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28
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 218 (13 self)
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For a copy with the handdrawn figures please email
A unified WittenReshetikhinTuraev invariant for integral homology spheres
, 2006
"... We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTurae ..."
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Cited by 40 (4 self)
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We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTuraev invariants of M. As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function ” defined on the set of roots of unity. That is, the τζ(M) for all roots of unity are determined by a “Taylor expansion ” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.
Claspers and finite type invariants of links
, 2000
"... We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operatio ..."
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Cited by 34 (3 self)
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We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operations of a certain kind called “Ck–moves”. We prove that two knots in the 3–sphere are Ck+1–equivalent if and only if they have equal values of Vassiliev–Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev–Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3–dimensional topology.
Integrals for braided Hopf algebras
 J. Pure Appl. Algebra
, 2000
"... Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided ve ..."
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Cited by 29 (3 self)
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Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with crossproduct and transmutation are studied. 1991 Mathematics Subject Classification. Primary 16W30, 18D15, 17B37; Secondary 18D35.
Braided Hopf Algebras
, 2005
"... The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics a ..."
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Cited by 19 (7 self)
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The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke’s theorem are found. Maschke’s theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange’s theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery’s “Hopf algebras and their actions on rings ” contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. S. Montegomery, M. Cohen, H.J. Schneider, W. Chin, J.R. Fisher and the author study the relation between algebra R and the crossed product R#σH of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R#σH is semiprime when R is semiprime for a finitedimensional semiprime Hopf algebra H. This is a famous semiprime problem. The
Bottom tangles and universal invariants
, 2006
"... A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite ..."
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Cited by 12 (2 self)
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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action ” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H, we define a braided functor J from B to the category ModH of left H–modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH. Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants.
Towards an Algebraic Characterization of 3dimensional Cobordisms. ArXiv: math.GT/0106253
"... (To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dim ..."
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Cited by 10 (0 self)
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(To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3dimensional relative cobordisms, the second is a
A FUNCTORIAL LMO INVARIANT FOR LAGRANGIAN COBORDISMS
, 2007
"... Abstract. Lagrangian cobordisms are threedimensional compact oriented cobordisms between oncepunctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology threespheres to a functor ..."
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Cited by 8 (2 self)
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Abstract. Lagrangian cobordisms are threedimensional compact oriented cobordisms between oncepunctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology threespheres to a functor
Invariants of spin threemanifolds from ChernSimons theory and finitedimensional Hopf algebras
 Adv. Math
"... Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed threemanifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum group situations correspon ..."
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Cited by 7 (3 self)
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Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed threemanifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the halfinteger level ChernSimons theories conjectured to give spin TQFTs by Dijkgraaf and Witten [10]. In particular, the spin invariants constructed by Kirby and Melvin [21] are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finitedimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg’s noninvolutory invariant of framed manifolds associated to that Hopf algebra.
Integrals, Quantum Galois Extensions and the Affineness Criterion for Quantum Yetter–Drinfel’d Modules
 J. Algebra
"... In this paper we shall generalize the notion of integral on a Hopf algebra introduced by Sweedler, by defining the more general concept of integral of a threetuple (H,A,C), where H is a Hopf algebra coacting on an algebra A and acting on a coalgebra C. We prove that there exists a total integral γ: ..."
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Cited by 5 (2 self)
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In this paper we shall generalize the notion of integral on a Hopf algebra introduced by Sweedler, by defining the more general concept of integral of a threetuple (H,A,C), where H is a Hopf algebra coacting on an algebra A and acting on a coalgebra C. We prove that there exists a total integral γ: C → Hom(C,A) of (H,A,C) if and only if any representation of (H,A,C) is injective in a functorial way, as a corepresentation of C. In particular, the quantum integrals associated to YetterDrinfel’d modules are defined. Let now A be an Hbicomodule algebra, HYDA the category of quantum YetterDrinfel’d modules and B = {a ∈ A ∑S−1(a<1>)a<−1> ⊗ a<0> = 1H ⊗ a}, the subalgebra of coinvariants of the Verma structure A ∈ HYDA. We shall prove the following affineness criterion: if there exists γ: H → Hom(H,A) a total quantum integral and the canonical map β: A ⊗B A → H ⊗ A, β(a ⊗B b) = S−1(b<1>)b<−1> ⊗ ab<0> is surjective (i.e. A/B is a quantum homogeneous space), then the induction functor − ⊗B A: MB → HYDA is an equivalence of categories. The affineness criteria proven by Cline, Parshall and Scott, and independently by Oberst (for affine algebraic groups schemes), Schneider (in the noncommutative case), are recovered as special cases. 0