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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 140 (14 self)
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For a copy with the handdrawn figures please email
HigherDimensional Algebra I: Braided Monoidal 2Categories
 Adv. Math
, 1996
"... We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give lon ..."
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Cited by 53 (9 self)
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We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `HigherDimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...
and Category: is quantum gravity algebraic
 Journal of Mathematical Physics
, 1995
"... ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretati ..."
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Cited by 51 (3 self)
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ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context. I.
On algebraic structures implicit in topological quantum field theories
 J. Knot Theory Ramifications
, 1999
"... In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamenta ..."
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Cited by 48 (2 self)
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In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamental objects in abstract algebra.
A categorification of the TemperleyLieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors
 Selecta Math. (N.S
, 1999
"... ..."
A diagrammatic approach to categorification of quantum groups I
, 2009
"... To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph. ..."
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Cited by 41 (5 self)
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To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph.
Higherdimensional algebra IV: 2Tangles
"... Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we p ..."
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Cited by 35 (10 self)
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Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2category is the ‘free semistrict braided monoidal 2category with duals on one unframed selfdual object’. By this universal property, any unframed selfdual object in a braided monoidal 2category with duals determines an invariant of 2tangles in 4 dimensions. 1
Diagrammatics, Singularities, and Their Algebraic Interpretations
 in ``10th Brazilian Topology Meeting, Sa~ o Carlos, July 22 26, 1996,'' Mathematica Contempora^ nea
, 1996
"... This series of lectures reviews the remarkable feature of quantum topology: There are unexpected direct relations among algebraic structures and the combinatorics of knots and manifolds. The 6j symbols, Hopf algebras, triangulations of 3manifolds, TemperleyLieb algebra, and braid groups are rev ..."
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Cited by 21 (2 self)
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This series of lectures reviews the remarkable feature of quantum topology: There are unexpected direct relations among algebraic structures and the combinatorics of knots and manifolds. The 6j symbols, Hopf algebras, triangulations of 3manifolds, TemperleyLieb algebra, and braid groups are reviewed in the first three lectures. In the second lecture, we discuss parentheses structures and 2categories of surfaces in 3space in relation to the TemperleyLieb algebras. In the fourth lecture, we give diagrammatics of 4 dimensional triangulations and their relations to the associahedron, a higher associativity condition. We prove that the 4dimensional Pachner moves can be decomposed in terms of singular moves, and lower dimensional relations. In our last lecture, we give a combinatorial description of knotted surfaces in 4space and their isotopies. MRCN: 57Q45 Key words: Reidemeister Moves, 2categories, Movie Moves, Knotted Surfaces 1 1 Introduction In this series of tal...
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...