Varieties of dialgebras and conformal algebras

On representations of dialgebras and conformal algebras

In this note, we observe a relation between dialgebras (in particular, Leibniz algebras) and conformal algebras. The purpose is to show how the methods of conformal algebras help solving problems on dialgebras, and, conversely, how the ideas of dialgebras work for conformal algebras.

Conformal representations of Leibniz algebras

In this note we present a more detailed and explicit exposition of the definition of a conformal representation of a Leibniz algebra. Recall (arXiv:math/0611501v3) that Leibniz algebras are exactly Lie dialgebras. The idea is based on the general fact that every dialgebra that belongs to a variety $\Var $ can be embedded into a conformal algebra of the same variety. In particular, we prove that an arbitrary (finite dimensional) Leibniz algebra has a (finite) faithful conformal representation. As a corollary, we deduce the analogue of the PBW-theorem for Leibniz algebras.

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra $V$ with conformal vector $ω$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of $(V,ω)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,ω)$, we shall understand the structure of the vertex operator algebra $(V,ω)$. At first, we define the set $\on{Sc}(V,ω)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,ω)$ is an affine algebraic variety…

Vertex algebras associated with hypertoric varieties

We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically and we construct sheaves of $\hbar$-adic vertex algebras over hypertoric varieties which localize the vertex algebras. We show when the vertex algebras are vertex operator algebras by giving explicit conformal vectors. We also show that …

具有阿贝尔共形部分的 chiral 代数

Chiral algebras with abelian conformal part

研究 chiral 代数变体的范畴方法，证明共形部分为阿贝尔的 chiral 代数类等价于由 Var 与 Com 的 Manin 黑积定义的变体中的微分代数类。

Moduli spaces of conformal structures on Heisenberg vertex algebras

This paper is a continuation to understand Heisenberg vertex algebras in terms of moduli spaces of their conformal structures. We study the moduli space of the conformal structures on a Heisenberg vertex algebra that have the standard fixed conformal gradation. As we know in Proposition 3.1 in Sect.3, conformal vectors of the Heisenberg vertex algebra $V_{\hatη}(1,0)$ that have the standard fixed conformal gradation is parameterized by a complex vector $h$ of its weight-one subspace. First, we classify all such conformal structures of the Heisenberg vertex algebra $V_{\hatη}(1,0)$ by describin…