Degenerate affine Hecke algebras and Yangians. (English. Russian original) Zbl 0599.20049

Funct. Anal. Appl. 20, 58-60 (1986); translation from Funkts. Anal. Prilozh. 20, No. 1, 69-70 (1986).
The Yangians is a new class of Hopf algebras introduced by the author in connection with the Yang-Baxter equation. Such an algebra \(Y({\mathfrak a})\) is defined for any finite-dimensional simple Lie algebra \(\mathfrak a\) over the complex numbers. Each finite-dimensional irreducible representation of \(Y({\mathfrak a})\) gives a quantum \(R\)-matrix, so those representations are of interest for mathematical physics. In the present paper, the case \({\mathfrak a}=\mathfrak{sl}(N)\) is studied. It is known that the representations of \({\mathfrak a}\) and the symmetric groups \(S_m\) are closely related. A similar relation between representations of \(Y({\mathfrak a})\) and certain algebras \(\Lambda_m\) is obtained.


20C08 Hecke algebras and their representations
20G05 Representation theory for linear algebraic groups
16T25 Yang-Baxter equations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
Full Text: DOI


[1] V. G. Drinfel’d, ”Hopf algebras and the quantum Yang?Baxter equation,” Dokl. Akad. Nauk SSSR,283, No. 5, 1060-1064 (1985).
[2] I. N. Bernshtein and A. V. Zelevinskii, ”Representations of the group GL(n, F), where F is a local Archimedean field,” Usp. Mat. Nauk,31, No. 3, 5-70 (1976).
[3] J. D. Rogawski, ”On modules over the Hecke algebra of a p-adic group,” Invent. Math.,79, No. 3, 443-465 (1985). · Zbl 0579.20037
[4] G. Lusztig, ”Some examples of square integrable representations of semisimple p-adic groups,” Trans. Am. Math. Soc.,277, No. 2, 623-653 (1983). · Zbl 0526.22015
[5] A. Borel, ”Admissible representations of a semisimple group over a local field with vectors fixed under an Twahori subgroup,” Invent. Math.,35, 233-259 (1976). · Zbl 0334.22012
[6] H. Matsumoto, Analyse Harmonique dans les Systems de Tits Bornoloques de Type Affine, Lect. Notes in Math., Vol. 590, Springer-Verlag, Berlin?New York (1977). · Zbl 0366.22001
[7] M. Jimbo, RIMS Preprint No. 517, Kyoto Univ. (1985).
[8] M. Jimbo, ”A q-difference analogue of U(g) and the Yang?Baxter equation,” Lett. Math. Phys.,10, No. 1, 63-69 (1985). · Zbl 0587.17004
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.