Browsing by Author "Mukhin, Eugene"

Now showing 1 - 5 of 5
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    Affinization of category 𝒪 for quantum groups
    (2014-05) Mukhin, Eugene; Young, C. A. S.; Department of Mathematical Sciences, School of Science
    Let be a simple Lie algebra. We consider the category of those modules over the affine quantum group whose -weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category . In particular, we define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.
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    Branching rules for quantum toroidal gl (n)
    (2013) Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, Eugene; Department of Mathematical Sciences, School of Science
    We construct an analog of the subalgebra Ugl(n) ⊗ Ugl(m) ⊂ Ugl(m + n) in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.
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    Finite Type Modules and Bethe Ansatz Equations
    (Springer, 2017-08) Feigin, Boris; Jimbo, Michio; Miwa, Tetsuji; Mukhin, Eugene; Department of Mathematical Sciences, School of Science
    We introduce and study a category OfinbObfin of modules of the Borel subalgebra UqbUqb of a quantum affine algebra UqgUqg, where the commutative algebra of Drinfeld generators hi,rhi,r, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional UqgUqg modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in OfinbObfin. Among them, we find the Baxter QiQi operators and TiTi operators satisfying relations of the form TiQi=∏jQj+∏kQkTiQi=∏jQj+∏kQk. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the QiQi operators acting in an arbitrary finite-dimensional representation of UqgUqg.
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    Integrals of motion from quantum toroidal algebras
    (IOP, 2017) Feigin, B.; Jimbo, M.; Mukhin, Eugene; Mathematical Sciences, School of Science
    We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the $({\mathfrak {gl}}_m, {\mathfrak {gl}}_n)$ duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine ${\mathfrak{sl}}_2$ .
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    On the Gaudin model associated to Lie algebras of classical types
    (AIP, 2016-10) Lu, Kang; Mukhin, Eugene; Varchenko, A.; Department of Mathematical Sciences, School of Science
    We derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.