Braid group actions, Baxter polynomials, and affine quantum groups

Abstract

It is a classical result in representation theory that the braid group ${\mathcal{B}}_{\mathfrak{g}}$ of a simple Lie algebra $\mathfrak{g}$ acts on any integrable representation of $\mathfrak{g}$ via triple products of exponentials in its Chevalley generators. In this article, we show that a modification of this construction induces an action of ${\mathcal{B}}_{\mathfrak{g}}$ on the commutative subalgebra $Y_{\hslash }^0(\mathfrak{g})\subset Y_{\hslash }(\mathfrak{g})$ of the Yangian by Hopf algebra automorphisms, which gives rise to a representation of the Hecke algebra of type $\mathfrak{g}$ on a flat deformation of the Cartan subalgebra $\mathfrak{h}[t]\subset \mathfrak{g}[t]$. By dualizing, we recover a representation of ${\mathcal{B}}_{\mathfrak{g}}$ constructed in the works of Y. Tan and V. Chari, which was used to obtain sufficient conditions for the cyclicity of any tensor product of irreducible representations of $Y_{\hslash }(\mathfrak{g})$ and the quantum loop algebra $U_q(L\mathfrak{g})$. We apply this dual action to prove that the cyclicity conditions from the work of Tan are identical to those obtained in the recent work of the third author and S. Gautam. Finally, we study the $U_q(L\mathfrak{g})$-counterpart of the braid group action on $Y_{\hslash }^0(\mathfrak{g})$, which arises from Lusztig's braid group operators and recovers the aforementioned ${\mathcal{B}}_{\mathfrak{g}}$-action defined by Chari.

44 pages. Updates: Theorem 3.5, Corollary 3.11 and Theorem 6.5 now include descriptions of the inverse modified braid group operators. In addition, Corollary 4.5 has been added and Remarks 4.2, 4.6 and 4.7 have been adjusted. The numbering of some statements has changed accordingly. To appear in Transactions of the American Mathematical Society

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Cited by in corpus (1)

• Bethe subalgebras in Yangians and the wonderful compactification