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19
On Finite Type 3Manifold Invariants I
 II, Math. Annalen
, 1996
"... . Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3spheres (ZHS for short). In the present paper we propose another definition of finite type invariants of ZHS and give equivalent reformu ..."
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Cited by 25 (8 self)
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. Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3spheres (ZHS for short). In the present paper we propose another definition of finite type invariants of ZHS and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra and are of finite type in in the sense of Ohtsuki and thus conclude that the associated graded algebra is a priori finite dimensional in each degree. We discover a new set of restrictions that Ohtsuki's invariants satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3manifold invariants with the (Vassiliev) knot invariants. Contents 1. Introduction 2 1.1. History 1.2. A review of Ohtsuki's definition 1.3. Variations for finite type 3manifold invariants 1.4. Statement of the results 1.5. Plan of the proof 1.6. Acknowledgmen...
Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories
 Advances in Math. 146
, 1998
"... Crane and Frenkel proposed a state sum invariant for triangulated 4manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double ..."
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Cited by 20 (5 self)
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Crane and Frenkel proposed a state sum invariant for triangulated 4manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4manifolds using CraneYetter cocycles as Boltzmann weights. Our invariant generalizes the 3dimensional invariants defined by Dijkgraaf and Witten and the invariants that are defined via Hopf algebras. We present diagrammatic methods for the study of such invariants that illustrate connections between Hopf categories and moves to triangulations. 1 Contents 1 Introduction 3 2 Quantum 2 and 3 manifold invariants 4 Topological lattice field theories in dimension 2 . . . . . . . . . . . . . . . . . . . 4 Pachner moves in dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 TuraevViro inv...
Surgery diagrams for contact 3– manifolds
 Turkish J. Math
"... Abstract. In two previous papers, the two firstnamed authors introduced a notion of contact rsurgery along Legendrian knots in contact 3manifolds. They also showed how (at least in principle) to convert any contact rsurgery into a sequence of contact (±1)surgeries, and used this to prove that a ..."
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Cited by 18 (5 self)
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Abstract. In two previous papers, the two firstnamed authors introduced a notion of contact rsurgery along Legendrian knots in contact 3manifolds. They also showed how (at least in principle) to convert any contact rsurgery into a sequence of contact (±1)surgeries, and used this to prove that any (closed) contact 3manifold can be obtained from the standard contact structure on S 3 by a sequence of such contact (±1)surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact rsurgery into (±1)surgeries. We use this to give explicit surgery diagrams for all contact structures on S 3 and S 1 × S 2, as well as all overtwisted contact structures on arbitrary closed, orientable 3manifolds. This amounts to a new proof of the LutzMartinet theorem that each homotopy class of 2plane fields on such a manifold is represented by a contact structure. 1.
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 17 (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.
A treatise on quantum Clifford Algebras
"... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very e ..."
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Cited by 13 (10 self)
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on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A ‘spinorial ’ generalization of the antipode is given. The nonexistence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially generated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordization is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory. MSC2000: 16W30 Coalgebras, bialgebras, Hopf algebras; 1502 Research exposition (monographs, survey articles);
New branching rules induced by plethysm
 J. Phys A: Math. Gen
, 2006
"... We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi, a symmetric tensor gij = gji an ..."
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Cited by 9 (6 self)
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We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = −fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function sπ ≡ {π} by the basic M series of complete symmetric functions and the L = M −1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras, and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains πgeneralized NewellLittlewood formulae, and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for H 1 3, H21, and H3, showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine, and in some instances non reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly
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 6 (2 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.
AN ANALOGUE OF RADFORD’S S 4FORMULA FOR FINITE TENSOR CATEGORIES
, 2004
"... Abstract. We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ: V ∗ ∗ → D ⊗ ∗ ∗ V ⊗D −1. This provides a categorical generalization of Radford’s S 4 formula fo ..."
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Cited by 3 (1 self)
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Abstract. We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ: V ∗ ∗ → D ⊗ ∗ ∗ V ⊗D −1. This provides a categorical generalization of Radford’s S 4 formula for finite dimensional Hopf algebras [R1], which was proved in [N] for weak Hopf algebras, in [HN] for quasiHopf algebras, and conjectured in general in [EO]. When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of [R2]. We also show that a factorizable braided tensor category is unimodular (i.e. D = 1). Finally, we apply our theory to prove that the pivotalization of a fusion category is spherical, and give a purely algebraic characterization of exact module categories defined in [EO]. 1.
EQUIVARIANT FRAMINGS, LENS SPACES AND CONTACT STRUCTURES
"... We construct a simple topological invariant of certain 3manifolds, including quotients of S3 by finite groups, based on the fact that the tangent bundle of an orientable 3manifold is trivialisable. This invariant is strong enough to yield the classification of lens spaces of odd, prime order. We a ..."
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Cited by 2 (2 self)
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We construct a simple topological invariant of certain 3manifolds, including quotients of S3 by finite groups, based on the fact that the tangent bundle of an orientable 3manifold is trivialisable. This invariant is strong enough to yield the classification of lens spaces of odd, prime order. We also use properties of this invariant to show that there is an oriented 3manifold with no universally tight contact structure. We generalise and sharpen this invariant to an invariant of a finite covering of a 3manifold. It is wellknown that the tangent bundle of an orientable 3manifold is trivialisable. This is in particular true for manifolds of the form M = S 3 /G, where G is a finite group acting without fixed points on S 3. These are the socalled topological spherical space forms. Using the fact that TM is trivialisable, we can define an invariant F(M) of M with a fixed orientation, which we call the equivariant framing of M. Namely, the homotopy classes of trivialisations of the tangent bundle of S 3 are a torseur of Z (i.e., a set on which Z acts freely and transitively), which can moreover be canonically identified with Z by using the Lie group structure of S 3 as the unit quaternions and identifying a leftinvariant framing with 0 ∈ Z. Now, find a trivialisation τ of TM and pull it back to one of TS 3. Under the above identification, this gives an element F(M,τ) ∈ Z. This certainly depends on τ, but we shall see that its reduction modulo G, when H1(M,Z2) = 0 (in particular when G  is odd), and modulo G  /2 otherwise, is welldefined. Observe that this is the same as the collection of homotopy classes of equivariant trivialisations with respect to the action of G on S 3. The above definition does not depend on the identification of S 3 with the universal cover of M, since two such identifications differ by an orientation preserving selfhomeomorphism of S 3, which must be isotopic to the identity. Notice that this definition makes essential use of the fact that we have a quotient of S 3, rather than a homology (or even homotopy) sphere. In the more general situation, where we have a quotient of a homology sphere by a finite cyclic group, we can use canonical 2framings, as introduced by Atiyah
Threedimensional 2framed TQFTs and surgery
"... Abstract. The notion of 2framed threemanifolds is defined. The category of 2framed cobordisms is described, and used to define a 2framed threedimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2f ..."
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Cited by 2 (2 self)
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Abstract. The notion of 2framed threemanifolds is defined. The category of 2framed cobordisms is described, and used to define a 2framed threedimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2framed threedimensional TQFT. These data and relations are expressed in the language of surgery.