Smash products of group weighted bound quivers and Brauer graphs

Abstract

Let $\mathbb{k}$ be a field, G a group, and (Q, I) a bound quiver. A map $W:Q_1\mathbf{\to }G$ is called a G-weight on Q, which defines a G-graded $\mathbb{k}$-category $\mathbb{k}(Q,W)$, and W is called homogeneous if I is a homogeneous ideal of the G-graded $\mathbb{k}$-category $\mathbb{k}(Q,W)$. Then we have a G-graded $\mathbb{k}$-category $\mathbb{k}(Q,I,W):=\mathbb{k}(Q,W)/I$. We can then form a smash product $\mathbb{k}(Q,I,W)\#G$ of $\mathbb{k}(Q,I,W)$ and G, which canonically defines a Galois covering $\mathbb{k}(Q,I,W)\#G\to \mathbb{k}(Q,I)$ with group G [we will see that all such Galois coverings to $\mathbb{k}(Q,I)$ have this form for some W]. First we give a quiver presentation $\mathbb{k}(Q_{G,W},I_{G,W})\cong \mathbb{k}(Q,I,W)\#G$ of the smash product $\mathbb{k}(Q,I,W)\#G$. Next if (Q, I, W) is defined by a Brauer graph with an admissible weight, then the smash product $\mathbb{k}(Q,I,W)\#G$ is again defined by a Brauer graph, which will be computed explicitly. The computation is simplified by introducing a concept of Brauer permutations as an intermediate one between Brauer graphs and Brauer bound quivers. This extends and simplifies the result by Green–Schroll–Snashall on the computation of coverings of Brauer graphs, which dealt with the case that G is a finite abelian group, while in our case G is an arbitrary group. In particular, it enables us to delete all cycles in Brauer graphs to transform it to an infinite Brauer tree.

Keywords:

• Coverings
• gradings
• smash products
• quiver presentations
• Brauer graphs

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 18D05
• 16W22
• 16W50

Acknowledgements

This work was announced at Workshop on Brauer Graph Algebras held in March, 2016 in Stuttgart, and completed to write as an article during my stay in Bielefeld in July–September, 2017. I would like to thank Steffen König, William Crawley-Boevey, Henning Krause, and Claus M. Ringel and all members of algebra seminars in both universities for their kind hospitality. Finally, I also would like to thank the referee for his/her careful reading of the manuscript and for some language corrections.

This work is partially supported by Grant-in-Aid for Scientific Research 25610003 and 25287001 from JSPS.

Additional information

Funding

This work is partially supported by Grant-in-Aid for Scientific Research 25610003 and 25287001 from JSPS.