Noncommutative Polygonal Cluster Algebras
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arXiv
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| Format: | Preprint |
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2024
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| author | Greenberg, Zachary Kaufman, Dani Niemeyer, Merik Wienhard, Anna |
| author_facet | Greenberg, Zachary Kaufman, Dani Niemeyer, Merik Wienhard, Anna |
| contents | We define a new family of noncommutative generalizations of cluster algebras called polygonal cluster algebras. These algebras generalize the noncommutative surfaces of Berenstein-Retakh, and are inspired by the emerging theory of $Θ$-positivity for the groups $\mathrm{Spin}(p,q)$. They are generated by mutations of quivers which we call ST-compatible, and which encode the order of the products that appear in the exchange relations. We show that these ST-compatible quivers can be represented by tilings of surfaces by polygons, a generalization of the description of surface type cluster algebras. As examples, we construct tilings which produce ST-compatible versions of the Del Pezzo quivers and the quivers first described by Le for Fock-Goncharov coordinates for Lie groups of type $B$. We show that polygonal cluster algebras have natural evaluations in Clifford algebras, which we use to produce noncommutative generalizations of the Somos sequences and to parameterize the $Θ$-positive semigroup of $\mathrm{Spin}(2,n)$. We indicate how this will be done for the semigroup in $\mathrm{Spin}(p,q)$ and how one will give coordinates for general $Θ$-positive representations into $\mathrm{Spin}(p,q)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_08813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noncommutative Polygonal Cluster Algebras Greenberg, Zachary Kaufman, Dani Niemeyer, Merik Wienhard, Anna Representation Theory Combinatorics Rings and Algebras 13F60 (Primary), 58B34, 22E40 (Secondary) We define a new family of noncommutative generalizations of cluster algebras called polygonal cluster algebras. These algebras generalize the noncommutative surfaces of Berenstein-Retakh, and are inspired by the emerging theory of $Θ$-positivity for the groups $\mathrm{Spin}(p,q)$. They are generated by mutations of quivers which we call ST-compatible, and which encode the order of the products that appear in the exchange relations. We show that these ST-compatible quivers can be represented by tilings of surfaces by polygons, a generalization of the description of surface type cluster algebras. As examples, we construct tilings which produce ST-compatible versions of the Del Pezzo quivers and the quivers first described by Le for Fock-Goncharov coordinates for Lie groups of type $B$. We show that polygonal cluster algebras have natural evaluations in Clifford algebras, which we use to produce noncommutative generalizations of the Somos sequences and to parameterize the $Θ$-positive semigroup of $\mathrm{Spin}(2,n)$. We indicate how this will be done for the semigroup in $\mathrm{Spin}(p,q)$ and how one will give coordinates for general $Θ$-positive representations into $\mathrm{Spin}(p,q)$. |
| title | Noncommutative Polygonal Cluster Algebras |
| topic | Representation Theory Combinatorics Rings and Algebras 13F60 (Primary), 58B34, 22E40 (Secondary) |
| url | https://arxiv.org/abs/2410.08813 |