Semicanonical bases and preprojective algebras II: A multiplication formula
By C. Geiss, Bernard Leclerc, J. Schröer. Compositio Mathematica, 2005
95 citations recorded by the source corpus.
Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a complex symmetric Kac–Moody Lie algebra. Lusztig has introduced a basis of $U(\mathfrak{n})$ called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $\mathfrak{n}$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important rôle in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky's theory of cluster algebras. It was inspired by recent results of Caldero and Keller.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/a212817af1f8ac2c6f977e25e67b48fb75466367
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Citations in the corpus
- 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.”
- Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type (2013, background): “We refer to [5,10,15] and see [20] for the detailed proofs.”
- From triangulated categories to cluster algebras (2005, background): “The proof of this proposition is inspired by [16].”
- Representations of Algebras and Related Topics (2011, background): “bjects in the module category modΠ2(∆), which helped them to categorify certain cluster algebras. Further developments of this link between cluster algebras and preprojective algebras can be found in [47], [48], [50], [51] (see also [49] for an overview). On generalized cluster categories 15 Theorem”
- Cluster algebras and representation theory (2010, background): “Before stating it, we recall that modΛ possesses a remarkable symmetry with respect to extensions, namely, Ext(1)Λ(X,Y ) is isomorphic to the dual of Ext 1 Λ(Y,X) functorially in X and Y (see [10, 25]).”
- Cluster algebra structures and semicanonical bases for unipotent groups (2007, methodology): “ show that rk(C) ≤Σ(M). If Mis a terminal KQ-module, then we prove that rk(CM) = Σ(M), see Corollary 7.4. Recall that for all X,Y ∈mod(Λ) we have dim Ext1 Λ(X,Y) = dim Ext1 Λ(Y,X), see [CB1] and also [GLS4]. Assume that Tis a rigid Λ-module in an additive subcategory C of mod(Λ) with rk(C) <∞. We”
- Kac–Moody groups and cluster algebras (2010, background): “t 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 Theorem 2.6 is given, expressing δXδY as a linear ”
- Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “We refer to [5,10,15] and see [20] for the detailed proofs.”