## A $$q$$-analogue of $$U(\mathfrak{gl}(N+1))$$, Hecke algebra, and the Yang-Baxter equation.(English)Zbl 0602.17005

The following problem about the Yang-Baxter equation (YBE) is studied: given a classical r-matrix (= solution to the classical YBE), is it possible to find the corresponding quantum R-matrix (= solution to the quantum YBE) having the former as its classical limit ? Note that now the classical r-matrices associated with a simple Lie algebra $$\mathfrak g$$ are classified. It has been realized that to construct R-matrices one has to “quantize” the Lie algebra structure itself. This was worked out by Drinfel’d for a class of rational solutions that generalize Yang’s one.
Here a $$q$$-analogue $$\hat U(\mathfrak g)$$ of the universal enveloping algebra $$U(\mathfrak g)$$ appears. In this letter, the structure and the representations of $$\hat U(\mathfrak g)$$ in the case $$\mathfrak g=\mathfrak{gl}(N+1)$$ are studied having in mind its applications to the YBE.

### MSC:

 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16T25 Yang-Baxter equations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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### References:

 [1] Kulish P. P. and Sklyanin E. K., Zapiski nauch. semin. LOMI 95, 129 (1980); English translation J. Soviet Math. 19, 1596 (1982). [2] Belavin A. A. and Drinfel’d V. G., Funct. Anal. Appl. 17, 220 (1983). · Zbl 0533.22014 [3] Kulish P. P. and Sklyanin E. K., in Integrable Quantum Field Theories, Lecture Notes in Physics 151, Springer-Verlag, New York, 1982, p. 61. [4] Sklyanin E. K., Funct. Anal. Appl. 16, 263 (1982). · Zbl 0513.58028 [5] Kulish P. P. and Reshetikhin N. Yu., Zapiski nauch. semin. LOMI, 101, 112 (1980); English translation J. Soviet Math. 23, 2435 (1983). [6] Drinfel’d V. G., Doklady Akad. Nauk SSSR 283, 1060 (1985). [7] Yang C. N., Phys. Rev. Lett. 19, 1312 (1967). · Zbl 0152.46301 [8] Jimbo M., Lett. Math. Phys. 10, 63 (1985). · Zbl 0587.17004 [9] Bourbaki N., Groupes et alg?bres de Lie, Chap. 4, Exerc. 22-24, Hermann, Paris, 1968. [10] Kulish P. P., Reshetikhin N. Yu., and Sklyanin E. K., Lett. Math. Phys. 5, 393 (1981). · Zbl 0502.35074 [11] Jimbo, M., RIMS preprint 506, Kyoto University (1985), to appear in Commun. Math. Phys. [12] Babelon O., Nucl. Phys. B230 [FS10], 241 (1984). [13] Babelon O., de Vega H. J., and Viallet C. M., Nucl. Phys. B190, 542 (1981). Cherednik, I. V., Theor. Math. Phys. 43, 356 (1980); Chudnovsky, D. V. and Chudnovsky, G. V., Phys. Lett. 79A, 36 (1980); Schultz, C. L., Phys. Rev. Lett. 46, 629 (1981). [14] Weyl H., Classical Groups, Princeton University Press, Princeton, 1953. [15] Curtis C. W., Iwahori N., and Kilmoyer R., I.H.E.S. Publ. Math. 40, 81 (1972). [16] Gyoja, A., ?A q-Analogue of Young Symmetrizer?, preprint, Osaka University (1985). · Zbl 0644.20012 [17] Baxter R. J., Exactly Solved Models, Academic Press, London, 1982. · Zbl 0538.60093 [18] Jones, V. F. R., ?Braid Groups, Hecke Algebras and Type II1 Factors?, to appear in Japan-U.S. Conf. Proc. 1983.
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