Semicanonical bases and preprojective algebras

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 and is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components and 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 with \ud . Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type . 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 ffflfi and by the corresponding elliptic root system of type ffi "! #!%$ &