nLab cluster algebra

Idea

A cluster algebra of rank $n$ is a commutative algebra equipped with some subsets of size $n$ 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 $A$ 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:

• Sergey Fomin, Andrei Zelevinsky: Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 2 (2002) 497-529 [math.RT/0104151, doi:10.1090/S0894-0347-01-00385-X]

• Sergey Fomin, Andrei Zelevinsky: Cluster algebras II: Finite type classifications, Invent. Math. 154 1 (2003) 63-121 [doi:10.1007/s00222-003-0302-y, arXiv:math/0208229]

• Arkady Berenstein, Sergey Fomin, Andrei Zelevinsky: Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J. 126 1 (2005) 1-52 [doi:10.1215/S0012-7094-04-12611-9, arXiv:math/0305434]

• Sergey Fomin, Andrei Zelevinsky: Cluster algebras IV: Coefficients, Compos. Math. 143 (2007) 112-164 [doi:10.1112/S0010437X06002521, arXiv:math/0602259, MR2295199]

See also:

• Wikipedia, Cluster algebra

• Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960 MR2681708

• Lots of links at Cluster algebras portal including to the Fomin’s course slides.

• description and conference info “Cluster Algebras and Lusztig’s Semicanonical Basis”, Oregon, June 2011, html

• Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435

• Yuji Kodama, Lauren Williams, KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. 108 (22) 8984–8989 arxiv/1105.4170 doi

• Bernard Leclerc, Cluster algebras and representation theory, arxiv/1009.4552

• Christof Geiss, Bernard Leclerc, Jan Schröer, Preprojective algebras and cluster algebras, arxiv/0804.3168; Kac-Moody groups and cluster algebras, arxiv/1001.3545

• Tomoki Nakanishi, Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case, Nagoya Math. J. 202 (2011) 23–43, MR2804544 doi

• Kentaro Nagao, Donaldson–Thomas theory and cluster algebras, Duke Math. J. 162(7): 1313–1367 doi

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:

• Michael Gekhtman, Anton Izosimov, Integrable systems and cluster algebras, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2403.07287]

Quantum cluster algebras

• Arkady Berenstein, Andrei Zelevinsky, Quantum cluster algebras, Advances in Mathematics 195 2 (2005) 405-455 [doi:10.1016/j.aim.2004.08.003, math.QA/0404446]

• K. R. Goodearl, M. T. Yakimov, Quantum cluster algebras and quantum nilpotent algebras

  Proc Natl Acad Sci USA 111(27):9696–9703 (2014) doi

• Fan Qin, Dual canonical bases and quantum cluster algebras, arXiv:2003.13674

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