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We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega,…
The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…
In the variety of all linear algebras over the infinite field the difference between geometric and automorphic equivalence of algebras can be big.
Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they…
We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is…
We provide a complete classification of unitary subalgebras of even rank-one lattice vertex operator algebras. As a consequence of the correspondence between vertex operator algebras and conformal nets, we also obtain a complete…
In this paper, we mainly investigate profound interconnections between combinatorial designs, linear codes, and Boolean functions.
We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank is bounded above by the effective central charge. We show that lattice vertex operator algebras may…
In the paper, developing the idea of V. Sokolov et all. (J.Math.Phys. 40 (1999)6473 we construct recursion operators and hereditary algebra of symmetries for many field and lattice systems.
The extensions of the affine vertex operator algebras $L_{sl_2}(k,0)$ and the preunitary vertex operator algebras with central charges $c<1$ are classified. In particular, the unitary vertex operator algebras with central charges $c<1$ are…
The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that…
Nonhamiltonian interaction of hamiltonian systems is considered. Dynamical equations are constructed by use of symmetric designs on Lie algebras. The results of analysis of these equations show that some class of symmetric designs on Lie…
In the paper the main attention is paid to conditions on algebras from a given variety which provide coincidence of their algebraic geometries. The main part here play the notions mentioned in the title of the paper.
We investigate a general structure theory for a vertex operator algebra. We discuss the center and blocks, the Jacobson radical and solvable radical and local vertex operator algebras. The main consequence of our structure theory is that if…
We construct supersymmetric irregular vertex operators of arbitrary rank, appearing in the colliding limit of primary fields. We find that the structure of the supersymmetric irregular vertices differs significantly from the bosonic case:…
Let V be a vertex operator algebra. We construct a sequence of associative algebras A_n(V) (n=0,1,2,...) such that A_{n}(V) is a quotient of A_{n+1}(V) and a pair of functors between the category of A_n(V)-modules which are not…
Unitary vertex operator algebras are introduced and studied. It is proved that most well-known rational vertex operator algebras are unitary. The classification of unitary vertex operator algebras with central charge c less than or equal to…
We present recent progress in theory of local conformal nets which is an operator algebraic approach to study chiral conformal field theory. We emphasize representation theoretic aspects and relations to theory of vertex operator algebras…
For a vertex operator algebra $V$ with conformal vector $\omega$, 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…
This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras…