Abstract
The structure of Hopf co-Poisson algebra on the universal enveloping algebra U(S7(2)) of Lie algebra S7(2) is determined with the help of a solution of the Yang-Baxter equation. Using this solution, a bracket on the dual space of Lie algebra ST(2) is also determined. This cobracket on ST(2) induces a deformation of the universal enveloping algebra U(S7(2)) which has a Hopf algebra structure, as we shall verify. This Hopf algebra is called the quantum group associated to a universal enveloping algebra.
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References
G.E. Arutyunov & P.B. Medvedev, Quantization of the extemal algebro on a Poisson-Li.e group, hep-th/9311096 (1993), 1-20.
V. Chari & A. Pressley, A guide to Quantum Groups, Cambridge University Press, Cambridge University, 1994.
J. Dixmier, Envel-Oping Algebms, Graduate Studies in Mathematics 11, American Mathematical Society, 1996.
V, Drinfeld, Quantum groups, ICM-86, 1986, 798-820.
V. Drinfeld, On sorne unsolved problems in qu.antum group theory, Lecture Notes in Mathematics 1510, SpringerVerlag, 1992, 1-8.
H. D. Doebner, J. D. Hennig & W. Lücke, Quantum Grou.ps, Lecture Notes in Physics 370, Springer-Verlag, 1990.
P. Etingof & David Kazhdan, Quantization of Poisson algebraic groups and Poi9son lwmogeneous spaces, q-alg 9510020 (1995), 1-9.
Berenice Guerrero, Sobre una estructura diferencial cuántica. Reporte interno No. 56, Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia, 1997.
Berenice Guerrero, Cuantización no estándar del grupo triangular ST(S), Lecturas Matemáticas 18 (1997), 23-44. [10] D. Gurevich & V. Ruhtsov, Yang-Baxter equation and deformation of associative and Lie algebras. Lectures Notes in Mathematics 1510,'Springer-Verlag, 1992, 9-46.
C. Kassel, Quantum Groups, Springer-Verlag, Berlin, 1995.
J. H. Lu & A. Weinstein, Poisson Lie groups, dressing trasnformations and Bruhat decompositions, J. Differential Geometry 31 (1990), 501-526.
S. Majid Foundations of Quantum Group Theory, Cambridge University Press, 1995.
L. A. Takthajan, Quantum groups and integrable models, Advanced Studies in Pure Mathematics 19 (1990), 435- 457.
L. A. Takthajan, Lectures on Quantum Groups, Nakai Institute Series in Mathematical Physics (1990), 193-225. (16] N. Yu. Reshetikhin, L. Takthajan & L. D. Fad
N. Yu. Reshetikhin, L. Takthajan & L. D. Fad deev, Quantization of Lie groups and Lie algebras1 Algebra and Analyis (1989), 178-206.
M. A. Semenov-Tian-Shnsky, Lectures on Rmatrices, PoUJson-Lie Groups and Integrable Systems, Pro ceedings of the CIMPA School 1991 Nice (France), World Scientific, 1994, 269-318.
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