Let Q be a finite quiver without oriented cycles, and let $Łambda$ 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 $mod(Łambda)$. 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 $S^* A(C_M)$ 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 $N^w := N (B_-wB_-)$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.
%0 Generic
%1 GeLeSc08-1
%A Geiß, Christof
%A Leclerc, Bernard
%A Schröer, Jan
%D 2008
%K (math.RA) (math.RT), Algebras Representation Rings Theory and
%T Cluster algebra structures and semicanoncial bases for unipotent groups
%U http://arxiv.org/abs/math/0703039
%X Let Q be a finite quiver without oriented cycles, and let $Łambda$ 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 $mod(Łambda)$. 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 $S^* A(C_M)$ 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 $N^w := N (B_-wB_-)$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.
@misc{GeLeSc08-1,
abstract = {Let Q be a finite quiver without oriented cycles, and let $\Lambda$ 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 $mod(\Lambda)$. 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 $N^w := N \cap (B_-wB_-)$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.},
added-at = {2009-01-26T10:26:14.000+0100},
author = {Gei{\ss}, Christof and Leclerc, Bernard and Schr{\"o}er, Jan},
biburl = {https://www.bibsonomy.org/bibtex/2c8d93e83c7af998c0304528c46897a54/demonet},
interhash = {da732992f9d9f17d23ddaf8e04c42714},
intrahash = {c8d93e83c7af998c0304528c46897a54},
keywords = {(math.RA) (math.RT), Algebras Representation Rings Theory and},
month = {April},
pdf = {http://arxiv.org/pdf/math/0703039},
timestamp = {2009-01-26T10:26:14.000+0100},
title = {Cluster algebra structures and semicanoncial bases for unipotent groups},
url = {http://arxiv.org/abs/math/0703039},
v1descr = {Thu, 1 Mar 2007 17:35:26 GMT (40kb)},
v1url = {http://www.arxiv.org/abs/math/0703039v1},
v2descr = {Mon, 7 Apr 2008 13:36:26 GMT (94kb)},
v2url = {http://www.arxiv.org/abs/math/0703039v2},
v3descr = {Wed, 23 Apr 2008 10:33:29 GMT (96kb)},
year = 2008
}