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Quantum cluster algebra structures on quantum nilpotent algebras. (English) Zbl 1372.16040

Mem. Am. Math. Soc. 1169, viii, 116 p. (2017).
The paper under review presents a new algebraic approach to quantum cluster algebras based on noncommutative ring theory. The paper proposes a general construction of quantum cluster algebra structures on a broad class of algebras. Initial clusters and mutations are constructed in a uniform and intrinsic way, in particular, avoiding any ad hoc constructions with quantum minors.
The main theorem of the paper asserts that every algebra in a very large, axiomatically defined class of quantum nilpotent algebras admits a quantum cluster algebra structure. Furthermore, for all such algebras, the latter equals the corresponding upper quantum cluster algebra.
This theorem has a broad range of applications and the required axioms are easy to verify. Many classical families of algebras fall within this axiomatic class. In particular, an application of this theorem gives an explicit quantum cluster algebra structures on the quantum Schubert cell algebras for all finite dimensional simple Lie algebras.

MSC:

16T20 Ring-theoretic aspects of quantum groups
13F60 Cluster algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14M15 Grassmannians, Schubert varieties, flag manifolds
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