Cluster tilting and complexity
Petter Andreas Bergh, Steffen Oppermann
TL;DR
This paper investigates the complexity of pairs of objects in cluster categories, revealing that the maximum complexity depends on the algebra's representation type, with specific bounds for finite, tame, and wild types.
Contribution
It establishes bounds on the maximal complexity in cluster categories and derived categories, linking these bounds to the algebra's representation type.
Findings
Maximal complexity is 1, 2, or infinite depending on the hereditary algebra type.
In cluster tilted algebras of finite or tame type, maximal complexity is 0 or 1.
Complexity bounds are characterized for different algebra classes.
Abstract
We study the notion of positive and negative complexity of pairs of objects in cluster categories. The first main result shows that the maximal complexity occurring is either one, two or infinite, depending on the representation type of the underlying hereditary algebra. In the the second result, we study the bounded derived category of a cluster tilted algebra, and show that the maximal complexity occurring is either zero or one whenever the algebra is of finite or tame type.
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