Cluster algebra structures and semicanonical bases for unipotent groups

By C. Geiss, Bernard Leclerc, J. Schröer. 2007

98 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ac114da69e88e96d5993e1e74ed27472903832c5

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In Algebra and Number Theory

1. B. Keller
2. Osamu Iyama
3. I. Reiten
4. Bernard Leclerc
5. C. Geiss
6. J. Schröer
7. K. Yamagata
8. Andrzej Skowronski

In Geometry and Topology

1. B. Keller
2. Osamu Iyama
3. Bernard Leclerc
4. I. Reiten
5. C. Geiss
6. J. Schröer
7. K. Yamagata
8. Andrzej Skowronski

In Mathematical Physics

1. Osamu Iyama
2. B. Keller
3. Bernard Leclerc
4. I. Reiten
5. C. Geiss
6. J. Schröer
7. Philipp Lampe
8. Aslak Bakke Buan

Citations in the corpus

• On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories (2007, methodology): “Our results are inspired by and closely related to the conjectures of [8] and the results of section 15 in [ 10 ].”
• A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, methodology): “Geiß-Leclerc-Schr¨oer ([15, Theorem 11.1]) showed that if M = I ⊕ τ ( I ) where I is the direct sum of all indecomposable injective representations, then there is an equivalence of triangulated categories C M ≃ C Q between C M and the cluster category C Q as defined in [1] to be the orbit category D…”
• Frobenius categories, Gorenstein algebras and rational surface singularities (2012, methodology): “Dynkin quivers, but they also hold for Dynkin quivers. Another family of Hom-finite stably 2-Calabi–Yau Frobenius categories with clustertilting object are constructed by Geiß, Leclerc and Schr¨oer in [GLS]. Precisely, for a terminal module Mover KQ(i.e. Mis preinjective and addMis closed under takin”
• Mutation of cluster-tilting objects and potentials (2008, methodology): “(See [ GLS2 ] for an alternative approach to the construction of a subclass of these 2-CY categories).”
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, unknown): “v)(n(wi)). By multiplying v1 2 -factors, we obtain a bar-invariant basis S(w)|w ∈Nl . Then, the rational quantum cluster algebra has the bar-invariant basis {S(w)}= {κ−1S(w)}. Conjecture 8.3.2 ([FZ02][GLS07]). Assume (C,i) is of type (i). Then the quantum cluster monomials of A(i) are contained in {”
• Cluster tilting for higher Auslander algebras (2008, background): “It also played an important role from the viewpoint of higher analogue of Auslander-Reiten theory [I3, I4, I5] in the study of rigid Cohen-Macaulay modules, Calabi-Yau algebras and categories, and non-commutative crepant resolutions [BIRSc, BIRSm, BIKR, GLS1, GLS2, GLS3 , IR, IY, KR1, KR2, KV, KZ, ”
• Positroid varieties and cluster algebras (2019, background): “Some other closely related cluster structures include double Bruhat cells [BFZ05, GY16], partial flag varieties [GLS08], and unipotent groups [GLS07].”
• Cluster structures on quantum coordinate rings (2011, methodology): “[N(w)]is spanned by the functions ϕX where X goes over the objects of a certain subcategory Cw of mod(Λ). This category was introduced by Buan, Iyama, Reiten and Scott in [BIRS], and independently in [GLS4] for adaptable w. Moreover, we have proved that C[N(w)] has a cluster algebra structure in the”