#### Volume 20, issue 3 (2020)

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Exponential functors, $R$–matrices and twists

### Ulrich Pennig

Algebraic & Geometric Topology 20 (2020) 1279–1324
##### Abstract

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 $SU\left(n\right)$ for a localisation of $K$–theory that generalises the one classified 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.

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##### Keywords
twisted $K$–theory, polynomial functors, unit spectrum, Fell bundles
##### Mathematical Subject Classification 2010
Primary: 19L50
Secondary: 55N15, 55R37
##### Publication
Received: 18 June 2018
Revised: 22 August 2019
Accepted: 21 September 2019
Published: 27 May 2020
##### Authors
 Ulrich Pennig School of Mathematics Cardiff University Cardiff United Kingdom