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67
Equational Reasoning With 2Dimensional Diagrams (preliminary Version)
 Term Rewriting, volume 909 of LNCS
, 1992
"... The significance of the 2dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of ndimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special ca ..."
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Cited by 20 (1 self)
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The significance of the 2dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of ndimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special case of 2dimensional calculus. Here, we propose a combinatorial definition of 2dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2presentable. We illustrate the translation of terms into diagrams and we explain the change from groups to quantum groups in a purely syntactical way. This paper should serve as a reference for our future work on symbolic computation, including a theory of 2dimensional rewriting and the design of software for interactive diagrammatic reasoning. New address: CNRS  Laboratoire de math'ematiques discr`etes, 163 avenue de Luminy  Case 930, 13288 Marseille Cedex 9, France. Email: lafont@lmd.univmrs.fr 2 1 FROM T...
Generalized TemperleyLieb Algebras And Decorated Tangles
 J. Knot Th. Ram
"... We give presentations, by means of diagrammatic generators and relations, of the analogues of the TemperleyLieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and D. This generalizes Kauffman's diagram calculus for the TemperleyLieb algebra. ..."
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Cited by 17 (7 self)
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We give presentations, by means of diagrammatic generators and relations, of the analogues of the TemperleyLieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and D. This generalizes Kauffman's diagram calculus for the TemperleyLieb algebra.
SOLUTIONS OF THE YANGBAXTER EQUATIONS FROM BRAIDEDLIE ALGEBRAS AND BRAIDED GROUPS
, 1993
"... We obtain an Rmatrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)Lie algebra or braidedLie algebra. The same result applies for every (super)Hopf algebra or braidedHopf algebra. We recover some known representations such as th ..."
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Cited by 16 (10 self)
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We obtain an Rmatrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)Lie algebra or braidedLie algebra. The same result applies for every (super)Hopf algebra or braidedHopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a nontrivial one on the ring k[x] of polynomials in one variable, regarded as a braidedline. Representations of the extended Artin braid group for braids in the complement of S 1 are also obtained by the same method.
Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1
"... Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor ..."
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Cited by 16 (0 self)
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Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor
A Compositional Distributional Model of Meaning
"... We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, namely Lambek’s pregroup semantics. A key observation is that the monoidal category of (finite dimensional) vector spaces, ..."
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Cited by 16 (0 self)
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We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, namely Lambek’s pregroup semantics. A key observation is that the monoidal category of (finite dimensional) vector spaces, linear maps and the tensor product, as well as any pregroup, are examples of compact closed categories. Since, by definition, a pregroup is a compact closed category with trivial morphisms, its compositional content is reflected within the compositional structure of any nondegenerate compact
Generalized BarrettCrane vertices and invariants of embedded graphs
 Journal of Knot Theory and its Ramifications 8
, 1999
"... ..."
Hopf (Bi)Modules and Crossed Modules in Braided Monoidal Categories
 J. Pure Appl. Algebra
, 1995
"... Hopf (bi)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra ..."
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Cited by 15 (1 self)
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Hopf (bi)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. A generalization of the MajidRadford criterion for a braided Hopf algebra to be a cross product is obtained as an application of these results. Keywords: Braided category, Braided Hopf algebra, Crossed Module, Hopf (Bi)Module Mathematical Subject Classification (1991): 16W30, 17B37, 18D10, 81R50 1 Introduction For bialgebras over a field k the smash product and the smash coproduct are investigated extensively in the literature [Rad, Mol]. Let H be a bialgebra, B be an Hright module algebra and an Hright comodule coalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Ra...
Generalized lattice gauge theory, spin foams and state sum invariants
, 2003
"... We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold ( ≤ 3, ≤ 4, any). ..."
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Cited by 14 (1 self)
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We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold ( ≤ 3, ≤ 4, any).
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 12 (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...
Selfadjunctions and matrices
 Journal of Pure and Applied Algebra
"... It is shown that the multiplicative monoids of TemperleyLieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronec ..."
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Cited by 10 (4 self)
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It is shown that the multiplicative monoids of TemperleyLieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. Thereby one obtains a representation of braid groups in matrices, which, though different and presumably new, is related to the standard representation of braid groups in TemperleyLieb algebras. Mathematics Subject Classification (2000): 57M99, 20F36, 18A40 1