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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Ivelina Bobtcheva and Maria Grazia Messia

Algebraic & Geometric Topology 3 (2003) 33–87

arXiv: math.QA/0206307

Abstract

The HKR (Hennings–Kauffman–Radford) framework is used to construct invariants of 4–thickenings of 2–dimensional CW complexes under 2–deformations (1– and 2– handle slides and creations and cancellations of 1–2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T4 of trace elements which define invariants of 4–thickenings under 2–deformations. In T4 two subsets are identified : T3 T4, which produces invariants of 4–thickenings normalizable to invariants of the boundary, and T2 T4, which produces invariants of 4–thickenings depending only on the 2–dimensional spine and the second Whitney number of the 4–thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR–type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

Keywords
Hennings' invariant, Hopf algebras, CW complexes, 4–thickenings
Mathematical Subject Classification 2000
Primary: 57N13
Secondary: 57M20, 57N10, 16W30
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Publication
Received: 22 July 2002
Revised: 27 November 2002
Accepted: 10 January 2003
Published: 27 January 2003
Authors
Ivelina Bobtcheva
Dipartimento di Scienze Matematiche
Università di Ancona
Via Brece Bianche 1
60131, Ancona
Italy
Maria Grazia Messia
Dipartimento di Scienze Matematiche
Università di Ancona
Via Brece Bianche 1
60131, Ancona
Italy