Exponential Functors, R-Matrices and Twists

TL;DR

This paper links polynomial exponential functors on complex inner product spaces to involutive R-matrices solving the Yang-Baxter equation, classifies associated characters via Thoma parameters, and constructs higher twists over SU(n) in K-theory.

Contribution

It establishes a classification of polynomial exponential functors through involutive R-matrices and Thoma parameters, and constructs new higher twists in K-theory based on these structures.

Findings

Polynomial exponential functors correspond to involutive R-matrices.

Thoma parameters classify the associated characters on $S_{ ext{infty}}$.

Constructed higher twists over $SU(n)$ generalize the basic gerbe.

Abstract

In this paper we show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang-Baxter equation (an involutive $R$-matrix), which determines an extremal character on $S_{\infty}$. These characters are classified by Thoma parameters, and Thoma parameters resulting from polynomial exponential functors are of a special kind. Moreover, we show that each $R$-matrix with Thoma parameters of this kind yield a corresponding polynomial exponential functor. In the second part of the paper we use these functors to construct a higher twist over $SU(n)$ for a localisation of $K$-theory that generalises the one given by the basic gerbe. We compute the indecomposable part of the rational characteristic classes of these twists in terms of the Thoma parameters of their $R$-matrices.