The Experts below are selected from a list of 7419651 Experts worldwide ranked by ideXlab platform
Geoffrey Mason - One of the best experts on this subject based on the ideXlab platform.
-
Regularity of Rational Vertex Operator Algebras
Advances in Mathematics, 1997Co-Authors: Chongying Dong, Geoffrey MasonAbstract:Rational vertex operator algebras, which play a fundamental role in rational conformal field theory (see [BPZ] and [MS]), single out an important class of vertex operator algebras. Most vertex operator algebras which have been studied so far are rational vertex operator algebras. Familiar examples include the moonshine module V ♮ ([B], [FLM], [D2]), the vertex operator algebras VL associated with positive definite even lattices L ([B], [FLM], [D1]), the vertex operator algebras L(l, 0) associated with integrable representations of affine Lie algebras [FZ] and the vertex operator algebras L(cp,q, 0) associated with irreducible highest weight representations for the discrete series of the Virasoro algebra ([DMZ] and [W]). A rational vertex operator algebra as studied in this paper is a vertex operator algebra such that any admissible module is a direct sum of simple ordinary modules (see Section 2). It is natural to ask if such complete reducibility holds for an arbitrary weak module (defined in Section 2). A rational vertex operator algebra with this property is called a regular vertex operator algebra. One motivation for studying such vertex operator algebras arises in trying to understand the appearance of negative fusion rules (which are computed by the Verlinde formula) for vertex operator algebras L(l, 0) for certain rational l (cf. [KS] and [MW]). In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the rational vertex operator algebras V , L(l, 0) for positive integers l, L(cp,q, 0) and VL for positive definite even lattices L are regular. Our result for L(l, 0) implies that any restricted integrable module of level l for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. This result is expected to be useful in comparing the construction of tensor product of modules for L(l, 0) in [F] based on Kazhdan-Lusztig’s approach [KL] with the construction of tensor product of modules [HL] in this special case. We should remark that VL in general is a vertex algebra in the sense of [DL] if L is not positive definite. In this case we establish the complete reducibility of any weak module. Since the definition of vertex operator algebra is by now well-known, we do not define vertex operator algebra in this paper. We refer the reader to [FLM] and [FHL] for their elementary properties. The reader can find the details of the constructions of V ♮ and VL in [FLM], and L(l, 0) and L(cp,q, 0) in [DMZ], [DL], [FLM], [FZ], [L1] and [W].
-
Regularity of rational vertex operator algebras
arXiv: Quantum Algebra, 1995Co-Authors: Chongying Dong, Geoffrey MasonAbstract:A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{\natural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Virasoro algebra and the vertex operator algebras $V_L$ associated with positive definite even lattices $L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable module of level $l$ for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. The space $V_L$ in general is a vertex algebra if $L$ is not positive definite. In this case we establish the complete reducibility of any weak module.
Chelsea Walton - One of the best experts on this subject based on the ideXlab platform.
-
maps from the enveloping algebra of the positive witt algebra to regular algebras
Pacific Journal of Mathematics, 2016Co-Authors: Chelsea WaltonAbstract:We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.
-
the universal enveloping algebra of the witt algebra is not noetherian
Advances in Mathematics, 2014Co-Authors: Chelsea WaltonAbstract:Abstract This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces. As a consequence of our main result, we also show that the enveloping algebras of many other (infinite dimensional) Lie algebras are not noetherian. These Lie algebras include the Virasoro algebra and all infinite dimensional Z -graded simple Lie algebras of polynomial growth.
Chongying Dong - One of the best experts on this subject based on the ideXlab platform.
-
quantum dimensions and quantum galois theory
Transactions of the American Mathematical Society, 2013Co-Authors: Chongying Dong, Xiangyu JiaoAbstract:The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed systematically. The quantum dimensions of the Heisenberg vertex operator algebra modules, the Virasoro vertex operator algebra modules and the lattice vertex operator algebra modules are computed. A criterion for simple current modules of a rational vertex operator algebra is given. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A full Galois theory for rational vertex operator algebras is established using the quantum dimensions.
