Taft algebra actions on preprojective algebras

TL;DR

The paper addresses finite-dimensional quantum symmetry by classifying inner-faithful actions of generalized Taft algebras $T_\lambda(r,m)$ on preprojective algebras $\Pi_Q$ of extended Dynkin quivers $Q=\widetilde{A_{n-1}}$. The authors distinguish rotation and reflection actions of the grouplike element, and provide a complete parametric framework in terms of data $(\mu_i,\mu_i^*,\gamma_i,\sigma)$ with descent constraints to $\Pi_Q$; explicit formulas are derived for both rotation and reflection cases, including special subcases such as when $n=4$, $d=2$. They also analyze invariants under rotation actions, showing that in favorable coprimality settings the invariant ring $ (\Pi_Q)^T $ coincides with the center $Z(\Pi_Q)$. Overall, the work extends finite-dimensional quantum symmetry from commutative polynomial rings to noncommutative Calabi--Yau algebras of global dimension two, enriching the landscape of Hopf actions on nonconnected graded algebras.

Abstract

We classify actions of generalized Taft algebras on preprojective algebras of extended Dynkin quivers of type $A$. This may be viewed as an extension of the problem of classifying actions on the polynomial ring in two variables. In cases where the grouplike element acts via rotation on the underlying quiver, we compute invariants of the Taft action and, in certain cases, show that the invariant ring is isomorphic to the center of the preprojective algebra.

Theorems & Definitions (47)

• Proposition 2.1
• Definition 2.2
• Theorem 2.3
• Corollary 2.5
• proof
• Definition 2.6
• Lemma 2.8
• proof
• Lemma 2.9
• proof