Representations of the Twisted Heisenberg–Virasoro Algebra and the Full Toroidal Lie Algebras

Jiang, Qifen; Jiang, Cuipo

doi:10.1142/s1005386707000120

Cited by 24 publications

(13 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A free-field realization of certain H-modules also appeared in [12]. The main difference with our approach is in the fact that we use screening operators to construct singular and cosingular vectors.…”

Section: Free-field Realization Of the Heisenberg-virasoro Vertex Algmentioning

confidence: 99%

“…A version of Proposition 2.7 appeared in Theorem 2.4 of [12]. In some special cases, the authors obtained the filtration (2.11) by a different method which was not based on screening operators.…”

Section: Free-field Realization Of the Heisenberg-virasoro Vertex Algmentioning

confidence: 99%

“…We should mention that a free field realization of certain indecomposable modules at level zero was constructed and analyzed in [12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

Adamović

Radobolja

2015

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

Abstract. We investigate the free field realization of the twisted Heisenberg-Virasoro algebra H at level zero. We completely describe the structure of the associated Fock representations. Using vertex-algebraic methods and screening operators we construct singular vectors in certain Verma modules as Schur polynomials. We completely solve the irreducibility problem for the tensor products of irreducible highest weight modules with intermediate series. We also determine the fusion rules for an interesting subcategory of H-modules. Finally, as an application we present a free-field realization of the W (2, 2)-algebra and interpret the W (2, 2)-singular vectors as H-singular vectors in Verma modules.

show abstract

Section: Free-field Realization Of the Heisenberg-virasoro Vertex Algmentioning

confidence: 99%

Section: Free-field Realization Of the Heisenberg-virasoro Vertex Algmentioning

confidence: 99%

See 1 more Smart Citation

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

Adamović

Radobolja

2015

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…In the following, to simplify notations, we shall always denote (H , [·, ·] q , α q ) by (H , [·, ·], α). The structure and representation theory of the twisted HeisenbergVirasoro algebra were studied by Kac et al [1], Jiang et al [5,16], Liu and Zhu [12], etc. However, we may mention that in our case, because q is non-zero and nonroot of unity, the q-deformed Heisenberg-Virasoro algebra of Hom-type is not a Lie algebra, the structural constants are not linearly dependent on their gradings (in this case, the algebra is called nonlinear), and this renders the computation of low-dimensional cohomology group of the q-deformed Heisenberg-Virasoro algebra of Hom-type nontrivial.…”

Section: Q-deformed Heisenberg-virasoro Algebra Of Hom-typementioning

confidence: 99%

Low-dimensional cohomology of q-deformed Heisenberg-Virasoro algebra of Hom-type

Cheng

Yang

2010

Front. Math. China

View full text Add to dashboard Cite

Hom-Lie algebras were introduced by J. Hartwig, D. Larsson, and S. Silvestrov as a generalized Lie algebra. When studying the homology and cohomology theory of Hom-Lie algebras, the authors find that the lowdimensional cohomology theory of Hom-Lie algebras is not well studied because of the Hom-Jacobi identity. In this paper, the authors compute the first and second cohomology groups of the q-deformed Heisenberg-Virasoro algebra of Hom-type, which will be useful to build the low-dimensional cohomology theory of Hom-Lie algebras.

show abstract

“…The study of the twisted Heisenberg-Virasoro algebra at level zero was initiated by Y. Billig [12] motivated by applications to the toroidal Lie algebras. New results on the representation theory were obtained in recent papers [2], [19], [24].…”

Section: Introductionmentioning

confidence: 99%

Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero

Adamović

Radobolja

2019

Commun. Contemp. Math.

View full text Add to dashboard Cite

This paper is a continuation of [2]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg-Virasoro algebra H at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for H is presented by combining a bosonic construction of Whittaker modules from [7] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible selfdual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of H-modules.In Section 2 we recall from [2] a free field realization of H, the definition of vertex algebra Π(0) and its modules Π(p, r). We study an extension of the Heisenberg-Virasoro vertex algebra Π(0) ⊂ Π(0) and present a structure of Π(0)-modules Π(p, r) in Section 3. By using certain relation in Π(0)-modules we recover formula 1.2 in Section 4. Then we consider a deformed action of H on these modules and obtain a family of modules Π(p, r) [cf. Theorem 5.2, Theorem 5.5] with the following properties:• Π(p, r) is a logarithmic H-module with the following action of the element L(0) of the Virasoro algebra:

show abstract

Representations of the Twisted Heisenberg–Virasoro Algebra and the Full Toroidal Lie Algebras

Abstract: An explicit construction of indecomposable modules for the twisted Heisenberg-Virasoro algebra and representations for the full toroidal Lie algebras are given.

Cited by 24 publications

References 16 publications

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

Low-dimensional cohomology of q-deformed Heisenberg-Virasoro algebra of Hom-type

Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero

Contact Info

Product

Resources

About