Cluster structures on quantum coordinate rings

By C. Geiss, Bernard Leclerc, J. Schröer. 2011

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We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac–Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory $${\fancyscript{C}_{w}}$$ of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities that can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.

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

View Paper: https://www.semanticscholar.org/paper/8b8210280974b76dd298ee7327754097c4eb37bb

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Weak separation and plabic graphs (2011, background): “et Iand J∈C; we will show that IkJ. Let Cq(G(k,n)) be the quantization of the coordinate ring of G(k,n) introduced in [1]; since pI and pJ are in a common cluster, they quasi-commute in this ring. By [6], the ring Cq(G(k,n)) is isomorphic to the ring of “quantum minors” of a k×(n−k) matrix (see [10]”