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Graded cluster algebras
Rings and Algebras · Submitted 24 Sep 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases.
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The abstract, sentence by sentence
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics---namely tropical frieze patterns---on the Auslander--Reiten quivers of the categories.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. | background 0.99 | 0.04 | 0.03 | asserted 1.00 |
| s1 | In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. | contribution 0.65 | 0.35 | 0.06 | asserted 0.53 |
| s2 | We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. | contribution 0.82 | 0.93 | 0.11 | asserted 1.00 |
| s3 | We then study gradings for finite type cluster algebras without coefficients, giving a full classification. | result 0.62 | 0.95 | 0.24 | asserted 1.00 |
| s4 | Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. | contribution 0.90 | 0.93 | 0.07 | asserted 1.00 |
| s5 | This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. | result 0.69 | 0.91 | 0.18 | asserted 0.98 |
| s6 | We also obtain interesting combinatorics---namely tropical frieze patterns---on the Auslander--Reiten quivers of the categories. | result 0.97 | 0.94 | 0.11 | asserted 1.00 |
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The paper
Original abstract
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics---namely tropical frieze patterns---on the Auslander--Reiten quivers of the categories.
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How the literature uses it
6 arXiv papers cite this in 22 sentences
Papers citing it · 9 sentences without a confident label
Contests it 1 paper
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Then for all $X ∈C$, the cluster character $C_X^T∈ℂ[x_1^±1,…,x_r^±1]$ is homogeneous of degree $_G(X)$. * For any distinguished triangle $X→Y →Z →ΣX$ of $C$, we have [equation] * The degree $_G$ is compatible with mutation in the sense that for every cluster-tilting object $U$ of $C$ with indecomposable summand $U_k$ we have [equation] where $U_k^*$, $M$ and $M'$ are as in the above description of exchange triangles in $C$. * The space of gradings for a generalised cluster category $(C,T)$ may be identified with $K_0(C)$, where $K_0(C)$ is the Grothendieck group of $C$ as a triangulated category.[This statement corrects [ref] for the case $=$, which replaces $K_0(C)$ by $K_0(C)$ itself.
Graded Frobenius cluster categories · Graded triangulated cluster categories · contest 0.77
Builds on it 2 papers
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The proofs in [ref] are given for $=^m$, but remain valid in the more general setting.
Graded Frobenius cluster categories · Graded triangulated cluster categories · 0.80
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We now adapt the grading introduced in [ref] to the geometric type.
A Combinatorial Formula for Certain Elements of Upper Cluster Algebras · Grading on $𝒜$ · 0.91
Cites it as background 5 papers
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Such cluster algebras are naturally called graded cluster algebras and the general theory of these is set out in work of the first author ( [ref] ).
Segre products of cluster algebras · Introduction · 0.96
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Given a complete set of projectives, it is natural to ask whether the associated grading might be standard, as defined in [ref] ; we briefly recall this definition and some related facts.
Graded Frobenius cluster categories · Graded Frobenius cluster categories · 0.91
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* friezes from matrix multiplication: [ref] , [ref] ; * additive friezes: $a+d=b+c$ or $a+d=b+c+1$, [ref] , [ref] , [ref] , [ref] ; * tropical friezes: $a+d=max(b+c,0)$, [ref] , [ref] , [ref] , [ref] , [ref] ; or $a+d=max(b,0)+max(c,0)$, [ref] ; * quantum friezes: $ad-q^1/2bc=1$, [ref] ; * NIM friezes: $a⊞d= b⊞c +1$, [ref] , [ref] ; * cross-ratio friezes: $(b-a)(c-d)/(a-c)(d-b)=-1$, [ref] ; * continuous friezes $F(x,y)$: $F∂^2/∂x ∂yF-∂/∂xF∂/ ∂yF=1$, [ref] ; * $2$-friezes: $[ b; a e d; c ] ⇒ad-bc=e$, [ref] , [ref] , [ref] ; * super-friezes or $Osp(1|2)$-tilings, [ref] * $_k$-friezes or $_k$-tilings, [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] ; * 3D-friezes: $T$-systems, [ref] , [ref] [ref] ; * multiplicative friezes on repetition quivers or from Cartan matrices, [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] ; * friezes and combinatorial models, [ref] [ref] , [re
Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics · Friezes in the literature and open questions · 0.99
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