Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

Details Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

2001-04-17

arxiv.org/abs/math/0104168v2

Commun.Math.Phys. 234 (2003) 101-127

Mathematics

Quantum Algebra

33 pages, latex, references updated, to appear in Commun. Math. Phys

Scientific paper

10.1007/s00220-002-0753-9

For a space X acted by a finite group $\G$, the product space $X^n$ affords a natural action of the wreath product $\Gn$. In this paper we study the K-groups $K_{\tG_n}(X^n)$ of $\Gn$-equivariant Clifford supermodules on $X^n$. We show that $\tFG =\bigoplus_{n\ge 0}K_{\tG_n}(X^n) \otimes \C$ is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on $\tFG$, when $\G$ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups.

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