A two-parameter quantization of GL(n). (English) Zbl 0723.17012
The quantum analogue \(GL_ q(n)\) is now a well-known Hopf algebra, as is the quantized universal enveloping algebra \(U_ q(gl(n))\). The author gives a two-parameter quantization \(GL_{\alpha,\beta}(n)\) which is \(GL_ q(n)\) for \(\alpha =\beta =q\). Similarly, he constructs \(U_{\alpha,\beta}\) such that \(U_ q(gl(n))\) is a quotient Hopf algebra of \(U_{q,q}\).

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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[1] R. Dipper and S. Donkin: Quantum GLn (preprint). · Zbl 0734.20018 · doi:10.1112/plms/s3-63.1.165
[2] L. D. Faddeev, N. Y. Reshetikhin, and L. A. Takhtajan: Quantization of Lie groups and Lie algebras. Algebraic Analysis. Academic Press, pp. 129-140 (1988). · Zbl 0677.17010
[3] M. Hashimoto and T. Hayashi: Quantum multilinear algebra (preprint). · Zbl 0776.17007 · doi:10.2748/tmj/1178227246
[4] G. Lusztig: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math., 70, 237-249 (1988). · Zbl 0651.17007 · doi:10.1016/0001-8708(88)90056-4
[5] Y. I. Manin: Quantum groups and non-commutative geometry. CRM Univ. de Montreal (1988). · Zbl 0724.17006
[6] B. Parashall and J.-P. Wang: Quantum linear groups. I, II (preprint). · Zbl 0724.17011
[7] M. Sweedler: Hopf Algebras. W. A. Benjamin, Inc., New York (1969). · Zbl 0194.32901
[8] E. Taft and J. Towber: Quantum deformation of flag schemes and Grassmann schemes. I (preprint). · Zbl 0739.17007 · doi:10.1016/0021-8693(91)90214-S
[9] M. Takeuchi: Some topics on GLq(n) (preprint). · Zbl 0760.16015 · doi:10.1016/0021-8693(92)90212-5
[10] H. Yamane: A P-B-W theorem for quantized universal enveloping algebra of type AN. Publ. RIMS Kyoto Univ., 25, 503-520 (1989). · Zbl 0694.17007 · doi:10.2977/prims/1195173355
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