Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 041, 20 pages      arXiv:2001.02164      https://doi.org/10.3842/SIGMA.2021.041

A Decomposition of Twisted Equivariant $K$-Theory

José Manuel Gómez and Johana Ramírez
Escuela de Matemáticas, Universidad Nacional de Colombia, Medellín, Colombia

Received July 13, 2020, in final form April 15, 2021; Published online April 21, 2021

Abstract
For $G$ a finite group, a normalized 2-cocycle $\alpha\in Z^{2}\big(G,{\mathbb S}^{1}\big)$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $\alpha$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $\alpha$-projective representations of $A$. This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant $K$-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups $D_{2n}$ with $n\ge 2$ an even integer.

Key words: twisted equivariant $K$-theory; $K$-theory; finite groups.

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