Algebras determined by $τ$-slices

TL;DR

The paper studies algebras determined by $\tau$-slices, a framework that extends strict laura algebras via $\tau$-tilting theory. It proves that every strict laura algebra is determined by $\tau$-slices and provides a non-strict $\tau$-slice algebra, including a wild example where all indecomposables lie in a cycle. Under mild containment conditions on $Gen T_i$ and $Cogen S_j$ relative to the left/right parts, the representation dimension satisfies $rep.dim. A \leq 3$, implying the finitistic dimension is finite. The construction uses an explicit Auslander generator $L = A ⊕ DA ⊕ (⊕_{i=1}^s T_i) ⊕ (⊕_{j=1}^t S_j) ⊕ (⊕_{Y ∈ 𝒴_A} Y)$, connecting the theory to tilted algebras via $\tau$-slices.

Abstract

In this paper we revisit the notion of strict laura algebras through the lens of $τ$-tilting theory to define the family of algebras determined by $τ$-slices. We show that the representation dimension of every algebra determined by $τ$-slices satisfying mild conditions is at most three.

Figures (1)

• Figure 1: A component of the Auslander-Reiten quiver of $A$

Theorems & Definitions (30)

• Theorem 1.1: Theorem \ref{['strict laura is stric cyclic laura']}
• Theorem 1.2: Theorem \ref{['teo principal']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Theorem 2.7
• Definition 2.8