A new realization of Yangians and quantized affine algebras.

*(English. Russian original)*Zbl 0667.16004
Sov. Math., Dokl. 36, No. 2, 212-216 (1988); translation from Dokl. Akad. Nauk SSSR 269, 13-17 (1987).

This paper presents a more detailed account of the following results mentioned among others in the author’s programme talk at ICM, Berkeley, 1986 [see the preceding review Zbl 0667.16003]: A deformation of the universal enveloping algebra \({\mathcal U}({\mathfrak a}[\lambda])\), where \({\mathfrak a}\) is a simple finite-dimensional Lie algebra - a quantum group or Yangian - is described in terms of defining relations that generalize the conventional Serre’s relations between Chevalley generators (i.e. the generators with respect to which all structure constants are integers) of the initial Lie algebra. These relations are more convenient than the ones given in the first papers where Yangians had been introduced.

Similar results are obtained for \({\mathfrak a}\) being replaced by its twisted loop (Kac-Moody) version. For finite-dimensional representations of Yangians the highest weight theorem is given with explicit conditions on the labels of highest weight for a representation to be of finite dimension.

Similar results are obtained for \({\mathfrak a}\) being replaced by its twisted loop (Kac-Moody) version. For finite-dimensional representations of Yangians the highest weight theorem is given with explicit conditions on the labels of highest weight for a representation to be of finite dimension.

Reviewer: D.Leites

##### MSC:

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

81T99 | Quantum field theory; related classical field theories |

17B35 | Universal enveloping (super)algebras |

22E70 | Applications of Lie groups to the sciences; explicit representations |