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Exponential functors, $R$–matrices and twists

Ulrich Pennig

Algebraic & Geometric Topology 20 (2020) 1279–1324

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. 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 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.

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