Bibliographic Details
| Title: |
Braid group actions, Baxter polynomials, and affine quantum groups. |
| Authors: |
Friesen, Noah1 (AUTHOR), Weekes, Alex1 (AUTHOR), Wendlandt, Curtis1 (AUTHOR) |
| Source: |
Transactions of the American Mathematical Society. Feb2025, Vol. 378 Issue 2, p1329-1372. 44p. |
| Subject Terms: |
*QUANTUM groups, *REPRESENTATION theory, *TENSOR products, *HECKE algebras, *HOPF algebras, *BRAID group (Knot theory) |
| Abstract: |
It is a classical result in representation theory that the braid group \mathscr {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 \mathscr {B}_\mathfrak {g} on the commutative subalgebra Y_\hbar ^{0}(\mathfrak {g})\subset Y_\hbar ^{}(\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 \mathscr {B}_\mathfrak {g} constructed in the works of Y. Tan [ Braid group actions and tensor products for Yangians , Preprint, arXiv: 1510.01533 , 2015] and V. Chari [Int. Math. Res. Not. 7 (2002), pp. 357–382], which was used to obtain sufficient conditions for the cyclicity of any tensor product of irreducible representations of Y_\hbar ^{}(\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 [Selecta Math. (N.S.) 29 (2023), no. 1, Paper No. 13] Finally, we study the U_q^{}(L\mathfrak {g})-counterpart of the braid group action on Y_\hbar ^{0}(\mathfrak {g}), which arises from Lusztig's braid group operators and recovers the aforementioned \mathscr {B}_\mathfrak {g}-action defined by Chari. [ABSTRACT FROM AUTHOR] |
|
Copyright of Transactions of the American Mathematical Society is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) |
| Database: |
Academic Search Complete |
