Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand-Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand-Kirillov dimension under some specific conditions and construct an example to show that this upper bound can not be sharpened any more.

Yang Zhang Xiangui Zhao

We introduce a category of stochastic maps (certain Markov kernels) on compact Hausdorff spaces, construct a stochastic analogue of the Gelfand spectrum functor, and prove a stochastic version of the commutative Gelfand-Naimark Theorem. This relates concepts from algebra and operator theory to concepts from topology and probability theory. For completeness, we review stochastic matrices, their relationship to positive maps on commutative $C^*$-algebras, and the Gelfand-Naimark Theorem. No knowledge of probability theory nor $C^*$-algebras is assumed and several examples are drawn from physics.

Arthur J. Parzygnat

The classical Gelfand-Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered-graded rings, skew $PBW$ extensions, etc. In this paper, we present complete proofs of the computation of this more general dimension for the mentioned algebraic constructions for algebras over commutative domains. The Gelfand-Kirillov dimension for modules and the Gelfand-Kirillov transcendence degree will be also considered. The obtained results can be applied in particular to algebras over the ring of integers, i.e, to arbitrary rings.

Oswaldo Lezama Helbert Venegas

Let G be a locally compact group and let K be a compact subgroup of Aut(G), the group of automorphisms of G. The pair $(G, K )$ is a Gelfand pair if the algebra $L^{1}_{K}(G)$ of K-invariant integrable functions on G is commutative under convolution. In \cite{toure2008lie}, the charactezations of this algebra in the nilpotent case were studied, which generalize some results obtained by C. Benson, J. Jenkins, G. Ratcliff in \cite{benson1990gel} and obtained a new criterion for Gelfand pairs. In this paper we describe the spherical function associated with this type of pair.

Cornelie Mitcha Malanda

Based on results of Digne-Michel-Lehrer (2003) we give two formulae for two-variable Green functions attached to Lusztig induction in a finite reductive group. We present applications to explicit computation of these Green functions, to conjectures of Malle and Rotilio, and to scalar products between Lusztig inductions of Gelfand-Graev characters.

François Digne Jean Michel

Humans interact with each other on a daily basis by developing and maintaining various social norms and it is critical to form a deeper understanding of how such norms develop, how they change, and how fast they change. In this work, we develop an evolutionary game-theoretic model based on research in cultural psychology that shows that humans in various cultures differ in their tendencies to conform with those around them. Using this model, we analyze the evolutionary relationships between the tendency to conform and how quickly a population reacts when conditions make a change in norm desirable. Our analysis identifies conditions when a tipping point is reached in a population, causing norms to change rapidly.

Soham De Dana S. Nau Xinyue Pan Michele J. Gelfand

This paper deals with Gelfand-type problems \begin{equation}\label{Gelfand10} \qquad\qquad\left\{\begin{array}{ll} - \Delta_m u = \lambda f(u), \quad&\hbox{in} \ \Omega, \ \lambda >0, \\[10pt] u =0, \quad&\hbox{on} \ \partial_m\Omega, \end{array} \right. \end{equation} in the framework of Random Walk Spaces, which includes as particular cases: Gelfand-type problems posed on locally finite weighted connected graphs and Gelfand-type problems driven by convolution integrable kernels. Under the same assumption on the nonlinearity $f$ as in the local case, we show there exists an extremal parameter $\lambda^* \in (0, \infty)$ such that, for $0 \leq \lambda < \lambda^*$, problem \eqref{Gelfand10} admits a minimal bounded solution $u_\lambda$ and there are not solution for $\lambda > \lambda^*$. Moreover, assuming $f$ is convex, we show that Problem \eqref{Gelfand10} admits a minimal bounded solution for $\lambda = \lambda^*$. We also show that $u_\lambda$ are stable, and, for $f$ strictly convex, we show that they are the unique stable solutions. We give simple examples that illustrate the many situations that can occur when solving Gelfand-type problems on weighted graphs.

J. M. Mazon A. Molino J. Toledo

We classify all noetherian Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero which are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\Ext^1_H(k,k)\neq 0$. The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.

K. R. Goodearl J. J. Zhang

We classify all non-affine Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero that are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\text{Ext}^1_H(k, k) \neq 0$. The affine ones were classified by the authors in 2010.

K. R. Goodearl J. J. Zhang

A Gelfand model for an algebra is a module given by a direct sum of irreducible submodules, with every isomorphism class of irreducible modules represented exactly once. We introduce the notion of a perfect model for a finite Coxeter group, which is a certain set of discrete data (involving Rains and Vazirani's concept of a perfect involution) that parametrizes a Gelfand model for the associated Iwahori-Hecke algebra. We describe perfect models for all classical Weyl groups, excluding type D in even rank. The representations attached to these models simultaneously generalize constructions of Adin, Postnikov, and Roichman (from type A to other classical types) and of Araujo and Bratten (from group algebras to Iwahori-Hecke algebras). We show that each Gelfand model derived from a perfect model has a canonical basis that gives rise to a pair of related $W$-graphs, which we call Gelfand $W$-graphs. For types BC and D, we prove that these $W$-graphs are dual to each other, a phenomenon which does not occur in type A.

Eric Marberg Yifeng Zhang