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22
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n < ..."
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Cited by 38 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
A Hopf laboratory for symmetric functions
 J. Phys. A: Math. Gen
, 2004
"... An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bialgebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focussed on Laplace pairing, Sweedler cohomology for 1 and 2cochains ..."
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Cited by 20 (14 self)
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An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bialgebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focussed on Laplace pairing, Sweedler cohomology for 1 and 2cochains, and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.
Jacobi identity for vertex algebras in higher dimensions
 J. MATH. PHYS
, 2006
"... Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. W ..."
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Cited by 10 (7 self)
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Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.
Generalized enrichment of categories
 Also Journal of Pure and Applied Algebra
, 1999
"... We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmultica ..."
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Cited by 10 (1 self)
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We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmulticategory extends the (more or less wellknown) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fcmulticategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and selfcontained, we also explain why, from one point of view, fcmulticategories are the natural structures in which to enrich categories.
General operads and multicategories
 Eprint math.CT/9810053
, 1997
"... Notions of ‘operad ’ and ‘multicategory ’ abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad ∗ on a category S, we define the term (S, ∗)multicategory, subject to certain conditions on S and ∗. Different choices ofS and ..."
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Cited by 9 (3 self)
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Notions of ‘operad ’ and ‘multicategory ’ abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad ∗ on a category S, we define the term (S, ∗)multicategory, subject to certain conditions on S and ∗. Different choices ofS and ∗ give some of the existing notions. We then describe the algebras for an (S, ∗)multicategory and, finally, present a tentative selection of further developments. Our approach makes possible concise descriptions of Baez and Dolan’s opetopes and Batanin’s operads; both of these are included.
Equivalence of Borcherds Gvertex algebras and axiomatic vertex algebras
, 1999
"... In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds ’ notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations ..."
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Cited by 7 (1 self)
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In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds ’ notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations of singularities to their inputs. In particular we show that this description of a vertex algebra allows us to present generalised notions of rationality, commutativity and associativity as natural consequences of the definition. Finally, we show that for a certain choice of vertex group, axiomatic vertex algebras correspond bijectively to algebras in the relaxed multilinear category of representations of a vertex group.
Quantum vertex F((t))algebras and their modules
, 903
"... This is a paper in a series to study vertex algebralike structures arising from various algebras including quantum affine algebras and Yangians. In this paper, on the basis of [Li6] we develop a theory of (weak) quantum vertex F((t))algebras with F a field of characteristic zero and t a formal var ..."
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Cited by 6 (5 self)
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This is a paper in a series to study vertex algebralike structures arising from various algebras including quantum affine algebras and Yangians. In this paper, on the basis of [Li6] we develop a theory of (weak) quantum vertex F((t))algebras with F a field of characteristic zero and t a formal variable, and we give a general construction of (weak) quantum vertex F((t))algebras and their modules. As an application, we associate quantum affine algebras with weak quantum vertex F((t))algebras and we construct an example of quantum vertex F((t))algebras from a certain quantum βγsystem. 1
Npoint locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebras
 J. Pure Appl. Algebra
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FIELD ALGEBRAS
, 2002
"... Dedicated to Ernest Borisovich Vinberg on the occasion of his 65th birthday. Abstract. A field algebra is a “noncommutative ” generalization of a vertex algebra. In this paper we develop foundations of the theory of field algebras. Roughly speaking, the notion of a vertex algebra [B1] is a generali ..."
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Cited by 1 (1 self)
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Dedicated to Ernest Borisovich Vinberg on the occasion of his 65th birthday. Abstract. A field algebra is a “noncommutative ” generalization of a vertex algebra. In this paper we develop foundations of the theory of field algebras. Roughly speaking, the notion of a vertex algebra [B1] is a generalization of the notion of a unital commutative associative algebra where the multiplication depends on a parameter. (In fact, in [B2] vertex algebras are described as “singular”