- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:39
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2016
3
Skein and cluster algebras of marked surfaces
Greg
Muller
University of Michigan, ANN ARBOR, UNITED STATES
Cluster algebra, quantum cluster algebra, skein algebra, quantum torus, triangulation of surfaces
This paper considers several algebras associated to an oriented surface $\Sigma$ with a finite set of marked points on its boundary. The first is the skein algebra $\mathsf {Sk}_q(\Sigma)$, which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results. When $\Sigma $ is triangulable, a quantum cluster algebra $\mathcal A_q(\Sigma)$ and quantum upper cluster algebra $\mathcal U_q(\Sigma)$ can be defined. These are algebras coming from the triangulations of $\Sigma$ and the elementary moves between them. Cluster algebras have been a subject of significant recent interest, due in part to their extraordinary positivity and Laurent properties. Natural inclusions $\mathcal A_q(\Sigma) \subseteq \mathsf {Sk}_q^o (\Sigma) \subseteq \mathcal U_q(\Sigma)$ are shown, where $\mathsf {Sk}_q^o(\Sigma)$ is a certain Ore localization of $\mathsf {Sk}_q(\Sigma)$. When $\Sigma$ has at least two marked points in each component, these inclusions are strengthened to equality, exhibiting a quantum cluster structure on $\mathsf {Sk}_q^o(\Sigma)$. The method for proving these equalities has the potential to show $\mathcal A_q = \mathcal U_q$ for other classes of cluster algebras. As a demonstration of this fact, a new proof is given that $\mathcal A_q = \mathcal U_q$ for acyclic cluster algebras.
Commutative rings and algebras
General
Associative rings and algebras
Manifolds and cell complexes
435
503
10.4171/QT/79
http://www.ems-ph.org/doi/10.4171/QT/79