Coxeter-Dynkin algebras of canonical type
Daniel Perniok
TL;DR
The paper generalizes Coxeter-Dynkin algebras of canonical type beyond the algebraically closed setting by starting from squid algebras and establishing two independent derived-equivalence routes to Coxeter-Dynkin and canonical algebras. It provides explicit tilting constructions via generalized APR-tilting and one-point-extensions, connecting A (squid) to B (Coxeter-Dynkin) and to C (canonical) through detailed tilting complexes and reflection functors. It also links these algebras to tubular symbols, extended affine root systems, and Weyl groups, and proves finite representation type in the domestic case, yielding a cohesive picture that ties representation theory to geometric and group-theoretic structures. Overall, the work broadens the canonical family, offers concrete derived-equivalences, and clarifies the representation-type landscape across the domestic, tubular, and wild regimes.
Abstract
We propose a definition of Coxeter-Dynkin algebras of canonical type generalising the definition as a path algebra of a quiver. Moreover, we construct two tilting objects over the squid algebra - one via generalised APR-tilting and one via one-point-extensions and reflection functors - and identify their endomorphism algebras with the Coxeter-Dynkin algebra. This shows that our definition gives another representative in the derived equivalence class of the squid algebra, and hence of the corresponding canonical algebra. Finally, we have a closer look at the Grothendieck group and the Euler form which illustrates the connection to Saito's classification of marked extended affine root systems. On the other hand, this enables us to prove that in the domestic case Coxeter-Dynkin algebras are of finite representation type.