Preprojective algebras and Lie theory

Bernard Leclerc

The preprojective algebra Λ of a quiver Q was introduced by Gelfand and Ponomarev (as a tool for studying the preprojective representations of Q). Lusztig has shown that Λ is deeply related to the positive part n of the symmetric Kac-Moody algebra attached to Q. In particular he has given a geometric description of the enveloping algebra U(n) in terms of constructible functions on varieties of Λ-modules. This yields a new basis of U(n) called the semicanonical basis.

Assume that Q is a Dynkin quiver. Dualizing Lusztig's approach, one gets a natural map φ from mod(Λ) to C[N] (where N is the unipotent group with Lie algebra n) and a dual semicanonical basis of C[N].

The algebra C[N] has a cluster algebra structure discovered by Berenstein, Fomin and Zelevinsky. In our joint work with Geiss and Schröer, we have shown that this cluster structure can be understood as the projection via φ of a similar structure in the category mod(Λ). This implies that the cluster monomials belong to the dual canonical basis of C[N], and thus are linearly independent as conjectured by Fomin and Zelevinsky.

Replacing mod(Λ) by appropriate 2-Calabi-Yau subcategories, the same approach allows to obtain new cluster algebras structures on coordinate rings of unipotent radicals of parabolic subgroups, and on multi-homogeneous coordinate rings of partial flag manifolds.

The plan of the 3 lectures will be as follows:

1) Lusztig's construction of U(n) and the semicanonical basis.

2) The cluster structure of C[N] and its categorification.

3) Coordinate rings of unipotent radicals and partial flag varieties.