The blocks of the Brauer algebra in characteristic zero

We determine the blocks of the Brauer algebra in characteristic zero. We also give information on the submodule structure of standard modules for this algebra

Universal Labeling Algebras as Invariants of Layered Graphs

In this work we will study the universal labeling algebra A(Gamma), a related algebra B(Gamma), and their behavior as invariants of layered graphs. We will introduce the notion of an upper vertex-like basis, which allows us to recover structural information about the graph Gamma from the algebra B(Gamma). We will use these bases to show that several classes of layered graphs are uniquely identified by their corresponding algebras B(Gamma)

Classical and Quantum Parts of the Quantum Dynamics: the Discrete-Time Case

In the study of open quantum systems modeled by a unitary evolution of a bipartite Hilbert space, we address the question of which parts of the environment can be said to have a "classical action" on the system, in the sense of acting as a classical stochastic process. Our method relies on the definition of the Environment Algebra, a relevant von Neumann algebra of the environment. With this algebra we define the classical parts of the environment and prove a decomposition between a maximal classical part and a quantum part. Then we investigate what other information can be obtained via this algebra, which leads us to define a more pertinent algebra: the Environment Action Algebra. This second algebra is linked to the minimal Stinespring representations induced by the unitary evolution on the system. Finally in finite dimension we give a characterization of both algebras in terms of the spectrum of a certain completely positive map acting on the states of the environment

One quiver to rule them all

To a formally smooth algebra A we associate a quiver setting (Q,a) containing enough information to reconstruct all the local quiver settings determining the etale local structure of finite dimensional representation schemes of A, see math.AG/9904171. Conjecturally, in a (yet to be developed) non-commutative etale topology, A is locally isomorphic to an algebra B which in turn is Morita equivalent (via the dimension vector a) to the path algebra CQ of the quiver Q