On Quinn’s invariants of 2-dimensional CW-complexes

Bobtcheva, Ivelina

doi:10.1090/conm/233/03420

Cited by 3 publications

(21 citation statements)

References 14 publications

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“…Moreover, when the Hopf algebra is the finite dimensional quantum enveloping algebra at root of unity of some simple Lie group, T s contains at least one more element z RT which corresponds to the Reshetikhin-Turaev invariant. This fact was first observed by Hennings, and then, for the quantum sl (2), z RT was made explicit by Kerler in [8] (for completeness, in the appendix we present the derivation of z RT ). In an analogous way (though it won't be done here), one can see that Quinn's invariant can be derived in the HKR-framework from a triangular Hopf algebra over Z/p and a central element z Q = 1 in it.…”

Section: 1mentioning

confidence: 54%

“…Such invariants were introduced by Quinn in [16] and studied in [2]. The input for their construction is a finite semisimple symmetric monoidal category, which is taken to be one of the Lie families described by Gelfand and Kazhdan in [4], obtained as subquotients of mod p representations of simple Lie algebras.…”

Section: 1mentioning

confidence: 99%

“…The only partial remedy we can offer, is that by weakening "slightly" the defining conditions on T , one can define a subset T Z , containing T , such that its elements are relatively easy to determine since the calculations are entirely restricted to the center of the algebra. We make this calculation explicit for the case of the quantum sl (2) and show that in this case T Z consists of 4 elements, three of which are exactly 1, Λ and z RT . This fact leads to the conjecture that T Z = T for the quantum sl (2).…”

Section: • Tmentioning

confidence: 99%

“…We make this calculation explicit for the case of the quantum sl (2) and show that in this case T Z consists of 4 elements, three of which are exactly 1, Λ and z RT . This fact leads to the conjecture that T Z = T for the quantum sl (2). Under this assumption we show that the invariant corresponding to the forth element in T Z is the ratio of the Hennings and the Reshetikhin-Turaev invariants.…”

Section: • Tmentioning

confidence: 99%

“…Section 6 defines the invariant of 4-thickenings and shows that, when the trace element is in T 2 , the invariant depends only on the two dimensional spine of the 4-thickening and its second Whitney number. Section 7 studies the reducibility of the invariant to a 3-manifold invariant and section 8 illustrates the construction with two examples: the case of a group algebra and the case of the quantum sl (2). At the end we list some open questions.…”

Section: • Tmentioning

confidence: 99%

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HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

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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 T 4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T 4 two subsets are identified : T 3 ⊂ T 4 , which produces invariants of 4-thickenings normalizable to invariants of the boundary, 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 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.

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Section: 1mentioning

confidence: 54%

Section: 1mentioning

confidence: 99%

Section: • Tmentioning

confidence: 99%

Section: • Tmentioning

confidence: 99%

Section: • Tmentioning

confidence: 99%

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HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

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Group categories and their field theories

Quinn¹

1999

Geometry &Amp; Topology Monographs

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A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of "topological quantum field theories" for group-categories, in hopes of finding clues to a better understanding of the general situation. Group-categories are classified in several ways extending results of Frölich and Kerler. Topological field theories based on homology and cohomology are constructed, and these are shown to include theories obtained from group-categories by Reshetikhin-Turaev constructions. Braidedcommutative categories most naturally give theories on 4-manifold thickenings of 2-complexes; the usual 3-manifold theories are obtained from these by normalizing them (using results of Kirby) to depend mostly on the boundary of the thickening. This is worked out for group-categories, and in particular we determine when the normalization is possible and when it is not.

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The reduction of quantum invariants of 4-thickenings

Bobtcheva

Quinn

2005

Fund. Math.

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Abstract. We study the sensibility of an invariant of 2-dimensional CW complexes in the case when it comes as a reduction (through a change of ring) of a modular invariant of 4-dimensional thickenings of such complexes: it is shown that if the Euler characteristic of the 2-complex is greater than or equal to 1, its invariant depends only on homology. To see what is happening when the Euler characteristic is smaller than 1, we use ideas of Kerler and construct, from any tortile category, an invariant of 4-thickenings which, if the category is modular, is a normalization of the Reshetikhin-Turaev invariant, and if the category is symmetric, depends only on the spine and the second Whitney number of the thickening. Then we show that the so(3) quantum invariant at a 5th root of unity has its reduction, and this reduction is able to distinguish complexes with the same homology groups when the Euler characteristic is smaller than 1.

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On Quinn’s invariants of 2-dimensional CW-complexes

Cited by 3 publications

References 14 publications

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Group categories and their field theories

The reduction of quantum invariants of 4-thickenings

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