Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$

Ishibashi, Tsukasa; Yuasa, Wataru

doi:10.48550/arxiv.2101.00643

Cited by 3 publications

(13 citation statements)

References 35 publications

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“…Indeed, a web containing these faces can be written as a linear combination of non-elliptic webs in the skein algebra (see Section 5) and hence not needed as a basis element. The first two faces in (2.2) are excluded as variants of boundary skein relations [IY21]. It is also related to the weakly reduced condition in [FS20].…”

Section: Organization Of the Papermentioning

confidence: 99%

“…When the marked surface has no punctures (hence the exchange matrix has full-rank), it is also expected to parametrize a linear basis of the quantum upper cluster algebra O q (A sl 3 ,Σ ) of Berenstein-Zelevinsky [BZ05]. On the other hand, a skein model for O q (A sl 3 ,Σ ) is investigated in [IY21] by the first named author and W. Yuasa. They study a skein algebra S q sl 3 ,Σ with appropriate "clasped" skein relations at marked points, and constructed an inclusion of its boundary-localization S q sl 3 ,Σ [∂ −1 ] into the quantum cluster algebra (and hence into O q (A sl 3 ,Σ )).…”

Section: Introductionmentioning

confidence: 99%

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Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Ishibashi¹,

Kano²

2022

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We study the space L x sl3 (Σ, Q) of rational unbounded sl 3 -laminations on a marked surface Σ. We introduce an sl 3 -analogue of the Thurston's shear coordinates associated with any decorated triangulation, which gives rise to a natural identificationWe also introduce the space L p sl3 (Σ, Q) of rational unbounded sl 3 -laminations with pinnings, which possesses the frozen coordinates as well. Then we give a tropical anologue of the amalgamation maps [FG06b] between them, which is indeed a procedure of gluing sl 3 -laminations with "shearings". We also investigate a relation to the graphical basis of the sl 3 -skein algebra [IY21], which conjecturally leads to a quantum duality map.

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Section: Organization Of the Papermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Ishibashi¹,

Kano²

2022

Preprint

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“…In this section, we give an explicit description of the formula (4.6) in the cases G = SL 2 , SL 3 , Sp 4 , and their description in terms of the skein algebras studied in [Mul16,IY21,IY]. Let us consider a simple Wilson line g [c] , and use the notations in Section 4.4.…”

Section: Examples and Skein Descriptionmentioning

confidence: 99%

“…give rise to the same cluster for s = t. The corresponding transformations of dashed diagonals connecting the decorated flags are shown in Figure 10. Recall the skein model studied in [IY21]. For any marked surface as in Section 3.1, the first author and W. Yuasa realized the quantum cluster algebra A q sl 3 ,Σ quantizing A sl 3 ,Σ inside the skew-field of fractions of a certain sl 3 -skein algebra S q sl 3 ,Σ consisting of sl 3 -webs (i.e., oriented trivalent graphs whose vertices are either sinks or sources ), and shown the inclusion…”

Section: Thus the Wilson Line Matrix Gmentioning

confidence: 99%

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$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems

Ishibashi¹,

Oya²,

Han³

2022

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For a finite-dimensional simple Lie algebra g admitting a non-trivial minuscule representation and a connected marked surface Σ with at least two marked points and no punctures, we prove that the cluster algebra Ag,Σ associated with the pair (g, Σ) coincides with the upper cluster algebra Ug,Σ. The proof is based on the fact that the function ring O(A × G,Σ ) of the moduli space of decorated twisted G-local systems on Σ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras Sg,Σ[∂ −1 ] [Mul16, IY21, IY] for g = sl2, sl3, or sp 4 are isomorphic to the cluster algebras Ag,Σ.• Muller [Mul13] proved A = U for locally acyclic cluster algebras. When g = sl 2 and Σ is unpunctured and contains at least two marked points, the cluster algebra A sl 2 ,Σ is locally acyclic, and hence A sl 2 ,Σ = U sl 2 ,Σ .1 In [BMS19], Bucher, Machacek, and Shapiro show that the equality A = U depends on the choice of ground ring. In this paper, we always choose the ground field C. By an easy exercise of linear algebra, all the results of the paper can be generalized to Q.

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Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$

Ishibashi¹,

Yuasa²

2022

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Continuing to our previous work [IY21] on the sl 3 -case, we introduce a skein algebra S q sp 4 ,Σ consisting of sp 4 -webs on a marked surface Σ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural Z q -form S Zq sp 4 ,Σ ⊂ S q sp 4 ,Σ , while the natural coefficient ring R of S q sp 4 ,Σ includes the inverse of the quantum integer [2] q . We prove that its boundary-localizationis included into a quantum cluster algebra A q sp 4 ,Σ that quantizes the function ring of the moduli space A × Sp4,Σ . Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21]. We also propose a characterization of cluster variables in the spirit of Fomin-Pylyavksyy [FP16] in terms of the sp 4 -webs, and give infinitely many supporting examples on a quadrilateral.

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Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$

Cited by 3 publications

References 35 publications

Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates

$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$

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