In this paper we define $varphi$-Connes module amenability of a dual Banach algebra $mathcal{A}$ where $varphi$ is a bounded $w_{k^*}$-module homomorphism from $mathcal{A}$ to $mathcal{A}$. We are mainly concerned with the study of $varphi$-module normal virtual diagonals. We show that if $S$ is a weakly cancellative inverse semigroup with subsemigroup $E$ of idemp...

these are notes from introductory survey lectures given at the institute for studies in theoretical physics and mathematics (ipm), teheran, in 2008 and 2010. we present the definition and the fundamental properties of fomin-zelevinsky’s cluster algebras. then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of ...

The left part LA of the module category of an artin algebra A consists of all indecomposables whose predecessors have projective dimension at most one. In this paper, we study the Auslander-Reiten components of A (and of its left support Aλ) which intersect LA and also the class E of the indecomposable Ext-injectives in the addditive subcategory addLA generated by LA.

Let K be a locally compact p-adic field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We investigate the category Cg of coadmissible modules over the p-adic Arens-Michael envelope Û(g) of U(g). Let p ⊆ g be a parabolic subalgebra. The main result gives a canonical equivalence between the classical parabolic BGG category of g relative to p and a certain expli...

Green's theorem states that the Hall algebra of category representations a quiver over finite field is twisted bialgebra. Considering instead categories orthogonal or symplectic leads to class modules algebra, called modules, which are also comodules. A module theoretic analogue theorem, describing compatibility and comodule structures, not known. In this paper we prove analogues in degenerate ...

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi-Yau triangulated categories, in particular stable categories of maximal Cohen-Macaul...

We investigate the (unbounded) derived category of a diierential Z-graded category (=DG category). As a rst application, we deduce a 'triangulatedanaloguè (4.3) of a theorem of Freyd's 5, Ex. 5.3 H] and Gabriel's 6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem`(8.2) generalizing results of 19] and 2...

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to pr...