Let g be a simple Lie algebra. We consider the category Ô of those modules over the affine quantum group Uq(ĝ) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category Ô. In particular, we develop the theory of q-characters a...

1.1. Overview. This will be an introductory course on the geometry of numbers. We will mostly adopt a classical approach, but here is the highbrow way of describing the goal of the course. For G a reductive group over Q (or really over any global field), we want to understand the size and shape of G (R)/G (Z) (for G a suitably nice flat affine Z-group of finite type with generic fiber G ), or t...

this paper, deals with some equivalence relations in fuzzy subgroups. further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.

we say that a finite group $g$ is conjugacy expansive if for anynormal subset $s$ and any conjugacy class $c$ of $g$ the normalset $sc$ consists of at least as many conjugacy classes of $g$ as$s$ does. halasi, mar'oti, sidki, bezerra have shown that a groupis conjugacy expansive if and only if it is a direct product ofconjugacy expansive simple or abelian groups.by considering a character analo...

let $g$ be a finite group‎. ‎we denote by $psi(g)$ the integer $sum_{gin g}o(g)$‎, ‎where $o(g)$ denotes the order of $g in g$‎. ‎here we show that‎ ‎$psi(a_5)< psi(g)$ for every non-simple group $g$ of order $60$‎, ‎where $a_5$ is the alternating group of degree $5$‎. ‎also we prove that $psi(psl(2,7))