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14
Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato
, 1999
"... The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the ..."
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Cited by 110 (6 self)
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The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the
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 <= 2 ..."
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Cited by 32 (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.
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.
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 9 (6 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.
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 6 (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.
Nikolov: Jacobi identity for vertex algebras in higher dimensions
 J. Math. Phys
"... Abstract. 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 f ..."
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Cited by 5 (5 self)
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Abstract. 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 3 (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.
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”
unknown title
, 2002
"... We present a new way of defining varieties of conformal algebras: considering them as pseudoalgebras. This method agrees with the known one (via coefficient algebras) and allows to obtain the identities defining Jordan, alternative and Mal’cev conformal algebras. The Tits—Kantor—Koecher construction ..."
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We present a new way of defining varieties of conformal algebras: considering them as pseudoalgebras. This method agrees with the known one (via coefficient algebras) and allows to obtain the identities defining Jordan, alternative and Mal’cev conformal algebras. The Tits—Kantor—Koecher construction is built for finite Jordan pseudoalgebras. It gives the classification of such pseudoalgebras. 1
GEOMETRIC VERTEX OPERATORS
, 2002
"... Abstract. We investigate relationships between (infinitedimensional) algebraic geometry of loop spaces, smooth families of partial linear operators, and classical vertex operators. Vertex operators are shown to be actions of birational transformations of infinitedimensional algebraic “varieties ” ..."
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Abstract. We investigate relationships between (infinitedimensional) algebraic geometry of loop spaces, smooth families of partial linear operators, and classical vertex operators. Vertex operators are shown to be actions of birational transformations of infinitedimensional algebraic “varieties ” M on appropriate line bundles on M. These vertex operators act in the vector space M (M) of meromorphic functions on M as partial operators: they are defined on a subspace (in an appropriate lattice of subspaces of M (M)), and send smooth families in such a subspace to smooth families. Axiomatizing this, we define conformal fields as arbitrary families of partial operators in M (M) which satisfy both these properties. The “variety ” M related to standard vertex operators is formed by rational functions of one variable z ∈ Z = P 1, changing the variety Z one obtains different examples of M, and multidimensional analogues of vertex operators. One can cover M (M) by “big smooth subsets”; these subsets are parameterized by appropriate projective bundles over Hilbert schemes of points on Z. We deduce conformal