#### DMCA

## The multiplication theorem and bases in finite and affine quantum cluster algebras

Citations: | 3 - 2 self |

### Citations

355 |
Tame algebras and integral quadratic forms
- Ringel
- 1984
(Show Context)
Citation Context ...r in the sense of [23],[7] and [10], respectively. 5.2. The case in affine types. An affine quiver is an acyclic quiver whose underlying diagram in an extended Dynkin diagram. One can refer to [12][8]=-=[20]-=- for the theory of representations of affine quivers. We recall some useful background concerning representation theory of affine quivers. In this section we always assume that Q is an affine quiver w... |

171 | From triangulated categories to cluster algebras
- Caldero, Keller
(Show Context)
Citation Context ...um cluster algebras of finite and affine types are constructed. Under the specialization q and coefficients to 1, these bases are the integral bases of cluster algebra of finite and affine types (see =-=[4]-=- and [11]). 1. Introduction Quantum cluster algebras were introduced by A. Berenstein and A. Zelevinsky [2] as a noncommutative analogue of cluster algebras [13][14] to study canonical bases. A quantu... |

153 |
Indecomposable Representations of Graphs and Algebras,
- Dlab, Ringel
- 1976
(Show Context)
Citation Context ...r quiver in the sense of [23],[7] and [10], respectively. 5.2. The case in affine types. An affine quiver is an acyclic quiver whose underlying diagram in an extended Dynkin diagram. One can refer to =-=[12]-=-[8][20] for the theory of representations of affine quivers. We recall some useful background concerning representation theory of affine quivers. In this section we always assume that Q is an affine q... |

135 | Cluster algebras as Hall algebras of quiver representations
- Caldero, Chapoton
(Show Context)
Citation Context ...omin and A. Zelevinsky [13][14]. Cluster algebras have a close link to quiver representations via cluster categories invented in [1]. The link is explicitly characterized by the Caldero-Chapoton map (=-=[3]-=-) and the Caldero-Keller multiplication theorems ([4],[5]). The Caldero-Chapoton map associates the objects in the cluster categories to some Laurent polynomials, in particular, sends rigid objects to... |

113 |
Tilting theory and cluster combinatorics’, Adv
- Buan, Marsh, et al.
(Show Context)
Citation Context ...ter algebras are exactly cluster algebras which were introduced by S. Fomin and A. Zelevinsky [13][14]. Cluster algebras have a close link to quiver representations via cluster categories invented in =-=[1]-=-. The link is explicitly characterized by the Caldero-Chapoton map ([3]) and the Caldero-Keller multiplication theorems ([4],[5]). The Caldero-Chapoton map associates the objects in the cluster catego... |

111 |
algebras, hereditary algebras and quantum groups
- Green, Hall
- 1995
(Show Context)
Citation Context ...ation, which is a generalization of [19, Proposition 5.3.2]. Theorem 3.3. Let M and N be kQ-modules. Then q[M,N ] 1 XNXM = q − 1 2 Λ((Ĩ−R̃)m,(Ĩ−R̃)n) ∑ E εEMNXE . Proof. We apply Green’s formula in =-=[15]-=-∑ E εEMNF E XY = ∑ A,B,C,D q[M,N ]−[A,C]−[B,D]−〈A,D〉FMABF N CDε X ACε Y BD. Then ∑ E εEMNXE = ∑ E,X,Y εEMNq − 1 2 〈Y,X〉FEXYX B̃y−(Ĩ−R̃)e = ∑ A,B,C,D,X,Y q[M,N ]−[A,C]−[B,D]−〈A,D〉− 1 2 〈B+D,A+C〉FMABF ... |

71 | Quantum cluster algebras - Berenstein, Zelevinsky - 2005 |

64 | Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc
- Sherman, Zelevinsky
(Show Context)
Citation Context ...ied in [9]. Moreover, under the specialization q = 1, the bases in [9] induce the canonical basis, semicanonical basis and generic basis of the cluster algebra of the Kronecker quiver in the sense of =-=[23]-=-,[7] and [10], respectively. 5.2. The case in affine types. An affine quiver is an acyclic quiver whose underlying diagram in an extended Dynkin diagram. One can refer to [12][8][20] for the theory of... |

45 | Laurent expansions in cluster algebras via quiver representations
- Caldero, Zelevinsky
(Show Context)
Citation Context ...tiplication theorems are essentially important to construct integral bases of cluster algebras. Following this link, some good bases have been constructed for finite and affine cluster algebras ([4], =-=[7]-=-, [10] and [11]). Naturally, one can study the quantum analogue of the link. Recently, Rupel ([22]) defined a quantum analog of the Caldero-Chapoton map (called the quantum CalderoChapoton map) and co... |

