We discuss the notion of a Batalin-Vilkovisky (BV) algebra and give several classical examples from differential geometry and Lie theory. We introduce the notion of a quantum operator algebra (QOA) as a generalization of a classical operator algebra. In some examples, we view a QOA as a deformation of a commutative algebra. We then review the notion of a vertex operator algebra (VOA) and show t...

An operator is uniform if its restriction to any infinite-dimensional invariant subspace is unitarily equivalent to itself. We show that a uniform operator having a proper infinite-dimensional invariant subspace resembles an analytic Toeplitz operator in the way that the weakly closed algebra generated by it and the identity operator is isomorphic to a subalgebra of the Calkin algebra; furtherm...

We give an example of an irrational C2-cofinite vertex operator algebra whose central charge is −2d for any positive integer d. This vertex operator algebra is given as the even part of the vertex operator superalgebra generated by d pairs of symplectic fermions, and it is just the realization of the c = −2-triplet algebra given by Kausch in the case d = 1. We also classify irreducible modules ...

Given a directed graph, there exists a universal operator algebra and universal C-algebra associated to the directed graph. In this paper we give intrinsic constructions of these objects. We provide an explicit construction for the maximal C-algebra of an operator algebra. We also discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isom...