Cluster algebra

Cluster algebras are a class of commutative rings introduced by Fomin & Zelevinsky (2002). A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.

Simple examples are given by the algebras of homogeneous functions on the Grassmannians. The Plücker coordinates provide some of the distinguished elements.

For the Grassmannian of planes in ℂⁿ, the situation is even more simple. In that case, the Plücker coordinates provide all the distinguished elements and the clusters can be completely described using triangulations of a regular polygon with n vertices.

References

• Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei (2005), "Cluster algebras. III. Upper bounds and double Bruhat cells", Duke Mathematical Journal, 126 (1): 1–52, MR 2110627
• Fomin, Sergey; Zelevinsky, Andrei (2002), "Cluster algebras. I. Foundations", Journal of the American Mathematical Society, 15 (2): 497–529, MR 1887642
• Fomin, Sergey; Zelevinsky, Andrei (2003), "Cluster algebras. II. Finite type classification", Inventiones Mathematicae, 154 (1): 63–121, MR 2004457
• Fomin, Sergey; Zelevinsky, Andrei (2007), "Cluster algebras. IV. Coefficients", Compositio Mathematica, 143 (1): 112–164, MR 2295199
• Fomin, Sergey; Reading, Nathan (2007), "Root systems and generalized associahedra", in Miller, Ezra; Reiner, Victor; Sturmfels, Bernd (eds.), Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Providence, R.I.: Amer. Math. Soc., arXiv:math/0505518, ISBN 9780821837368, MR 2383126
• Zelevinsky, Andrei (2007), "What Is . . . a Cluster Algebra?" (PDF), AMS Notices, 54 (11): 1494–1495.