40 | On the quiver Grassmannian in the acyclic case - Caldero, Reineke |

35 |
Hall polynomials for the representation-finite hereditary algebras,
- Ringel
- 1990
(Show Context)
Citation Context ...lgebra of finite type. It is the good bases in a cluster algebra of finite type in [4] by specializing q and coefficients to 1 and the existence of Hall polynomials for representation direct algebras =-=[21]-=-. Theorem 4.6. Let Q is a simple-laced Dynkin quiver with Q0 = {1, 2, · · · , n}. Then the set B(Q) := {XM |M =M0⊕PM [1] withM0 is kQ-module, PM projective kQ̃-module, M rigid object in CQ̃} is a ZP-b... |

34 | Quantum Cluster Variables via Serre Polynomials, eprint
- Qin
(Show Context)
Citation Context ...S MING DING AND FAN XU Abstract. We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in =-=[19]-=-. Moreover some ZP-bases in quantum cluster algebras of finite and affine types are constructed. Under the specialization q and coefficients to 1, these bases are the integral bases of cluster algebra... |

32 | Acyclic cluster algebras via Ringel-Hall algebras, Preprint available at the author’s home
- Hubery
(Show Context)
Citation Context ...lects the information and the difficulty to prove the more general quantum analog of the Caldero-Keller multiplication theorems. The main idea in the proof of the multiplication theorem is taken from =-=[17]-=-. Moreover, we construct some good ZP-bases in quantum cluster algebras of finite and affine types. By specializing q and coefficients to 1, these bases induce the good bases for cluster algebras of f... |

22 | Generic variables in acyclic cluster algebras
- Dupont
(Show Context)
Citation Context ...cation theorems are essentially important to construct integral bases of cluster algebras. Following this link, some good bases have been constructed for finite and affine cluster algebras ([4], [7], =-=[10]-=- and [11]). Naturally, one can study the quantum analogue of the link. Recently, Rupel ([22]) defined a quantum analog of the Caldero-Chapoton map (called the quantum CalderoChapoton map) and conjectu... |

15 | On a quantum analogue of the Caldero-Chapoton Formula
- Rupel
(Show Context)
Citation Context ...ollowing this link, some good bases have been constructed for finite and affine cluster algebras ([4], [7], [10] and [11]). Naturally, one can study the quantum analogue of the link. Recently, Rupel (=-=[22]-=-) defined a quantum analog of the Caldero-Chapoton map (called the quantum CalderoChapoton map) and conjectured that cluster variables could be expressed as images of indecomposable rigid objects unde... |

14 | Integral bases of cluster algebras and representations of tame quivers, arXiv:0901.1937v1 [math.RT - Ding, Xiao, et al. |

11 |
More lectures on representations of quivers
- Crawley-Boevey
- 1992
(Show Context)
Citation Context ...iver in the sense of [23],[7] and [10], respectively. 5.2. The case in affine types. An affine quiver is an acyclic quiver whose underlying diagram in an extended Dynkin diagram. One can refer to [12]=-=[8]-=-[20] for the theory of representations of affine quivers. We recall some useful background concerning representation theory of affine quivers. In this section we always assume that Q is an affine quiv... |

10 |
Universal PBW-Basis of Hall-Ringel Algebras and
- Guo, Peng
- 1997
(Show Context)
Citation Context ...M,N ] 1 XNXM . This completes the proof. Remark 3.4. Theorem 3.3 is similar to the multiplication formula in dual Hall algebras. It is reasonable to conjecture that it provides some PBW-type basis (=-=[16]-=-) in the corresponding quantum cluster algebra. Let M,N be kQ−modules and assume that dimkExt 1 kQ̃ (M,N) = dimkHomkQ̃(N, τM) = 1. Then there are two “canonical” exact sequences ε : 0 −→ N −→ E −→M −→... |

6 | Bases of the quantum cluster algebra of the Kronecker quiver. arXiv:1004. 2349v4 [math.RT
- Ding, Xu
(Show Context)
Citation Context ... with |K| = q2. In this way, we obtain a quantum cluster algebra of Kronecker type without coefficients. The multiplication and the bar-invariant bases in this algebra have been thoroughly studied in =-=[9]-=-. Moreover, under the specialization q = 1, the bases in [9] induce the canonical basis, semicanonical basis and generic basis of the cluster algebra of the Kronecker quiver in the sense of [23],[7] a... |

5 |
Equivalence between cluster categories, J. Algebra 304 (2006), 832-850
- Zhu
(Show Context)
Citation Context ...Q̃) and A|k|(Q̃ ′), denoted by Φi : A|k|(Q̃)→ A|k|(Q̃ ′). Let Σ+i : mod(Q̃) −→ mod(Q̃ ′) be the standard BGP-reflection functor and R+i : CQ̃ −→ CQ̃′ be the extended BGP-reflection functor defined by =-=[24]-=-: R+i : X 7→ Σ+i (X), if X 6≃ Si is a kQ-module, Si 7→ Pi[1], Pj [1] 7→ Pj [1], if j 6= i, Pi[1] 7→ Si. By Rupel [22], the following holds: Theorem 4.1. [22, Theorem 2.4, Lemma 5.6] For any X... |

4 | polynomials for affine quivers’, Represent. Theory 14 - Hubery, ‘Hall - 2010 |