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

Jimbo, Michio

Lett. Math. Phys. 11, 247-252 (1986).

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.

Reviewer: L. N. Vaserstein (University Park)

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

Keywords:

Yang-Baxter equation; r-matrix; quantum R-matrix; universal enveloping algebra

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.