The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸˜, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸˜.

Key Words:

• Tame weakly shod algebras
• Simply connected and strongly simply connected algebras
• Orbit graphs
• First Hochschild cohomology group

Mathematics Subject Classification:

• 16G20
• 16G70

Acknowledgment

The authors wish to thank Diane Castonguay and Rosana Vargas for their useful comments. The first author gratefully acknowledges partial support from the NSERC of Canada.

Notes

^#Communicated by C. Cibils.