---
_id: '63135'
abstract:
- lang: eng
  text: 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.
author:
- first_name: Daniel
  full_name: Perniok, Daniel
  id: '64242'
  last_name: Perniok
citation:
  ama: Perniok D. Coxeter-Dynkin algebras of canonical type. <i>Journal of Pure and
    Applied Algebra</i>. 2026;230(5). doi:<a href="https://doi.org/10.1016/j.jpaa.2026.108250">10.1016/j.jpaa.2026.108250</a>
  apa: Perniok, D. (2026). Coxeter-Dynkin algebras of canonical type. <i>Journal of
    Pure and Applied Algebra</i>, <i>230</i>(5). <a href="https://doi.org/10.1016/j.jpaa.2026.108250">https://doi.org/10.1016/j.jpaa.2026.108250</a>
  bibtex: '@article{Perniok_2026, title={Coxeter-Dynkin algebras of canonical type},
    volume={230}, DOI={<a href="https://doi.org/10.1016/j.jpaa.2026.108250">10.1016/j.jpaa.2026.108250</a>},
    number={5}, journal={Journal of Pure and Applied Algebra}, author={Perniok, Daniel},
    year={2026} }'
  chicago: Perniok, Daniel. “Coxeter-Dynkin Algebras of Canonical Type.” <i>Journal
    of Pure and Applied Algebra</i> 230, no. 5 (2026). <a href="https://doi.org/10.1016/j.jpaa.2026.108250">https://doi.org/10.1016/j.jpaa.2026.108250</a>.
  ieee: 'D. Perniok, “Coxeter-Dynkin algebras of canonical type,” <i>Journal of Pure
    and Applied Algebra</i>, vol. 230, no. 5, 2026, doi: <a href="https://doi.org/10.1016/j.jpaa.2026.108250">10.1016/j.jpaa.2026.108250</a>.'
  mla: Perniok, Daniel. “Coxeter-Dynkin Algebras of Canonical Type.” <i>Journal of
    Pure and Applied Algebra</i>, vol. 230, no. 5, 2026, doi:<a href="https://doi.org/10.1016/j.jpaa.2026.108250">10.1016/j.jpaa.2026.108250</a>.
  short: D. Perniok, Journal of Pure and Applied Algebra 230 (2026).
date_created: 2025-12-16T14:20:09Z
date_updated: 2026-04-23T13:00:59Z
department:
- _id: '602'
doi: 10.1016/j.jpaa.2026.108250
external_id:
  arxiv:
  - '2509.17887'
intvolume: '       230'
issue: '5'
language:
- iso: eng
publication: Journal of Pure and Applied Algebra
status: public
title: Coxeter-Dynkin algebras of canonical type
type: journal_article
user_id: '64242'
volume: 230
year: '2026'
...
---
_id: '66859'
abstract:
- lang: eng
  text: Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear
    $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts
    that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A})
    \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such categories induces
    an algebra isomorphism of the corresponding double Hall algebras $\mathsf{DH}_\mathcal{A}
    \!\longrightarrow\! \mathsf{DH}_\mathcal{B}$. It turns out that a counting formula
    for certain distinguished triangles in $\mathcal{D}^b(\mathcal{A})$, on which
    Cramer's proof relies, is incorrect in general. We give a corrected proof of Cramer's
    theorem which preserves the overall strategy of his approach.
author:
- first_name: Igor
  full_name: Burban, Igor
  id: '72064'
  last_name: Burban
- first_name: Daniel
  full_name: Perniok, Daniel
  id: '64242'
  last_name: Perniok
citation:
  ama: Burban I, Perniok D. Double Hall algebras and derived equivalences revisited.
    Published online 2026.
  apa: Burban, I., &#38; Perniok, D. (2026). <i>Double Hall algebras and derived equivalences
    revisited</i>.
  bibtex: '@article{Burban_Perniok_2026, title={Double Hall algebras and derived equivalences
    revisited}, author={Burban, Igor and Perniok, Daniel}, year={2026} }'
  chicago: Burban, Igor, and Daniel Perniok. “Double Hall Algebras and Derived Equivalences
    Revisited,” 2026.
  ieee: I. Burban and D. Perniok, “Double Hall algebras and derived equivalences revisited.”
    2026.
  mla: Burban, Igor, and Daniel Perniok. <i>Double Hall Algebras and Derived Equivalences
    Revisited</i>. 2026.
  short: I. Burban, D. Perniok, (2026).
date_created: 2026-08-26T12:08:41Z
date_updated: 2026-08-26T12:15:11Z
external_id:
  arxiv:
  - '2608.24331'
language:
- iso: eng
status: public
title: Double Hall algebras and derived equivalences revisited
type: preprint
user_id: '64242'
year: '2026'
...