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Zbl 1131.17006
Geiss, Christof; Leclerc, Bernard; Schröer, Jan
Semicanonical bases and preprojective algebras. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 38, No. 2, 193-253 (2005). ISSN 0012-9593

Summary: 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 $\rho _{Z^{\prime}}$ and $\rho_{Z^{\prime\prime}}$ is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components $Z^{\prime}$ and $Z^{\prime\prime}$ 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 $\Bbb A_n$ with $n\leqslant 4$. Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type $\Bbb A_5$. 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 $\Bbb E_8^{(1,1)}$. This article is the first of a series by the authors investigating the connections between preprojective algebras, universal enveloping algebras and cluster algebras [Nagoya Math. J. 182, 241--258 (2006; Zbl 1137.17021), Invent. Math. 165, No. 3, 589--632 (2006; Zbl 1167.16009), J. Lond. Math. Soc., II. Ser. 75, No. 3, 718--740 (2007; Zbl 1135.17007), Ann. Inst. Fourier 58, No. 3, 825--876 (2008; Zbl 1151.16009), Rigid modules over preprojective algebras. II: The Kac-Moody case'', preprint, arxiv.org/abs/math/0703039].

MSC 2000:: *17B37 Quantum groups and related deformations
16G20 Representations of quivers and partially ordered sets

Citations: Zbl 1137.17021; Zbl 1167.16009; Zbl 1135.17007; Zbl 1151.16009

Cited in: Zbl 1132.17004

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