nLab
Karoubi K-theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Karoubi defined K-theory classes given by Clifford-algebra module bundles, where a Cl nCl_n-module represents a class in K nK^n and represents the trivial class if it extends to a Cl n+1Cl_{n+1}-module.

Idea

We have a sequence of Clifford algebras Cl nCl_n which are generated by nn anticommuting square roots of ±1\pm 1. The sequence is periodic up to Morita equivalence; Cl 8Cl_8 is (16)\mathbb{R}(16), the algebra of 16×1616 \times 16 real matrices, which is Morita equivalent to \mathbb{R}, and from then on it repeats every 8 with extra matrix dimensions thrown in.

Here we treat Clifford algebras as 2\mathbb{Z}_2-graded algebras: while Cl 6Cl_6 is Morita equivalent to (8)\mathbb{R}(8) as an algebra, it is not so as a graded algebra.

It turns out that K n(X)K^n(X) can be represented geometrically by ‘bundles of Clifford modules’ over XX. Start with K 0K^0; we know that elements of K 0(X)K^0(X) are ‘formal differences’ VWV - W of vector bundles over XX. We can model the formal difference VWV - W with an honest geometric object by using the 2\mathbb{Z}_2-graded vector bundle VWV \oplus W, where VV is even and WW is odd. Such a thing should represent the zero class in K-theory just when VV and WW are isomorphic; this can be rephrased as saying that there exists an odd operator ee on VWV \oplus W (hence, taking VV to WW and vice versa) such that e 2=1e^2 = 1. But this just says that VWV \oplus W has an action of the first Clifford algebra Cl 1Cl_1.

More generally, Karoubi proved that for any nn, K nXK^{-n}X can be represented by Cl nCl_n-module bundles on XX modulo those such that the Cl nCl_n-action extends to a Cl n+1Cl_{n+1}-action. When n=0n = 0 this is what we had above, since a Cl 0Cl_0-module is just a vector space. This allows us to deduce Bott periodicity for K-groups from the algebraic periodicity (up to Morita equivalence) of Clifford algebras.

Remarks

Chris Douglas et al. are proposing that this description of K-theory has a good categorification that might be relevant for tmf.

Here is a report by Mike Shulman on a talk by Chris Douglas on this topic, from which also part of the above text is taken.

References

  • Karoubi, Twisted K-theory old and new, (pdf)

  • Karoubi, Donovan, Graded Brauer groups and KK-theory with local coefficients (pdf)

Relation to twisted K-theory:

  • Max Karoubi, Clifford modules and twisted K-theory, Proceedings of the International Conference on Clifford algebras (ICCA7) (arXiv:0801.2794)

Revised on June 8, 2012 13:41:26 by Urs Schreiber (131.130.239.35)