1,721,043 research outputs found
Finite vs Infinite W algebras
No full textIn Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the lambda-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_G V, an associative algebra which "controls" G-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu_H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu_H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits
Freely generated vertex algebras and non-linear Lie conformal algebras
No full textWe introduce the notion of a non–linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique, up to isomorphism, non–linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras
On the integral representation of the q-gamma and the q-beta functions
No full textWe study q-integral representations of the q-gamma and the q-beta functions. This study leads to a very interesting q-constant. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series
Structure of some -graded Lie superalgebras of vector fields. Dedicated to the memory of Claude Chevalley.
No full text
Denominator identities and Lie superalgebras (extended abstract)
No full textWe provide formulas for the Weyl-Kac denominator and superdenominator of a basic classical Lie superalgebra for a distinguished set of positive roots. © 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebras
No full textWe describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero
A Lie conformal superalgebra and duality of representations for E(4,4)
Full text linkWe construct a duality functor in the category of continuous
representations of the Lie superalgebra E(4, 4), the only
exceptional simple linearly compact Lie superalgebra, for
which it wasn’t known. This is achieved by constructing a
Lie conformal superalgebra of type (4, 4) for which E(4, 4) is
the annihilation algebra. Along the way we obtain an explicit
realization of E(4, 4) by vector fields on a (4|4)-dimensional
supermanifold
Solitons in affine and permutation orbifolds
No full textWe consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to prove a number of conjectures: in the affine orbifold case we clarify the issue of “fixed point resolutions”; in the permutation orbifold case we determine all irreducible representations of the orbifold, and we also determine the fusion rules in a nontrivial case, which imply an integral property of chiral data for any completely rational conformal net
Corrigendum to “Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting” (Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting (2008) 217(6) (2485–2562), (S0001870807003088), (10.1016/j.aim.2007.10.005))
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