Exponential functors, R–matrices and twists

Pennig, Ulrich

doi:10.2140/agt.2020.20.1279

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Exponential functors, $R$–matrices and twists

Ulrich Pennig

Algebraic & Geometric Topology 20 (2020) 1279–1324

DOI: 10.2140/agt.2020.20.1279

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 $SU (n)$ 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.

Keywords

twisted $K$–theory, polynomial functors, unit spectrum, Fell bundles

Mathematical Subject Classification 2010

Primary: 19L50

Secondary: 55N15, 55R37

References

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

Volume 20, issue 3 (2020)