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We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either "annihilator algebras" --- algebras whose socle is a principal ideal --- or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories. Keywords: topological quantum field theory, frobenius algebra, two-dimensional cobordism, category theory 1. Introduction Topological Quantum Field Theories (TQFT's) were first described axiomatically by Atiyah in [1]. Since then, much work has been done ...

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, 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 2-category 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 2-categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `Higher-Dimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...

smolin@phys.psu.edu y permanent address I. Introduction In the last years significant progress has been made towards the construction of a quantum theory of gravity in several different directions. Three of these directions, in particular, have involved the use of new ideas and mathematical structures that seem, in different ways, well suited to the problem of describing the geometry of spacetime quantum mechanically. These are string theory[1], topological quantum field theory[2, 3, 4, 5, 6, 7], and non-perturbative quantum gravity, based on the loop representation [8, 9, 10, 11, 12, 13, 14]. Furthermore, despite genuine differences, there are a number of concepts shared by these approaches, which suggests the possibility of a deeper relation between them[15, 54]. These include the common use of one dimensional rather than pointlike excitations, as well as the appearance of structures associated with knot theory, spin networks and duality. There are also senses in which each deve...

: 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. INTRODUCTION The histories of mathematics and theoretical physics are so intimately interwoven that neither can really be understood in isolation from the other. There is no example of a fundamental advance in theoretical physics which did not involve a change in the mathematical structure in which the physical theory is formulated. On the other hand, a new construction in mathematics very often represents a distillation of a physical concept. Differential four dimensional manifolds with lorentzian metrics, for example, can be thought of as structures within which Einstein's falling elevators can coexist. Turning points in the development...

Recent developments in higher-dimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2-categories, and a center construction for monoidal 2-categories which gives a braided monoidal 2-category. I give generalized center constructions for braided and sylleptic monoidal 2-categories which give sylleptic and symmetric monoidal 2-categories respectively, and I correct some errors in the original center construction for monoidal 2-categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3-types of arc-connected, simply connected spaces, and from higher-dimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...

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.

Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. 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 4-manifolds using Crane-Yetter cocycles as Boltzmann weights. Our invariant generalizes the 3-dimensional 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 Turaev-Viro inv...

Abstract: We study the relations between the invariants τRT, τHKR, and τL of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how τL specializes to τRT for semisimple categories and to τHKR for Tannakian categories. We give arguments for that τL is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, Q, for a quasi-triangular Hopf algebra, A, that allows us to apply the Hennings algorithm directly, in order to compute τRT, which is originally obtained from the semisimple trace-subquotient of A − mod. Moreover, we generalize Hennings ’ rules to the context of cobordisms, in order to obtain a TQFT for connected surfaces compatible with τHKR. As an application we show that, for lens spaces and A = Uq(sl2), the ratio of τHKR and τRT is the order of the first homology group. In the course of this paper we also outline the topology and the algebra that enter invariance proofs, which contain no reference to 2-handle slides, but to other moves that are local. Finally, we give a list of open questions regarding cellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their relations among

A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1 + 1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit.