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

Citations: | 2 - 2 self |

### BibTeX

@MISC{Bobtcheva_hkr-typeinvariants,

author = {Ivelina Bobtcheva and Maria Grazia Messia},

title = {HKR-type invariants of 4-thickenings of 2dimensional CW-complexes},

year = {}

}

### OpenURL

### 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

### Citations

483 |
Introduction to Quantum Groups
- Lusztig
- 1993
(Show Context)
Citation Context ...orphisms G → π1(P). 8.2 The quantum enveloping algebra of sl(2) The quantum deformation slq(2) of the classical algebra sl(2) has been defined for first time in [10] (see also [19]). Later Lusztig in =-=[15]-=- defined a quantum deformation of any Lie algebra, as an algebra over Q(v), using slightly different set of generators from [19] in order to obtain highest weight theory analogous of that of the class... |

121 |
Stipsicz, 4-manifolds and Kirby calculus, Graduate
- Gompf, I
- 1999
(Show Context)
Citation Context ...ifold together with a decomposition as a handlebody with a single 0-handle and a number of 1- and 2-handles. Then M can be represented by describing the attaching maps of the 1- and 2-handles in S 3 (=-=[12, 13]-=-). The attaching map of an 1-handle is a pair of 3- balls in S 3 or equivalently it can be described as a unknot of framing 0 in S 3 ( figure 1). In this last case the result of attaching the one hand... |

79 |
The topology of 4-manifolds
- Kirby
- 1989
(Show Context)
Citation Context ...ifold together with a decomposition as a handlebody with a single 0-handle and a number of 1- and 2-handles. Then M can be represented by describing the attaching maps of the 1- and 2-handles in S 3 (=-=[12, 13]-=-). The attaching map of an 1-handle is a pair of 3- balls in S 3 or equivalently it can be described as a unknot of framing 0 in S 3 ( figure 1). In this last case the result of attaching the one hand... |

40 | On Witten’s 3-manifold invariants - Walker - 1991 |

38 | Non-Semisimple Topological Quantum Field Theories for - Kerler, Lyubashenko - 2001 |

37 |
Lectures on Axiomatic Topological Quantum Field Theory, Geometry and Quantum Field theory
- Quinn
- 1995
(Show Context)
Citation Context ... under the AC-moves, a multiplicative invariant would be considered potentially interesting for the AC-conjecture if its value for < ∅ | 1 > is not a unit. Such invariants were introduced by Quinn in =-=[16]-=- and studied in [2, 3]. 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-Kazhdan, obtained as subq... |

33 |
Free groups and handlebodies
- Andrews, Curtis
- 1965
(Show Context)
Citation Context ...e semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center. 1 Introduction 1.1 The (generalized) Andrews-Curtis conjecture =-=[1]-=- asserts that any simple homotopy equivalence of 2-complexes can be obtained by deformation through 2-complexes (expansions and collapses of disks of dimension at most three), to which we refer here a... |

33 |
The trace function and Hopf algebras
- Radford
- 1994
(Show Context)
Citation Context ...onal, the Hopf algebra isomorphism A ≃ A ∗∗ implies that one can define a left (right) integral for A as an element Λ ∈ A, such that a.Λ = ǫ(a)Λ (Λa = ǫ(a)Λ) for any a ∈ A. 3.3 The following results (=-=[21, 18, 17]-=-) concern the existence of integrals when A is a finite-dimensional Hopf algebra over a field k. (a) The subspaces of left (right) integrals for ∫ ∗ L , ∫ ∗ R ⊂ A∗ and left (right) integrals ∫ , ⊂ A a... |

28 | Mapping class group actions on quantum doubles
- Kerler
(Show Context)
Citation Context ...42.7 Let J : Z(A) → Z(A), be defined as J(z) = (λ ⊗ 1)(z ⊗ 1)R 21 R = ∑ λ(zβiαj)αiβj. This operator is related to the image of one of the generators, S, in the action of the torus group on Z(A) (see =-=[8]-=-, (2.55)) and it is essential in understanding when the invariant of the 4-thickening reducesto an invariant of the border. Let Z⋆(A) denote the algebra which has as a vector space Z(A) and the ⋆ prod... |

27 |
Invariants of links and 3–manifolds obtained from Hopf algebras
- Hennings
- 1996
(Show Context)
Citation Context ... are 2actually invariants of 4-dimensional thickenings of the 2-complex (but we shall indicate when the value of the invariant depends “mainly” on the spine) 1 , and on another hand the framework in =-=[5, 10]-=- is used, so that the invariants are constructed directly from a finite dimensional unimodular ribbon Hopf algebra, without additional assumptions for its semisimplicity or for the structure of its re... |

