Semicanonical bases and preprojective algebras

By C. Geiss, Bernard Leclerc, J. Schröer. 2004

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Abstract We study the multiplicative properties of the dual of Lusztig's semicanonical basis. The elements of this basis are naturally indexed by the irreducible components of Lusztig's nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras. We prove that the product of two dual semicanonical basis vectors ρ Z ′ and ρ Z ″ is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components Z ′ and Z ″ is again an irreducible component. It follows that the semicanonical basis and the canonical basis coincide if and only if we are in Dynkin type A n with n ⩽ 4 . Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type A 5 . We show that in this case the multiplicative properties of the dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type ( 6 , 3 , 2 ) and by the corresponding elliptic root system of type E 8 ( 1 , 1 ) .

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a2de22040b25a8fcd5f792f78faded30701b7c0e

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Citations in the corpus

Cluster algebra structures and semicanonical bases for unipotent groups (2007, background): “re to a cluster structure on C[B− K\G] (see [GLS6, §10]). 23. Open problems 23.1. It is known that the dual canonical basis B∗ and the dual semicanonical basis S∗ of M∗ ≡U(n)∗ gr do not coincide, see [GLS1]. But one might at least hope that both bases have some interesting elements in common: 118 CH”
Kac–Moody groups and cluster algebras (2010, unknown): “Λd →C be the map which sends X ∈Λd to χc(F i,X). It follows from the deﬁnition of ⋆that d i = 1i1 ⋆···⋆1im. The C-vector space Md is spanned by the maps d i. We have δX(d i) = χc(F i,X). Theorem 2.6 ([GLS1]). For X,Y ∈nil(Λ) we have δXδY = δX⊕Y . In [GLS4] a more complicated formula than the one in ”
Cluster algebras and triangulated surfaces. Part I: Cluster complexes (2006, background): “The connection between cluster combinatorics and extended affine root systems was first noticed by Geiss, Leclerc, and Schröer [21].”
Cluster characters for cluster categories with infinite-dimensional morphism spaces (2010, unknown): “Schröer in [16], [17], [18] and [20] where the authors use the category of modules over preprojective algebras of acyclic type.”
n-angulated categories (2010, result): “act sequences and thus they also have their shadows in the derived category, for example in the form of the higher octahedra described in Remark 1.1.14 of [4]. Cluster tilting theory (as developed in [2, 12, 20] and many other articles) and in particular Iyama’s higher Auslander-Reitentheory (see [1”
On the cluster category of a marked surface without punctures (2010, background): “Cluster categories associated to quivers are object of intense investigation (see for example [Geiss et al. 2005; Caldero et al. 2006; Buan et al. 2006; 2008; Caldero and Chapoton 2006; Geiß et al. 2006; Assem et al. 2008]) and are understood explicitly in terms of representations of the quiver.”
On differential graded categories (2006, background): “[6], [26], [55]) to the study of cluster algebras [47], [48], [13], [49].”
Auslander–Reiten theory revisited (2008, background): “More generally, we have the following result [47, 45, 61].”
Cluster algebras and quantum affine algebras (2009, background): “uster monomials of Aℓ are classes of simple objects of Cℓ. To ﬁnish the proof of Conjecture 13.2 one would still have to e xplain why all classes of real simple objects are cluster monomials. 13.8 In [GLS1] it was shown that if N is of type Ar (r 6 4)the dual canonical and dual semicanonical basis o”
Representations of Algebras and Related Topics (2011, background): “ot of ingredients: the quantum aﬃne Schur-Weyl duality, Ariki’s theorem for type Aaﬃne Hecke algebras [1], the coincidence of Lusztig’s dual canonical and dual semicanonical bases of C[N] in type A 4 [18], and the results of [19] on cluster algebras and dual semicanonical bases. This proof could be ”
Generic Variables in Acyclic Cluster Algebras (2010, background): “ that for d,d′ ∈ NQ0, if Ext1 CQ (d,d′) = 0 then, Ext1 kQ(d,d ′) = 0 and Ext1 kQ(d ′,d) = 0. We now prove an analogue of Geiss-Leclerc-Schröer multiplicative property for the dual semicanonical basis [GLS05] : Lemma 3.5. Let Q be an acyclic quiver. Fix d,d′ ∈ ZQ0 and assume that Ext1 CQ (d,d′) = 0. ”