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Register a new userRepresentations of quantum toroidal $gl_n$
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We define and study representations of quantum toroidal $gl_n$ with natural bases labeled by plane partitions with various conditions. As an application, we give an explicit description of a family of highest weight representations of quantum affine $gl_n$ with generic level.
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Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case.
In this article, we develop a process to symmetrize the irreducible admissible representation of $GL_N(mathbb{Q}_p)$, as a consequence we obtain a more geometric understanding of the coefficient $m(mathbf{b}, mathbf{a})$ appearing in the decomposition of parabolic inductions, which allows us to prove a conjecture posed by Zelevinsky.
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 (gl(m),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 sl(2).
We construct an analog of the subalgebra $Ugl(n)otimes Ugl(m)$ of $Ugl(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.
We establish the method of Bethe ansatz for the XXZ type model obtained from the R-matrix associated to quantum toroidal gl(1). We do that by using shuffle realizations of the modules and by showing that the Hamiltonian of the model is obtained from a simple multiplication operator by taking an appropriate quotient. We expect this approach to be applicable to a wide variety of models.
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