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## Triangular bases in quantum cluster algebras

Citations: | 4 - 4 self |

### Citations

353 |
Canonical bases arising from quantized enveloping algebras
- Lusztig
- 1990
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Citation Context ...rview of these approaches with relevant references can be found in [17]. In this paper we develop a new approach. It is in fact much closer to Lusztig’s original way of constructing a canonical basis =-=[15]-=- (and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra). The key ingredient of our approach is a version of Lusztig’s Lemma generalizing [7, Theorem 1.2] (see also [16]). He... |

238 | Cluster algebras - Fomin, Zelevinsky |

214 | Cluster algebras II: finite type classification - Fomin, Zelevinsky |

171 | From triangulated categories to cluster algebras
- Caldero, Keller
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Citation Context ...the order ✁ is the usual one: 1✁ 2✁ · · ·✁ n. Thus, we have (3.1) bij ≤ 0 for 1 ≤ i < j ≤ n. For a vector a = (a1, . . . , am) ∈ Z m, we abbreviate (3.2) a>n = ∑ n<i≤m aiei, a ≤n = ∑ 1≤i≤n aiei . 1In =-=[3]-=- the exchange matrix B was assumed to be skew-symmetric; the general case can be deduced from this one by the standard argument involving folding. TRIANGULAR BASES IN QUANTUM CLUSTER ALGEBRAS 9 Note t... |

144 | Cluster algebras III. Upper bounds and double Bruhat cells - Berenstein, Fomin, et al. |

109 | Hecke algebras with unequal parameters
- Lusztig
- 2003
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Citation Context ...asis [15] (and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra). The key ingredient of our approach is a version of Lusztig’s Lemma generalizing [7, Theorem 1.2] (see also =-=[16]-=-). Here is this version (a proof will be given in Section 7). Theorem 1.1. Let A be a free Z[v, v−1]-module with a basis {Ea : a ∈ L} indexed by a partially ordered set (L,≺) such that, for any a ∈ L,... |

71 | Quantum cluster algebras
- Berenstein, Zelevinsky
- 2005
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Citation Context ...atisfies (1.5) Ca − Ea ∈ ⊕a′≺avZ[v]Ea′ , hence the elements Ca for a ∈ L form a Z[v, v −1]-basis in A. We will apply Theorem 1.1 in the situation where A is a quantum cluster algebra (in the sense of =-=[1]-=-) with an acyclic quantum seed. To state the main results we need to recall some terminology and notation from [1]. Recall that a (labeled) quantum seed is specified by the following data: • Two posit... |

48 | The quantum dilogarithm and representations of quantum cluster varieties,” math/0702397
- Fock, Goncharov
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Citation Context ...ome frozen variables in an appropriate way; and then we show that the principal quantization A•(B) can also be embedded in A (2)(B). Note that the principal quantization appears (in some disguise) in =-=[8]-=- in the context of the “quantum symplectic double” (we are grateful to A. Goncharov for bringing this connection to our attention). It remains to be seen whether the above reduction procedure makes se... |

45 | Laurent expansions in cluster algebras via quiver representations
- Caldero, Zelevinsky
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Citation Context ...ons of the Kronecker quiver (see [18]). In this case the canonial triangular basis coincides with the natural quantum version of the dual semicanonical basis introduced for the commutative setting in =-=[4]-=-; this version was also discovered and studied in [14]. Proof of Proposition 6.1. The fact that all cluster monomials belong to the canonical triangular basis B in A, follows from Corollary 1.7 (in ou... |

29 | Quantum Schubert cells and representations at roots of 1, in: Algebraic groups and Lie groups - Concini, Procesi - 1997 |

26 |
Quiver varieties and quantum cluster algebras, in preparation
- Kimura, Qin
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Citation Context ...ollary 1.7. The canonical triangular basis in A contains all cluster monomials associated to acyclic quantum seeds of A. Remark 1.8. F. Qin (private communication) has informed us that the results of =-=[13]-=- imply the following: if A has an acyclic quantum seed such that the exchange matrix B is skew-symmetic, and the directed graph Γ(B) is bipartite then B contains all cluster monomials associated to qu... |

23 | A quantum cluster algebra of kronecker type and the dual canonical basis,” 1002.2762
- Lampe
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Citation Context ... less technical situation. In Section 6 we illustrate our results in the special case when m = n = 2 and B̃ = B = ( 0 −2 2 0 ) . This cluster algebra and various bases in it were studied in detail in =-=[6, 14]-=-. We make use of the calculations in [6], and we show (in Proposition 6.1) that in this special case our canonical triangular basis coincides with the dual canonical basis studied in [14], and also wi... |

20 | IC bases and quantum linear groups - Du - 1994 |

18 | Quantum unipotent subgroup and dual canonical basis,
- KIMURA
- 2012
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Citation Context ...ith respect to frozen variables of the coordinate ring of a quantum unipotent cell (in an appropriate Kac-Moody group) associated with the square of a Coxeter element. Then the technique developed in =-=[11, 12]-=- implies that, under the same assumptions as above (B skew-symmetric, and Γ(B) bipartite), the canonical triangular basis 6 ARKADY BERENSTEIN AND ANDREI ZELEVINSKY becomes identified with the dual can... |

15 | Generic bases for cluster algebras from the cluster category. arXiv:1111.4431v2 [math.RT
- Plamondon
- 2011
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Citation Context .... To this end a lot of recent activity has been directed towards various constructions of “natural” bases in cluster algebras. An overview of these approaches with relevant references can be found in =-=[17]-=-. In this paper we develop a new approach. It is in fact much closer to Lusztig’s original way of constructing a canonical basis [15] (and the pioneering construction of the Kazhdan-Lusztig basis in a... |

15 | On a quantum analogue of the Caldero-Chapoton Formula
- Rupel
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Citation Context ...studied in [14], and also with one of the bases constructed in [6] with the help of the quantum Caldero-Chapoton characters associated with indecomposable representations of the Kronecker quiver (see =-=[18]-=-). The concluding Section 7 contains the proof of Lusztig’s Lemma (Theorem 1.1). 2. Proof of Theorem 1.4 In this section we prove Theorem 1.4. We freely use the terminology and notation from Section 1... |

8 |
Cluster structures on quantum coordinate rings
- Geiss, Leclerc, et al.
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Citation Context ...xchange matrix B is skew-symmetic, and the directed graph Γ(B) is bipartite then B contains all cluster monomials associated to quantum seeds of A (not only acyclic ones). Furthermore, the results in =-=[11]-=- imply that each acyclic quantum cluster algebra A (with an appropriate choice of frozen variables) can be realized as the localization with respect to frozen variables of the coordinate ring of a qua... |

6 | Bases of the quantum cluster algebra of the Kronecker quiver. arXiv:1004. 2349v4 [math.RT
- Ding, Xu
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Citation Context ... less technical situation. In Section 6 we illustrate our results in the special case when m = n = 2 and B̃ = B = ( 0 −2 2 0 ) . This cluster algebra and various bases in it were studied in detail in =-=[6, 14]-=-. We make use of the calculations in [6], and we show (in Proposition 6.1) that in this special case our canonical triangular basis coincides with the dual canonical basis studied in [14], and also wi... |