Pimsner algebras and Gysin sequences from principal circle actions

TL;DR

This paper explores Pimsner algebras associated with principal circle actions, establishing Gysin sequences relating their KK-theories, and computes explicit KK-theory for quantum lens spaces over quantum weighted projective lines.

Contribution

It introduces a Gysin sequence for Pimsner algebras from circle actions and explicitly computes KK-theory for quantum lens spaces, providing new insights into noncommutative principal bundles.

Findings

Established a Gysin-like sequence relating KK-theories of Pimsner algebras and base algebras.

Explicitly computed KK-theory for quantum lens spaces over quantum weighted projective lines.

Identified natural generators for the KK-theory of these quantum spaces.

Abstract

A self Morita equivalence over an algebra B, given by a B-bimodule E, is thought of as a line bundle over B. The corresponding Pimsner algebra O_E is then the total space algebra of a noncommutative principal circle bundle over B. A natural Gysin-like sequence relates the KK-theories of O_E and of B. Interesting examples come from O_E a quantum lens space over B a quantum weighted projective line (with arbitrary weights). The KK-theory of these spaces is explicitly computed and natural generators are exhibited.