Cluster algebra structures and semicanonical bases for unipotent groups

Christof Geiss; Bernard Leclerc; Jan Schröer

paper

Cluster algebra structures and semicanonical bases for unipotent groups

arXiv:math/0703039

Abstract

Let Q be a finite quiver without oriented cycles, and let $Λ$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_{M}$ of $m o d (Λ)$ . We show that $C_{M}$ is a Frobenius category,and that its stable category is a Calabi-Yau category of dimension 2. Then we develop a theory of mutations of maximal rigid objects of $C_{M}$ , analogous to the mutations of clusters in Fomin and Zelevinsky's theory of cluster algebras. We show that $C_{M}$ yields a categorification of a cluster algebra $A (C_{M})$ , which is not acyclic in general. We give a realization of $A (C_{M})$ as a subalgebra of the graded dual of the enveloping algebra $U(\n)$, where $\n$ is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra $\g$ associated to the quiver Q. Let $S^{*}$ be the dual of Lusztig's semicanonical basis $S$ of $U(\n)$. We show that all cluster monomials of $A (C_{M})$ belong to $S^{*}$ , and that $S^{*} \cap A (C_{M})$ is a basis of $A (C_{M})$ . Next, we prove that $A (C_{M})$ is naturally isomorphic to the coordinate ring of the finite-dimensional unipotent subgroup $N (w)$ of the Kac-Moody group $G$ attached to $\g$. Here w = w(M) is the adaptable element of the Weyl group of $\g$ which we associate to each terminal representation M of Q. Moreover, we show that the cluster algebra obtained from $A (C_{M})$ by formally inverting the generators of the coefficient ring is isomorphic to the coordinate ring of the unipotent cell $N^{w} := N \cap (B_{-} w B_{-})$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.

Some minor typos corrected. Problem 23.1 of v2 is now solved (see Sections 22.8, 22.9), 121 pages. v4: typo in arxiv title corrected

Cluster algebra structures and semicanonical bases for unipotent groups

Abstract

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