Representations of quantum toroidal algebra \(U_q(\text{sl}_{n+1,\text{tor}}) (n\geq 2)\).

*(English)*Zbl 1028.17011From the introduction: After the work of Drinfeld on finite-dimensional representations of the Yangian, V. Chari and A. Pressley studied finite-dimensional representations of the quantum affine algebra in a series of papers. Among their results, those related to our work are as follows. First [V. Chari and A. Pressley, Commun. Math. Phys. 142, 261-283 (1991; Zbl 0739.17004), A guide to quantum groups (Cambridge University Press) (1994; Zbl 0839.17009) and CMS Conf. Proc. 16, 59-78 (1995; Zbl 0855.17009)] they proved that irreducible finite-dimensional representations are characterizable by Drinfeld polynomials as in the Yangian case. Moreover they showed that the existence of \(R\) matrices acting on their tensor products was proven by utilizing the Drinfeld polynomials associated to the tensor products. Then possible \(U_q' (\widehat {\text{sl}_{n+1}})\) module structures on irreducible finite-dimensional \(U_q (\text{sl}_{n+1})\) modules were shown to be only those via the homomorphisms \(U_q' (\widehat {\text{sl}_{n+1}})\to U_q (\text{gl}_{n+1})\) by Jimbo. Moreover, minimal affinizations of representations of quantum groups of nonaffine type were studied.

In this paper, we apply their method to highest weight representations of the quantum toroidal algebra \(U_{q,\kappa} (\text{sl}_{n+1,\text{tor}})\) (\(\kappa\) is the parameter contained in the algebra). Many of the results obtained by Chari and Pressley for \(U_q' (\widehat{\text{sl}_{n+1}})\) \((n\geq 2)\) can be generalized to our case almost verbatim. In this analysis, we use the automorphism of the quantum toroidal algebra obtained in [K. Miki, Lett. Math. Phys. 47, 365-378 (1999; Zbl 1022.17009)]. Our main results are the proofs of the following facts: (i) some class of irreducible highest weight representations of the quantum toroidal algebra are characterized by Drinfeld polynomials, (ii) there exist solutions of the Yang-Baxter equation which depend on a spectral parameter and act on the tensor product of irreducible highest weight representations characterized by Drinfeld polynomials, (iii) no toroidal action can be defined on integrable highest weight representations of \(U_q (\widehat {\text{sl}_{n+1}})\) with level \(>1\), (iv) if the parameter \(\kappa\) is not equal to \(q^{\pm(n+1)}\), then toroidal module structures can be defined on irreducible integrable highest weight representations of \(U_q (\widehat {\text{gl}_{n+1}})\) with level \(c>1\) if and only if \(\kappa= q^{\pm(n+1+2c)}\). Moreover, these structures are those via the homomorphisms from the quantum toroidal algebra to a completion of \(U_q (\widehat {\text{gl}_{n+1}})\).

Note that our result clarifies the relation between the level 1 representation of the quantum toroidal algebra of Y. Saito (1998) and the one by M. Varagnolo and E. Vasserot (1998) and Y. Saito, K. Takemura and D. Uglov (1998).

In this paper, we apply their method to highest weight representations of the quantum toroidal algebra \(U_{q,\kappa} (\text{sl}_{n+1,\text{tor}})\) (\(\kappa\) is the parameter contained in the algebra). Many of the results obtained by Chari and Pressley for \(U_q' (\widehat{\text{sl}_{n+1}})\) \((n\geq 2)\) can be generalized to our case almost verbatim. In this analysis, we use the automorphism of the quantum toroidal algebra obtained in [K. Miki, Lett. Math. Phys. 47, 365-378 (1999; Zbl 1022.17009)]. Our main results are the proofs of the following facts: (i) some class of irreducible highest weight representations of the quantum toroidal algebra are characterized by Drinfeld polynomials, (ii) there exist solutions of the Yang-Baxter equation which depend on a spectral parameter and act on the tensor product of irreducible highest weight representations characterized by Drinfeld polynomials, (iii) no toroidal action can be defined on integrable highest weight representations of \(U_q (\widehat {\text{sl}_{n+1}})\) with level \(>1\), (iv) if the parameter \(\kappa\) is not equal to \(q^{\pm(n+1)}\), then toroidal module structures can be defined on irreducible integrable highest weight representations of \(U_q (\widehat {\text{gl}_{n+1}})\) with level \(c>1\) if and only if \(\kappa= q^{\pm(n+1+2c)}\). Moreover, these structures are those via the homomorphisms from the quantum toroidal algebra to a completion of \(U_q (\widehat {\text{gl}_{n+1}})\).

Note that our result clarifies the relation between the level 1 representation of the quantum toroidal algebra of Y. Saito (1998) and the one by M. Varagnolo and E. Vasserot (1998) and Y. Saito, K. Takemura and D. Uglov (1998).

##### MSC:

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

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

##### Keywords:

Yangian; quantum affine algebra; minimal affinizations; quantum groups; highest weight representations; quantum toroidal algebra; Drinfeld polynomials; Yang-Baxter equation; integrable highest weight representations; level 1 representation
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##### References:

[1] | Drinfeld, Sov. Math. Dokl. 36 pp 212– (1987) |

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

[3] | V. Chari and A. Pressley,A Guide to Quantum Groups(Cambridge University Press, Cambridge, England, 1994). · Zbl 0839.17009 |

[4] | V. Chari and A. Pressley, ”Quantum affine algebras and their representations,”Representations of Groups(Banff, AB, 1994), pp. 59–78, CMS Conf. Proc.16(American Mathematical Society, Providence, RI, 1995). · Zbl 0855.17009 |

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

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[16] | Saito, Publ. RIMS. Kyoto Univ. 34 pp 155– (1998) |

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