Abstract:
Let $A$ be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of $A$ with coefficients in the bimodule $A$ vanishes if and only if $A$ is representation finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of $Q$ equals the number of indecomposable non uniserial projective injective $A$-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups of $A$ vanish.

Abstract:
We give a simple combinatorial criterion allowing to recognize whether a string (or, more generally, a special biserial) algebra is a laura algebra or not. We also show that a special biserial algebra is laura if and only if it has a finite number of isomorphism classes of indecomposable modules which have projective dimension and injective dimension greater than or equal to two, solving a conjecture ok Skowronski for special biserial algebras.

Abstract:
It is shown that for the restricted Zassenhaus algebra $\mathfrak{W}=\mathfrak{W}(1,n)$, $n>1$, defined over an algebraically closed field $\mathbb{F}$ of characteristic 2 any projective indecomposable restricted $\mathfrak{W}$-module has maximal possible dimension $2^{2^n-1}$, and thus is isomorphic to some induced module $\mathrm{ind}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))$ for some torus of maximal dimension $\mathfrak{t}$. This phenomenon is in contrast to the behavior of finite-dimensional simple restricted Lie algebras in characteristic $p>3$.

Abstract:
For modules over a finite-dimensional algebra, there is a canonical one-to-one correspondence between the projective indecomposable modules and the simple modules. In this purely expository note, we take a straight-line path from the definitions to this correspondence. The proof is self-contained.

Abstract:
We recall a construction of Mackaay, Pan and Tubbenhauer of the algebras $K^{\epsilon}$ which allow to understand the $sl_3$ homology for links in a local way (i.e. for tangles). Then, by studying the combinatorics of the Kuperberg bracket, we give a large family of non-elliptic webs whose associated projective $K^{\epsilon}$-modules are indecomposable.

Abstract:
We compute the Loewy structure of the indecomposable projective modules for the group algebra FG, where G is the alternating group on 10 letters and F is an algebraically closed field of characteristic 3.

Abstract:
Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with Frobenius kernel $G_{(1)}$. It is known that when $p\ge 2h-2$, where $h$ is the Coxeter number of $G$, the projective indecomposable $G_{(1)}$-modules (PIMs) lift to $G$, and this has been conjectured to hold in all characteristics. In this paper, we explore the lifting problem via extensions of algebraic groups, following the work of Parshall and Scott who in turn build upon ideas due to Donkin. We prove various results which augment this approach, and as an application demonstrate that the PIMs lift to $G_{(1)}H$, for particular closed subgroups $H \le G$ which contain a maximal torus of $G$.

Abstract:
It is proven each ring $R$ for which every indecomposable right module is pure-projective is right pure-semisimple. Each commutative ring $R$ for which every indecomposable module is pure-injective is a clean ring and for each maximal ideal $P$, $R_P$ is a maximal valuation ring. Complete discrete valuation domain of rank one are examples of non-artinian semi-perfect rings with pure-injective indecomposable modules.

Abstract:
We describe the structure of the Grothendieck ring of projective modules of basic Hopf algebras using a positive integer determined by the composition series of the principal indecomposable projective module.

Abstract:
The paper studies lower bounds for the dimensions of projective indecomposable modules for Chevalley groups G in defining characteristic p. The main result extending earlier one by Malle and Weigel (2008) determines the modules in question of dimension equal to the order of a Sylow p-subgroup of G. We also substantially generalize a result by Ballard (1978) on lower bounds for the dimensions of projective indecomposable modules and find lower bounds in some cases where Ballard's bounds are vacuous.