Speaker:	Robert Weston (Heroit Watt University)
Title:	Cyclic Representations of Quantum Affine sl(2) and its Borel Subalgebras at Roots of Unity and Q-operators
Date (JST):	Tue, Jun 10, 2025, 15:30 - 17:00
Place:	Seminar Room A
Abstract:	In this talk, I will revisit the finite-dimensional representations of quantum affine sl(2) that occur at q a root of unity which are not related to the standard integer-dimension representations of sl(2). These representations depend upon two points in two copies of a higher-genus algebraic curve. The R-matrix which is the intertwiner of two such representations factors into four terms - each of which encodes a Boltzmann weight of the Chiral Potts model as has long been known. In this work I will investigate this phenomenon further and show that, viewed as Borel subalgebra representations, the finite-dimensional representations themselves factorize into two very simple representations, each of which depends on a single point in the algebraic curve. This factorization will be used to both explain the R-matrix factorization and show that the transfer matrix factorises into two operators that play the role of Q-operators in the roots of unity case. The functional relations satisfied by these Q-operators will be obtained from the study of short-exact-sequences of representations. The situation parallels the generic q case, with the key difference being that only finite-dimensional representations appear in the roots-of-unity case.

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