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Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.
A theorem of Feigin, Frenkel and Reshetikhin provides expressions for the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in terms of elements of the center of the affine vertex algebra at the critical level. In our recent work, ex
We describe a categorification of the Double Affine Hecke Algebra ${mathcal{H}kern -.4emmathcal{H}}$ associated with an affine Lie algebra $widehat{mathfrak{g}}$, a categorification of the polynomial representation and a categorification of Macdonald
We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent.
The convolution ring $K^{GL_n(mathcal{O})rtimesmathbb{C}^times}(mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed
The crystals for a finite-dimensional complex reductive Lie algebra $mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{mathfrak{g}}$, constructed using t