# Algèbres Amassées Affines

Abstract : We introduce generic variables in acyclic cluster algebras $\mathcal A(Q)$. We give an explicit description of these variables in terms of AR-theory of the path algebra $kQ$ and prove that they form a $\mathbb Z$-basis in a certain class of cluster algebras, including affine type $\tilde A$.

We introduce generalized Chebyshev polynomials in order to study variables associated to regular modules. In particular, this allows to prove cluster multiplication formulas for regular modules over the path algebra $kQ$.

We give a simplified proof to a theorem of Buan, Marsh and Reiten interpreting the denominators of cluster variables in terms of tilting theory in the cluster category. We also study compatibility between the Caldero-Chapoton map and extended BGP functors.

Finally, we realize non simply laced cluster algebras as subalgebras of certain quotients of simply laced cluster algebras endowed with a group of automorphisms.
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Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00338684
Contributor : Grégoire Dupont <>
Submitted on : Friday, November 14, 2008 - 1:06:07 AM
Last modification on : Wednesday, July 8, 2020 - 12:43:13 PM
Long-term archiving on: : Tuesday, October 9, 2012 - 3:25:44 PM

### Identifiers

• HAL Id : tel-00338684, version 1

### Citation

Grégoire Dupont. Algèbres Amassées Affines. Mathématiques [math]. Université Claude Bernard - Lyon I, 2008. Français. ⟨tel-00338684⟩

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