25 | editors. Two-dimensional homotopy and combinatorial group theory, volume 197 - Hog-Angeloni, Metzler - 1993 |

4 |
Tortile tensor categories, Journal of Pure and Applied Algebra 93
- Shum
- 1994
(Show Context)
Citation Context ...undotted component is equal to ¯ci modulo 2. 5.6 We will assume that the reader is familiar with the notion of a framed tangle, which intuitively is a slice of a framed link. A good reference is Shum =-=[20]-=-, where it is called double tangle. In the future we will omit the word framed, since all tangles with which we will work will be such. A tangle with n incoming and m outgoing ends will be called an n... |

3 | On Quinn’s invariants of - Bobtcheva - 1999 |

3 |
On some aspects of Chern–Simons gauge theory
- Garoufalidis
- 1992
(Show Context)
Citation Context ...=0 2nj(j+1) [2j + 1] 2 ; λ(z ∗ RT θn ) = −pn(v − v −1 ); λ(zRTθ n ∑ ) = q−1 v j=0 2nj(j+1) [2j + 1] 2 . ]). Hence, The Gauss sum F(n) = ∑q−1 j=0 v2nj(j+1) [2j + 1] 2 has been evaluated for example in =-=[4]-=- and is equal to where ) g1v − n2 +1 2n ( n/2 F(n) = p (v − v−1 − ) 1 n ( ) a p is the Legendre symbol. In conclusion: [ Z ∂ [zH](L(1,n)) = n Z [z ∗ RT ](L(1,n)) = n; Z [zRT](L(1,n)) = ( n p ( n p ] p... |

3 |
Probleme des einfachen Homotopietyps in niederen Dimensionen und ihre Behandlung mit Hilfsmitteln der topologischen Quantenfeldtheorie , dissertation, Johann Wolfgang Goethe-Universitat
- Mueller
- 2000
(Show Context)
Citation Context ... the invariants come from a representation of the free group on the generators into a subgroup of GLN(Z/p) for some N, and in this representation every word has order p. Consequently, it was shown in =-=[14]-=- that any invariant possessing this property can’t detect counterexamples to the AC - conjecture. The goal of the present is to construct a more general framework for producing invariants of 2-complex... |

1 |
F.Quinn, On the connection between quantum invariants of 2complexes and 3-manifolds
- Bobtcheva
(Show Context)
Citation Context ... a multiplicative invariant would be considered potentially interesting for the AC-conjecture if its value for < ∅ | 1 > is not a unit. Such invariants were introduced by Quinn in [16] and studied in =-=[2, 3]-=-. 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-Kazhdan, obtained as subquotients of mod p repr... |

1 |
Invariants of 3-manifolds derived from finite-dimensional Hopf algebras, Journal of knot theory and its ramifications 4
- Kaufmann
(Show Context)
Citation Context ... are 2actually invariants of 4-dimensional thickenings of the 2-complex (but we shall indicate when the value of the invariant depends “mainly” on the spine) 1 , and on another hand the framework in =-=[5, 10]-=- is used, so that the invariants are constructed directly from a finite dimensional unimodular ribbon Hopf algebra, without additional assumptions for its semisimplicity or for the structure of its re... |

1 |
S.Sawin Centrality and the KRH invariant, Journal of knot theory and its ramifications 7
- Kaufmann
(Show Context)
Citation Context ...−1 (w (i)) → w ′ (n−i) = S−1 (w (i)). where we have used 3.1 (d). 6.6 Proof of (B). The proof is based on the following observation which is a version of the centrality result of the HKR-invariant in =-=[11]-=-. Let T be extended k −l tangle with n+m closed and r open components and let T ′ be the extended tangle obtained from T by embracing all incoming ends (figure 11 (a)) with a dotted component x ′, and... |

1 |
The order of the antipode of finite-dimesional Hopf algebras is finite
- Radford
(Show Context)
Citation Context ...onal, the Hopf algebra isomorphism A ≃ A ∗∗ implies that one can define a left (right) integral for A as an element Λ ∈ A, such that a.Λ = ǫ(a)Λ (Λa = ǫ(a)Λ) for any a ∈ A. 3.3 The following results (=-=[21, 18, 17]-=-) concern the existence of integrals when A is a finite-dimensional Hopf algebra over a field k. (a) The subspaces of left (right) integrals for ∫ ∗ L , ∫ ∗ R ⊂ A∗ and left (right) integrals ∫ , ⊂ A a... |