nLab cluster algebra

Contents

Contents

Idea

A cluster algebra of rank nn is a commutative algebra equipped with some subsets of size nn called the clusters. Some of these clusters are related by sequences of operations called mutations. A cluster variable is an element of any of the clusters.

This theory is motivated by the total positivity phenomena in Lie theory and existance of good bases of some algebras and their representation spaces, for example canonical and dual canonical bases.

A matrix AA over the field of real numbers is totally positive (resp. totally nonnegative) if every minor (= determinant of any submatrix) is a positive (resp. nonnegative) real number. Total positivity implies a number of remarkable properties; for example all eigenvalues are distinct and positive.

George Lusztig discovered that total positivity is closely related to some phenomena in the theory of Lie groups and quantum groups. Later, Sergey Fomin and Andrei Zelevinsky studied the canonical bases for quantum groups and discovered the combinatorics of simple transformations and defined associated classical and quantum cluster algebras to such situations. In particular, Stasheff associahedra are associated to these cluster algebras. Remarkably, they found an unusual algebraic geometry related to cluster algebras, possessing new, and at the beginning mysterious, Laurent phenomenon. Later, the cluster algebras appeared also in the connection to the representations of quivers, tilting theory and the wall crossing phenomenon, with the applications in representation theory and the study of triangulated categories.

Quantum cluster algebras are noncommutative.

References

Cluster algebras

The original articles:

See also:

The connections to the exact WKB method a la Voros are studied in

  • Kohei Iwaki, Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) 474009 arxiv/1401.7094; Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv:1409.4641

Generalized cluster algebras are studied in

  • Leonid Chekhov, Michael Shapiro, Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables, Int. Math. Res. Notices (2014) 2746-2772 arxiv/1111.3963 doi
  • Tomoki Nakanishi, Structure of seeds in generalized cluster algebras, arxiv/1409.5967

Review:

Quantum cluster algebras

Given any quantum cluster algebra arising from a quantum unipotent subgroup of symmetrizable Kac-Moody type, we verify the quantization conjecture in full generality that the quantum cluster monomials are contained in the dual canonical basis after rescaling.

  • Christof Geiss, Bernard Leclerc, Jan Schröer, Cluster structures on quantum coordinate rings, Selecta Math. 19 (2013), 337–397 arxiv/1104.0531

  • Yoshiyuki Kimura, Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math. 52(2): 277–331 (2012) doi

category: algebra

Last revised on July 10, 2024 at 13:45:05. See the history of this page for a list of all contributions to it.