Theta Series for Quantum Loop Algebras and Yangians

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 \({\mathcal {O}}\) 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 \({\mathcal {O}}\). 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.