Towards Vertex Algebras of Krichever-Novikov Type, Part I

The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.

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 $V_{\hat{\eta}}(1,0)$ that have the standard fixed conformal gradation is parameterized by a complex vector $h$ of its weight-...

We show how the recently again discussed $N$-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multi-point versions of Krichever--Novikov type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representa...

Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty d...

An overview of the authors' ideas about the process of completing a $p$-adically normed space in the setting of vertex operator algebras. We focus in particular on the $p$-adic Heisenberg VOA and its connections with $p$-adic modular forms.

We classify all the Heisenberg and conformal vectors and determine the full automorphism group of the free bosonic vertex algebra without gradation. To describe it we introduce a notion of inner automorphisms of a vertex algebra.

We give a survey on the developments in a certain theory of quantum vertex algebras, including a conceptual construction of quantum vertex algebras and their modules and a connection of double Yangians and Zamolodchikov-Faddeev algebras with quantum vertex algebras.

This paper is the first in a series of papers developing a functional-analytic theory of vertex (operator) algebras and their representations. For an arbitrary Z-graded finitely-generated vertex algebra (V, Y, 1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is a canonical linear map from the tensor product of V and V to the algebraic compl...

We give an introductory account of Khovanov's categorification of the Heisenberg algebra, and construct a combinatorial model for it in a 2-category of spans of groupoids. We also treat a categorification of $U(sl_n)$ in a similar way. These give rise to standard representations on vector spaces through a linearization process.

A report at Arbeitstagung, June 1999, Bonn, on the joint work with V.Gorbounov, F.Malikov and A.Vaintrob.