Serre cyclotomic algebras

TL;DR

This paper introduces Serre cyclotomic algebras, a new class of differential graded algebras that generalize known algebraic structures and connect to categorical diagonalization, with applications to entropy and complexity.

Contribution

It defines Serre cyclotomic algebras, relates them to existing theories, and analyzes their properties including entropy and global complexity.

Findings

Path algebras of affine type are Serre cyclotomic.

Homologically smooth graded gentle algebras can be Serre cyclotomic.

Trivial extensions of Serre cyclotomic algebras have finite global complexity.

Abstract

We introduce a class of proper differential graded algebras which we call Serre cyclotomic. They generalize fractionally Calabi-Yau algebras and categorify de la Pe\~na's algebras of cyclotomic type. Path algebras of affine type and (higher) canonical algebras are examples of Serre cyclotomic algebras. Our definition is related to Elias-Hogancamp's theory of categorical diagonalization. We compute the categorical entropy of their Serre functors, in the sense of Dimitrov-Haiden-Katzarkov-Kontsevich, and use this to determine which graded path algebras and which homologically smooth graded gentle algebras are Serre cyclotomic. Finally, we show that trivial extension algebras of Serre cyclotomic algebras have finite global complexity.