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Representations of quantum toroidal algebra $$U_q(\text{sl}_{n+1,\text{tor}}) (n\geq 2)$$. (English) Zbl 1028.17011
From 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).

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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##### References:
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