# zbMATH — the first resource for mathematics

A q-difference analogue of $$U({\mathfrak g})$$ and the Yang-Baxter equation. (English) Zbl 0587.17004
The author introduces a q-difference analogue of the universal enveloping algebra of a simple Lie algebra. Its representations are studied for the case of sl(2,$${\mathbb{C}})$$ and then the theory is applied to determine the trigonometric solutions of the Yang-Baxter equation related to sl(2,$${\mathbb{C}})$$ in an arbitrary finite-dimensional irreducible representation.
Reviewer: T.Ratiu

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B35 Universal enveloping (super)algebras 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text:
##### References:
 [1] KulishP. P. and ReshetikhinN. Yu.,J. Soviet Math. 23, 2435 (1983). Russian originalZapiski nauch. semin. LOMI 101, 112 (1980). [2] KulishP. P. and SklyaninE. K.,J. Soviet Math. 19, 1596 (1982). Russian originalZapiski nauch. semin. LOMI 95, 129 (1980). · Zbl 0553.58039 [3] KulishP. P., ReshetikhinN. Yu., and SklyaninE. K.,Lett. Math. Phys. 5, 393 (1981). · Zbl 0502.35074 [4] KacV. G.,Infinite Dimensional Lie Algebras, Birkh?user, Boston, 1983. [5] Andrews, G. E.,The Theory of Partitions, Addison-Wesley, 1976. · Zbl 0371.10001 [6] Jimbo, M., RIMS preprint506 (1985), to appear inCommun. Math. Phys.
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.