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Cluster structures on quantum coordinate rings,
 Selecta Math. (N.S.)
, 2013
"... Abstract We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric KacMoody 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 C w of the module category of t ..."
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Abstract We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric KacMoody 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 C w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a qanalogue 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.
Skein algebras and cluster algebras of marked surfaces
"... Abstract. This paper defines several algebras associated to an oriented surface Σ with a finite set of marked points on the boundary. The first is the skein algebra Skq(Σ), which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defi ..."
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Abstract. This paper defines several algebras associated to an oriented surface Σ with a finite set of marked points on the boundary. The first is the skein algebra Skq(Σ), which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results. When Σ is triangulable, the quantum cluster algebra Aq(Σ) and quantum upper cluster algebra Uq(Σ) can be defined. These are algebras coming from the triangulations of Σ and the elementary moves between them.
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, 2006
"... Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories SubQ for Q an injective Λmodule, and we introduce a mutation operation between complete rigid modules in SubQ. This yields clu ..."
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Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories SubQ for Q an injective Λmodule, and we introduce a mutation operation between complete rigid modules in SubQ. This yields cluster algebra structures on the coordinate rings of the partial
CLUSTER ALGEBRAS, REPRESENTATION THEORY, AND POISSON GEOMETRY
"... 1.1. Cluster algebras. Cluster algebras were introduced in 2000 by S. Fomin and A. Zelevinsky [26] as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables ..."
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1.1. Cluster algebras. Cluster algebras were introduced in 2000 by S. Fomin and A. Zelevinsky [26] as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. Both the generators and the relations among them are not given from the outset, but are produced by an elementary iterative process called seed mutation. This procedure appears to be closely related to constructions in various other fields, such as Poisson geometry, Teichmüller theory, representation theory of finite dimensional associative algebras and Lie theory and Coxeter groups. The theory of cluster algebras was further developed in the subsequent papers [27, 28, 3, 4, 29, 16, 17]. Remarkably, in the last two papers of this series superpotentials borrowed from mathematical physics play a prominent role. By now cluster algebras form a very active area of research which has obtained its own AMS classification number 13F60. A thematic semester in 2012 at the Mathematical Sciences Research Institute in Berkeley will be devoted to cluster algebras (more details on activities related to cluster algebras can be found on Fomin’s cluster algebra portal at