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The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field F whose characteristic is not equal to 2 are studied. The irreducible modules of vertex operator algebra…
The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $V$ of central charge $24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $V_1$ if $V_1$ is a Lie…
A catalogue of explicit realizations of representations of (super) Lie algebras and quantum algebras in Fock space is presented.
In this paper we develop a formalism for working with twisted realizations of vertex and conformal algebras. As an example, we study realizations of conformal algebras by twisted formal power series. The main application of our technique is…
In this paper we introduce the concepts of a Rota-Baxter operator and a differential operator with weights on an $n$-algebra. We then focus on Rota-Baxter 3-Lie algebras and show that they can be derived from Rota-Baxter Lie algebras and…
Primary fields of the $q$-deformed Virasoro algebra are constructed. Commutation relations among the primary fields are studied. Adjoint actions of the deformed Virasoro current on the primary fields are represented by the shift operator…
The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator…
The main goal of this paper is to give some construction results of BiHom-post-Lie algebras which are a generalization of both post-Lie-algebras and Hom-post-Lie algebras. They are the algebraic structures behind the $\mathcal{O}$-operator…
Rota-Baxter operators, $\mathcal{O}$-operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with applications in mathematical physics. This paper introduces the notions of a…
In this paper, we establish some basic properties of certain operators (element of centroids, averaging operators, derivations, Nijenhuis operators, Rota-Baxter operators) on (compatible) ternary Leibniz algebras and give the classification…
This paper is to study what we call twisted regular representations for vertex operator algebras. Let $V$ be a vertex operator algebra, let $\sigma_1,\sigma_2$ be commuting finite-order automorphisms of $V$ and let…
We expose the elliptic quantum groups in the Drinfeld realization associated with both the affine Lie algebra \g and the toroidal algebra \g_tor. There the level-0 and level \not=0 representations appear in a unified way so that one can…
We obtain explicit expressions for all genus one chiral n-point functions for free bosonic and lattice vertex operator algebras. We also consider the elliptic properties of these functions.
We introduce and study twist vertex operators for a (lower-bounded generalized) twisted modules for a grading-restricted vertex (super)algebra. We prove duality, weak associativity, a Jacobi identity, a generalized commutator formula,…
We determine the decomposition of V_{\sqrt{2}D_l} into a sum of irreducible T-modules for general l where D_l is the root lattice of type D_l and T is the tensor product of l+1 Virasoro vertex operator algebras with central charges…
In this paper, we define differential graded vertex operator algebras and the algebraic structures on the associated Zhu algebras and $C_2$-algebras. We also introduce the corresponding notions of modules, and investigate the relations…
The method of construction of auxiliary representations for a given Lie algebra is discussed in the framework of the BRST approach. The corresponding BRST charge turns out to be non -- hermitian. This problem is solved by the introduction…
We define an integral intertwining operator among modules for a vertex operator algebra to be an intertwining operator which respects integral forms in the modules, and we show that an intertwining operator is integral if it is integral…
Let $\Gamma$ be a generic subgroup of the multiplicative group $\mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $\Gamma$, called twisted $\Gamma$-Lie algebras, which is a natural generalization of…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.