Theta series for quantum loop algebras and Yangians

TL;DR

This paper introduces Theta series for quantum loop algebras and Yangians, proving their polynomiality and applying these results to R-matrices and module decompositions in representation theory.

Contribution

It defines Theta series from T-series, proves their polynomiality, and extends these concepts to Yangians, providing new tools for studying quantum algebra representations.

Findings

Each weight component of a Theta series is polynomial.

Decomposition formula for R-matrices involving irreducible modules.

Extension of Theta series to Yangians with polynomiality proved.

Abstract

We introduce and study a family of power series, which we call Theta series, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel--Hernandez, which are leading terms of transfer matrices of certain infinite-dimensional irreducible modules over the upper Borel subalgebra in the category O of Hernandez--Jimbo. We prove that each weight component of a Theta series is polynomial. As applications, we establish a decomposition formula and a polynomiality result for R-matrices between an irreducible module and a finite-dimensional irreducible module in category O. We extend T-series and Theta series to Yangians by solving difference equations determined by the truncation series of Gerasimov--Kharchev--Lebedev--Oblezin. We prove polynomiality of Theta series by interpreting them as associators for triple tensor product modules over shifted Yangians.