Cluster Algebras
Philipp Lampe
December 4, 2013 2 Contents
1 Informal introduction 5 1.1 Sequences of Laurent polynomials ...... 5 1.2 Exercises ...... 7
2 What are cluster algebras? 9 2.1 Quivers and adjacency matrices ...... 9 2.2 Quiver mutation ...... 14 2.3 Cluster algebras attached to quivers ...... 24 2.4 Skew-symmetrizable matrices, ice quivers and cluster algebras ...... 29 2.5 Exercises ...... 31
3 Examples of cluster algebras 33 3.1 Sequences and Diophantine equations attached to cluster algebras ...... 33 3.2 The Kronecker cluster algebra ...... 40 3.3 Cluster algebras of type A ...... 44 3.4 Exercises ...... 50
4 The Laurent phenomenon 53 4.1 The proof of the Laurent phenomenon ...... 53 4.2 Exercises ...... 57
5 Solutions to exercises 59
Bibliography 63
3 4 CONTENTS Chapter 1
Informal introduction
1.1 Sequences of Laurent polynomials
In the lecture we wish to give an introduction to Sergey Fomin and Andrei Zelevinsky’s theory of cluster algebras. Fomin and Zelevinsky have introduced and studied cluster algebras in a series of four influential articles [FZ, FZ2, BFZ3, FZ4] (one of which is coauthored with Arkady Berenstein). Although their initial motivation comes from Lie theory, the definition of a cluster algebra is very elementary. We will give the precise definition in Chapter 2, but to give the reader a first idea let a and b be two undeterminates and let us consider the map
b + 1 F : (a, b) 7→ b, . a
Surprisingly, the non-trivial algebraic identity F5 = id holds. This equation has a long and colour- ful history. The equation in this form dates back to Arthur Cayley [Ca] who remarks that this equation was already essentially known to Carl-Friedrich Gauß [Ga] in the context of spherical geometry and Napier’s rules. Latin speaking Gauß refers to the phenomenon as pentagramma mirificum, which the author would translate as marvelous pentagram. But not only the periodicity F5 = id after five steps is remarkable. Another interesting feature is the Laurent phenomenon: all terms occurring on the way are actually Laurent polynomials in the initial values a and b. Figure 1.1 illustrates the five Laurent polynomials.
(a, b) F F