Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

Gekhtman, Michael; Наканиши, Томоки; Rupel, Dylan

doi:10.1093/integr/xyx005

Cited by 14 publications

(12 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Throughout this paper, we shall consider cluster algebras with geometric coefficients in the sense of [FZ07]. The cluster algebra we defined is the same as in [FZ07], following the nice presentation of [GNR17]. Furthermore, our convention is compatible with the different formalism [GNR17] [GHK15], so that we can easily use results and arguments form these works.…”

Section: Preliminariesmentioning

confidence: 99%

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Fan¹

2017

Duke Math. J.

123

View full text Add to dashboard Cite

Abstract. We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parametrized by tropical points as expected in the Fock-Goncharov conjecture.As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez-Leclerc and Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Fan¹

2017

Duke Math. J.

123

View full text Add to dashboard Cite

show abstract

“…Let us recall the Hamiltonian formalism for the mutation maps following [GNR17] (see also [FG09a] [GHKK18]). Recall that the Euler dilogarithm function is given by…”

Section: Mutation Birational Maps Denote [ ]mentioning

confidence: 99%

Twist automorphisms and Poisson structures

Kimura¹,

Fan²,

Wei³

2022

Preprint

View full text Add to dashboard Cite

We introduce twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their compatibility with Poisson structures. The twist automorphisms always permute wellbehaved bases for cluster algebras. We explicitly construct Poisson twist automorphisms of Donaldson-Thomas type and for principal coefficients. Contents1. Introduction 1.1. Background 1.2. Main results 1.3. Contents 1.4. Convention 2. Preliminaries 2.1. Seeds and tori 2.2. Poisson structures 2.3. Mutation birational maps 2.4. Cluster algebras 2.5. Cluster varieties 2.6. Transition matrices 2.7. Cluster expansions, g-vectors, c-vectors 2.8. Canonical mutation signs 2.9. Principal coefficients 2.10. Degrees and pointedness 2.11. Injective-reachable 3. Twist endomorphisms for upper cluster A-algebras 3.1. Similar seeds and variation maps 3.2. Existence of variation maps 3.3. Change of seeds 3.4. Twist endomorphisms 4. Twist endomorphisms for upper cluster X-algebras 4.1. Variation maps 4.2. Change of seeds 4.3. Twist endomorphisms 1 2 YOSHIYUKI KIMURA, FAN QIN, AND QIAOLING WEI 5. Poisson twist automorphisms in special cases 31 5.1. Twist automorphism of Donaldson-Thomas type 31 5.2. Twist automorphism for Principal coefficients 33 6. Permutations on bases 35 6.1. Bases for U A 35 6.2. Bases for U X 36 7. Examples 38 7.1. Twist automorphisms on unipotent cells 38 7.2. Dehn twists for surface cases 39 7.3. Once-punctured digon 40 8. Quantization 42 8.1. Quantum upper cluster •-algebras 42 8.2. Quantum twist endomorphisms 42 References 43

show abstract

“…It was shown in [9] that each single mutation can be written as a nite-time evolution of a continuous Hamiltonian ow (see also [11] for a similar phenomenon). A simple computation shows that the same holds for µ 1 and µ 2 individually in our general situation.…”

Section: Conjectures and Further Questionsmentioning

confidence: 99%

Discrete Dynamical Systems From Real Valued Mutation

Machacek¹,

Ovenhouse²

2021

Preprint

View full text Add to dashboard Cite

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the 2-dimensional situation.

show abstract

Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

Cited by 14 publications

References 29 publications

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Twist automorphisms and Poisson structures

Discrete Dynamical Systems From Real Valued Mutation

Contact Info

Product

Resources

About