-
Regularity of Rational Vertex Operator Algebras
Advances in Mathematics, 1997Co-Authors: Chongying Dong, Geoffrey MasonAbstract:Rational vertex operator algebras, which play a fundamental role in rational conformal field theory (see [BPZ] and [MS]), single out an important class of vertex operator algebras. Most vertex operator algebras which have been studied so far are rational vertex operator algebras. Familiar examples include the moonshine module V ♮ ([B], [FLM], [D2]), the vertex operator algebras VL associated with positive definite even lattices L ([B], [FLM], [D1]), the vertex operator algebras L(l, 0) associated with integrable representations of affine Lie algebras [FZ] and the vertex operator algebras L(cp,q, 0) associated with irreducible highest weight representations for the discrete series of the Virasoro algebra ([DMZ] and [W]). A rational vertex operator algebra as studied in this paper is a vertex operator algebra such that any admissible module is a direct sum of simple ordinary modules (see Section 2). It is natural to ask if such complete reducibility holds for an arbitrary weak module (defined in Section 2). A rational vertex operator algebra with this property is called a regular vertex operator algebra. One motivation for studying such vertex operator algebras arises in trying to understand the appearance of negative fusion rules (which are computed by the Verlinde formula) for vertex operator algebras L(l, 0) for certain rational l (cf. [KS] and [MW]). In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the rational vertex operator algebras V , L(l, 0) for positive integers l, L(cp,q, 0) and VL for positive definite even lattices L are regular. Our result for L(l, 0) implies that any restricted integrable module of level l for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. This result is expected to be useful in comparing the construction of tensor product of modules for L(l, 0) in [F] based on Kazhdan-Lusztig’s approach [KL] with the construction of tensor product of modules [HL] in this special case. We should remark that VL in general is a vertex algebra in the sense of [DL] if L is not positive definite. In this case we establish the complete reducibility of any weak module. Since the definition of vertex operator algebra is by now well-known, we do not define vertex operator algebra in this paper. We refer the reader to [FLM] and [FHL] for their elementary properties. The reader can find the details of the constructions of V ♮ and VL in [FLM], and L(l, 0) and L(cp,q, 0) in [DMZ], [DL], [FLM], [FZ], [L1] and [W].
-
Regularity of rational vertex operator algebras
arXiv: Quantum Algebra, 1995Co-Authors: Chongying Dong, Geoffrey MasonAbstract:A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{\natural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Virasoro algebra and the vertex operator algebras $V_L$ associated with positive definite even lattices $L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable module of level $l$ for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. The space $V_L$ in general is a vertex algebra if $L$ is not positive definite. In this case we establish the complete reducibility of any weak module.
Sergei Silvestrov - One of the best experts on this subject based on the ideXlab platform.
-
quasi hom lie algebras central extensions and 2 cocycle like identities
arXiv: Rings and Algebras, 2004Co-Authors: Daniel Larsson, Sergei SilvestrovAbstract:This paper begins by introducing the concept of a quasi-hom-Lie algebra which is a natural generalization of hom-Lie algebras introduced in a previous paper by the authors. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is as a generalization-deformation of the Witt algebra $\Witt$ of derivations on the Laurent polynomials $\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-algebras which can be used to deform and generalize the Virasoro algebra by centrally extending the deformed Witt type algebras constructed here. In addition, we give a number of other interesting examples of quasi-hom-Lie algebras, among them a deformation of the loop algebra.
-
quasi hom lie algebras central extensions and 2 cocycle like identities
Preprints in Mathematical Sciences1999-01-01+01:00; (3) (2004), 2004Co-Authors: Daniel Larsson, Sergei SilvestrovAbstract:This paper introduces the notion of a quasi-hom-Lie algebra, or simply, a qhl-algebra, which is a natural generalization of hom-Lie algebras introduced in a previous paper [J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using sigma-derivations, math. QA/0408064]. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by maps, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is generalizations-deformations of the Witt algebra delta of derivations on the Laurent polynomials C[t,t(-1)]. We also develop a theory of central extensions for qhl-algebras which can be used to deform and generalize the Virasoro algebra by centrally extending the deformed Witt type algebras constructed here. In addition, we give a number of other interesting examples of quasi-hom-Lie algebras, among them a deformation of the loop algebra. (Less)
Iyama Osamu - One of the best experts on this subject based on the ideXlab platform.
-
Higher preprojective algebras, Koszul algebras, and superpotentials
'Wiley', 2020Co-Authors: Grant Joseph, Iyama OsamuAbstract:In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is -hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a -hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra
-
Higher preprojective algebras, Koszul algebras, and superpotentials
'Wiley', 2020Co-Authors: Grant Joseph, Iyama OsamuAbstract:In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.Comment: 36 pages; v3 is post referee repor