Quantization of Deformed Cluster Poisson Varieties
Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties ) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42 (6), 865–930 2009 ). Meanwhile, families of deformations of cluster X -varieties were introduced in Bossinger et al. (Compos. Math. 156...
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| Published in | Algebras and representation theory Vol. 27; no. 1; pp. 381 - 427 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.02.2024
Springer Nature B.V |
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| Abstract | Fock and Goncharov described a quantization of cluster
X
-varieties (also known as
cluster Poisson varieties
) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér.
42
(6), 865–930
2009
). Meanwhile, families of deformations of cluster
X
-varieties were introduced in Bossinger et al. (Compos. Math.
156
(10), 2149–2206,
2020
). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of
X
-varieties to the families of Bossinger et al. (Compos. Math.
156
(10), 2149–2206,
2020
). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of
A
-varieties (Berenstein and Zelevinsky, Adv. Math.
195
(2), 405–455,
2005
). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci.
111
(27), 9712–9716,
2014
), we compute a counter-example to quantum positivity of the quantum theta basis. |
|---|---|
| AbstractList | Fock and Goncharov described a quantization of cluster X-varieties (also known as cluster Poisson varieties) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42(6), 865–930 2009). Meanwhile, families of deformations of cluster X-varieties were introduced in Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of X-varieties to the families of Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A-varieties (Berenstein and Zelevinsky, Adv. Math. 195(2), 405–455, 2005). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci. 111(27), 9712–9716, 2014), we compute a counter-example to quantum positivity of the quantum theta basis. Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties ) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42 (6), 865–930 2009 ). Meanwhile, families of deformations of cluster X -varieties were introduced in Bossinger et al. (Compos. Math. 156 (10), 2149–2206, 2020 ). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of X -varieties to the families of Bossinger et al. (Compos. Math. 156 (10), 2149–2206, 2020 ). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A -varieties (Berenstein and Zelevinsky, Adv. Math. 195 (2), 405–455, 2005 ). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci. 111 (27), 9712–9716, 2014 ), we compute a counter-example to quantum positivity of the quantum theta basis. |
| Author | Frías-Medina, Juan Bosco Magee, Timothy Cheung, Man-Wai Mandy |
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| Keywords | Cluster varieties Quantum cluster algebras Poisson structure Toric degenerations |
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| Snippet | Fock and Goncharov described a quantization of cluster
X
-varieties (also known as
cluster Poisson varieties
) in Fock and Goncharov (Ann. Sci. Éc. Norm.... Fock and Goncharov described a quantization of cluster X-varieties (also known as cluster Poisson varieties) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér.... |
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| SubjectTerms | Associative Rings and Algebras Clusters Commutative Rings and Algebras Deformation Mathematics Mathematics and Statistics Non-associative Rings and Algebras |
| Title | Quantization of Deformed Cluster Poisson Varieties |
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