On Simple Connectedness of Minimal Representation-Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Purely infinite -algebras associated to étale groupoids

Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if all the non-zero positive elements of $C_{0}(G^{(0)})$ are infinite in $C_{r}^{\ast }(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if every non-zero projection in $C_{0}(G^{(0)})$ is infinite in $C_{r}^{\ast }(G)$. We then show how this result applies to $k$-graph $C^{\ast }$-algebras. Finally, we investigate strongly purely infinite groupoid $C^{\ast }$-algebras.

ON THE GEOMETRY OF ORBIT CLOSURES FOR REPRESENTATION-INFINITE ALGEBRAS

For the Kronecker algebra, Zwara found in [14] an example of a module whose orbit closure is neither unibranch nor Cohen-Macaulay. In this paper, we explain how to extend this example to all representation-infinite algebras with a preprojective component.