arXiv subjects

Daniel Perniok

Publications and source records attributed to Daniel Perniok.

2 recordsLinked to original sources

arXiv2026

Double Hall algebras and derived equivalences revisited

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.

math.RT↗

arXiv2025

Coxeter-Dynkin algebras of canonical type

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.

math.RT↗