Kac–Moody groups and cluster algebras

By C. Geiss, Bernard Leclerc, J. Schröer. 2010

196 citations recorded by the source corpus.

Abstract Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g . For each Weyl group element w, a subcategory C w of mod ( Λ ) was introduced by Buan, Iyama, Reiten and Scott. It is known that C w is a Frobenius category and that its stable category C w is a Calabi–Yau category of dimension two. We show that C w yields a cluster algebra structure on the coordinate ring C [ N ( w ) ] of the unipotent group N ( w ) : = N ∩ ( w − 1 N − w ) . Here N is the pro-unipotent pro-group with Lie algebra the completion n ˆ of n . One can identify C [ N ( w ) ] with a subalgebra of U ( n ) gr ⁎ , the graded dual of the universal enveloping algebra U ( n ) of n . Let S ⁎ be the dual of Lusztigʼs semicanonical basis S of U ( n ) . We show that all cluster monomials of C [ N ( w ) ] belong to S ⁎ , and that S ⁎ ∩ C [ N ( w ) ] is a C -basis of C [ N ( w ) ] . Moreover, we show that the cluster algebra obtained from C [ N ( w ) ] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C [ N w ] of regular functions on the unipotent cell N w of the Kac–Moody group with Lie algebra g . We obtain a corresponding dual semicanonical basis of C [ N w ] . As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1d2c759862b11f23c6e19c617439cd660c446a08

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Citations in the corpus

Monoidal categorification of cluster algebras (2014, background): “In a series of papers [6, 7, 8], Geiß, Leclerc and Schröer showed that the quantum unipotent coordinate algebra Aq(n(w)), associated with a symmetric quantum group Uq(g) and its Weyl group element w, has a skew-symmetric quantum cluster algebra structure whose initial cluster consists of quantum min”
Preprojective algebras and MV polytopes (2010, unknown): “morphism can be made functorial in T∈Π(Q)-mod; in other words, we believe that N(−wωin) is isomorphic to Π(Q)⊗KQ M(−wαin). This fact can probably be deduced from the work of Geiß, Leclerc and Schröer [8]. Anyway, this would generally not give a description of all the modules N(γ) with γa chamber wei”
Affine Mirković-Vilonen polytopes (2011, methodology): “Moreover, it follows from Crawley-Boevey and Schröer’s theory of canonical decomposition [18] that each Xk is the general point of an irreducible component Zk ⊆ Λ(nkδ).”
Cluster algebras and derived categories (2012, background): “He is very grateful to B. Leclerc for correcting an error in a previous version of example 4.3 and to H. Nakajima for a helpful conversation on the results of [30].”
Geometric conditions for □-irreducibility of certain representations of the general linear group over a non-archimedean local field (2016, background): “Oh is in the context of KLR algebras (named after Khovanov, Lauda and Rouquier) but it is easy to translate special cases of their results to the language of local non-archimedean groups – see §2. In [GLS11] a very elegant conjectural characterization of real representations was given by Geiss–Lecle”
Cluster structures on strata of flag varieties (2014, methodology): “ngs to Eu(mod(Λ)), we can 11 write V =Y~X where Y ∈add(⊕iQi) is the projective cover of V in mod(Λ). Using Lemma 3.2(b) again, we see that Z is a quotient of Eu(Y)∈add(Iw), so Z ∈Cw. It was proved in [GLS5] that Cw gives a categorical model for a cluster algebra structure on the unipotent cell Ne,w ”
A survey on maximal green sequences (2019, background): “quiver Q associated with the pair pR,wrq. It serves to obtain a cluster structure on the coordinate algebra of the unipotent cell associated with w in the Kac–Moody group determined by R, cf. [5] and [56]. It is closely related to the (upper) cluster structures on Bruhat cells obtained in [9]. In se”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “For example, Fac Iw and Sub /Iw are Frobenius and, moreover, stable 2-CY categories which have cluster tilting objects [14,21].”