Quantum toroidal algebra \(U_q(\widehat{\text{sl}}_{2,\text{tor}})\) and \(R\) matrices.

*(English)*Zbl 1032.17028From the introduction: In [K. Miki, Representation of quantum toroidal algebra \(U_q(sl_{n+1,\text{tor}})\) \((n\geq 2)\), J. Math. Phys. 41, 7079-7089 (2000; Zbl 1028.17011)] based on the work of V. Chari and A. Pressley [Commun. Math. Phys. 142, 261-283 (1991; Zbl 0739.17004) and CMS Conf. Proc. 16, 59-78 (1995; Zbl 0855.17009)] the author studied representations of the quantum toroidal algebra \(U_q(sl_{n+1, \text{tor}})\) and proved the existence of \(R\) matrices acting on the tensor product of a certain class of representations. In this analysis, he used an isomorphism of \(U_q'(sl_{n+1, \text{tor}})\) obtained by K. Miki [Lett. Math. Phys. 47, 365-378 (1999; Zbl 1022.17009)]. Since this result had been proved only for the case \(n\geq 2\), the analysis there was restricted to this case.

In this paper, for the algebra \(U_q(sl_{2, \text{tor}})\) he proves the existence of a similar isomorphism and shows that \(R\) matrices exist if we consider tensor product representations via similar comultiplications used in Miki (2000) (loc. cit.). This part is essentially the same as the results in the authors cited papers. Moreover he shows that \(R\) matrices exist for other comultiplications of \(U_q(sl_{2, \text{tor}})\). (This result is not restricted to the case \(n=1.)\) In particular, he obtains the explicit expressions of the latter \(R\) matrices acting on the tensor product of level 1 integrable highest weight representations of \(U_q(\widehat{sl}_2)\) as trigonometric \(R\) matrices were obtained as intertwiners of the tensor products of finite dimensional representations of the quantum affine algebras by M. Jimbo [Commun. Math. Phys. 102, 537-547 (1986; Zbl 0604.58013)].

In this paper, for the algebra \(U_q(sl_{2, \text{tor}})\) he proves the existence of a similar isomorphism and shows that \(R\) matrices exist if we consider tensor product representations via similar comultiplications used in Miki (2000) (loc. cit.). This part is essentially the same as the results in the authors cited papers. Moreover he shows that \(R\) matrices exist for other comultiplications of \(U_q(sl_{2, \text{tor}})\). (This result is not restricted to the case \(n=1.)\) In particular, he obtains the explicit expressions of the latter \(R\) matrices acting on the tensor product of level 1 integrable highest weight representations of \(U_q(\widehat{sl}_2)\) as trigonometric \(R\) matrices were obtained as intertwiners of the tensor products of finite dimensional representations of the quantum affine algebras by M. Jimbo [Commun. Math. Phys. 102, 537-547 (1986; Zbl 0604.58013)].

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

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##### References:

[1] | Miki, J. Math. Phys. 41 pp 7079– (2000) |

[2] | Chari, Commun. Math. Phys. 142 pp 261– (1991) |

[3] | V. Chari and A. Pressley,Representations of Groups(Banff, AB, 1994), pp. 59–78; · Zbl 0855.17009 |

[4] | CMS Conf. Proc., Vol. 16(Amer. Math. Soc., Providence, RI, 1995). |

[5] | Miki, Lett. Math. Phys. 47 pp 365– (1999) |

[6] | Jimbo, Commun. Math. Phys. 102 pp 537– (1986) |

[7] | Drinfeld, Soviet Mathematics Doklady 36 pp 212– (1987) |

[8] | Ginzburg, Mathematical Research Letters 2 pp 147– (1995) · Zbl 0914.11040 · doi:10.4310/MRL.1995.v2.n2.a4 |

[9] | Varagnolo, Commun. Math. Phys. 182 pp 469– (1996) |

[10] | Saito, Publications of the Research Institute for Mathematical Sciences 34 pp 155– (1998) |

[11] | Moody, Geom. Dedicata 35 pp 283– (1990) |

[12] | Beck, Commun. Math. Phys. 165 pp 555– (1994) |

[13] | G. Lusztig,Introduction to Quantum Groups(BirkhĂ¤user, Boston, 1993). · Zbl 0788.17010 |

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