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Noncommutative analogues of classical operations on symmetric functions are investigated , and applied to the description of idempotents and nilpotents in descent algebras. Its is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various(More)
Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D, E. Lusztig has introduced a basis of U (n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as n. We prove a formula for the product of two elements of(More)
Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring C[N ] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of(More)
Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQ-module M (these are certain preinjective CQ-modules), we attach a natural subcategory CM of mod(Λ). We show that CM is a Frobenius category, and that its stable category C M is a Calabi-Yau category of dimension two. Then we develop a(More)