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Semicanonical bases and preprojective algebras II: A multiplication formula

Published online by Cambridge University Press:  20 September 2007

Christof Geiss
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 México D.F., México christof@math.unam.mx
Bernard Leclerc
Affiliation:
Laboratoire LMNO, Université de Caen, F-14032 Caen cedex, France leclerc@math.unicaen.fr
Jan Schröer
Affiliation:
Mathematisches Institut, Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany schroer@math.uni-bonn.de
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Abstract

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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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2007