Abstract

The HKR (Hennings-Kaufmann-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW-complexes under 1- and 2- handle slides and cancellations (2-deformations). 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 T 4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T 4 are identified two subsets: T 3 ⊂ T 4, which produces invariants of 4-thickenings normalizable to invariants of the border, and T 2 ⊂ T 4, 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 show 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. 1

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