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Bakalov, Bojko; Kač, Victor G.

doi:10.1155/s1073792803204232

Cited by 127 publications

(31 citation statements)

References 6 publications

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“…Kac also introduced conformal algebras, which can be defined easily. Then he proved, using field algebras, that vertex algebras form a subclass of conformal algebras, see [4]. This allows to give an easier definition of vertex algebras, with fewer axioms.…”

Section: 4mentioning

confidence: 99%

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Left-symmetric algebras, or pre-Lie algebras in geometry and physics

2006

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Abstract. In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

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Section: 4mentioning

confidence: 99%

“…e 2 · e 2 = 2e 2 e 3 · e 4 = e 2 e 1 · e 4 = −e 4 e 2 · e 3 = e 3 e 4 · e 3 = e 2 e 2 · e 4 = e 4 and the other products equal to zero. Then rad(A) = span{e 1 }.…”

Section: 3unclassified

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

2006

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“…Nonlocal vertex algebras (or field algebras in the sense of [1]) are analogues and generalizations of associative unital algebras. In a study [7] on regular representations for a Möbius quantum vertex algebra V, we proved that the regular representation space has a canonical module structure for a certain twisted tensor product of V and V. This motivated us to develop a theory of twisted tensor products of nonlocal vertex algebras and their modules (see [6,10,11,13]).…”

Section: Introductionmentioning

confidence: 99%

Generalized twistors of nonlocal vertex algebras

Sun

Wang

2016

Front. Math. China

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“…(Nonlocal vertex algebras are the same as weak G 1 -vertex algebras in [19] and are also essentially field algebras in [1].) While vertex algebras are analogs of commutative associative algebras, nonlocal vertex algebras are analogs of noncommutative associative algebras.…”

Section: Introductionmentioning

confidence: 99%

Vertex F-algebras and their ϕ-coordinated modules

2011

Journal of Pure and Applied Algebra

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a b s t r a c tIn this paper, for every one-dimensional formal group F we formulate and study a notion of vertex F -algebra and a notion of φ-coordinated module for a vertex F -algebra where φ is what we call an associate of F . In the case that F is the additive formal group, vertex F -algebras are exactly ordinary vertex algebras. We give a canonical isomorphism between the category of vertex F -algebras and the category of ordinary vertex algebras. Meanwhile, for every formal group we completely determine its associates. We also study φ-coordinated modules for a general vertex Z-graded algebra V with φ specialized to a particular associate of the additive formal group and we give a canonical connection between V -modules and φ-coordinate modules for a vertex algebra obtained from V by Zhu's change-of-variables theorem.

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Abstract: Abstract. A field algebra is a "non-commutative" generalization of a vertex algebra. In this paper we develop foundations of the theory of field algebras.

Cited by 127 publications

References 6 publications

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Generalized twistors of nonlocal vertex algebras

Vertex F-algebras and their ϕ-coordinated modules

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