Representations of the Schrodinger algebra and Appell systems

We investigate the structure of the Schrodinger algebra and its representations in a Fock space realized in terms of canonical Appell systems. Generalized coherent states are used in the construction of a Hilbert space of functions on which certain commuting elements act as self-adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space is found. Then Appell systems connected with certain evolution equations, analogs of the classical heat equation, on this algebra are computed.Comment: 23 pages, LaTe

Zeon Algebra, Fock Space, and Markov Chains

Fock spaces over zeons are introduced. Trace identities and a noncommutative "integration-by-parts" formula are developed. As an application, we find a new criterion, without involving powers of the transition matrix, for a Markov chain to be ergodic

A hierarchical structure of transformation semigroups with applications to probability limit measures

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels. This kernel hierarchy produces a set of tools that provides direct access to computations of interest in probability limit theorems; in particular, finding certain factors of idempotent limit measures. In addition, when considering transformation semigroups that arise naturally from edge colorings of directed graphs, as in the road-coloring problem, the hierarchy produces simple techniques to determine the rank of the kernel and to decide when a given kernel is a right group. In particular, it is shown that all kernels of rank one less than the number of vertices must be right groups and their structure for the case of two generators is described.Comment: 35 pages, 4 figure