Lecture Notes Introduction to Cluster Algebra

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1 Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification using Dynkin diagrams. It follows [Fomin-Reading]. 5.1 Reflection groups Let V be Euclidean space. Definition 5.1. A reflection is a linear map s : V V that fixes a hyperplane, and reverse the direction of the normal vector of the hyperplane. A finite reflection group is a finite group generated by some reflections in V. α β s α (β) = β 2 α,β α,α α Figure 1: Orthogonal reflection Center for the Promotion of Interdisciplinary Education and Research/ Department of Mathematics, Graduate School of Science, Kyoto University, Japan ivan.ip@math.kyoto-u.ac.jp 1

2 Example 5.2. Symmetry of the pentagon, given by the reflection group I 2 (5) (Dihedral group) generated by s and t. Products of odd number of generators are again reflection through some hyperplane (indicated in the picture). Products of even number of generators are rotations by certain angles. Figure 2: The reflection group I 2 (5) In general I 2 (m) is a group with 2m elements, generated by reflections s, t such that (st) m = 1 and s 2 = t 2 = 1. Definition 5.3. The set H of all reflecting hyperplanes is called a Coxeter arrangement. It cuts V into connected components called regions. Lemma 5.4. Fix an arbitrary region R 1. Then the map w R w := w(r 1 ) is a bijection between reflection group W and the set of regions. Reflections by the facet hyperplanes of R 1 generates W. Let a hyperplane H α given by {v : (v, α) = 0} for some vector α (normal vector). Let s α be the reflection in the hyperplane H α. Then Lemma 5.5. s α is orthogonal linear transformation (i.e. preserving inner product), and 2(α, β) s α (β) = β (α, α) α 5.2 Root system Definition 5.6. A finite root system is a finite non-empty collection Φ of nonzero vectors in V called roots such that (1) For all α Φ, span(α) Φ = {±α}. 2

3 (2) If α Φ, then s α (Φ) = Φ. In particular if α Φ, then α Φ. Lemma 5.7. Any reflection group W correpsond to a root system Φ W : The roots correspond to the normal vector of the reflecting hyperplanes. Definition 5.8. The simple roots Π Φ are the roots normal to the facet hyperplanes of R 1 and pointing into the half-space containing R 1. The rank of Φ is n= dim(span(φ)) = #{simple roots}. Π = {α i : i I} for some index set I = [n] := {1, 2,..., n}. The set of positive roots Φ + (resp. negative roots Φ ) are the roots α = i I c iα i such that all c i 0. (resp. c i 0). Lemma 5.9. Φ is disjoint union of Φ + and Φ. We also assume (3) (Crystallographic condition) s α (β) = β c αβ α with c αβ = 2 (α,β) (α,α) Z. It means that any simple root coordinates of any root are integers. Crystallographic reflection group W is also called the Weyl group of Φ. Lemma c αβ {0, ±1, ±2, ±3}. Proof. Let the angle between α and β be θ. Then (α, β) = α β cos θ, hence and c αβ = 2 β α cos θ Z c αβ c βα = 4 cos θ 2 Z which forces cos θ = 0, 1 2, 2 2, 3 β 2, 1. If we assume β α, then α = 1, 2, 3 respectively. (When cos θ = 0 there is no restriction. When cos θ = 1, β = ±α). Definition The ambient space Q R (Φ) := R-span(Φ). Root systems Φ and Φ are isomorphic if there is an isometry of Q R (Φ) Q R (Φ ) that sends Φ to some dilation cφ of Φ. Definition The Cartan matrix of Φ is an integer matrix C = (c ij ) i,j I where c ij := c αiα j = 2 (αi,αj) (α with α i,α i) i Π. Lemma Root systems Φ and Φ are isomorphic iff they have same Cartan matrix up to simultaneous rearrangement of rows and columns (i.e. reindexing). Example These are all 4 Cartan matrices of rank 2 and their corresponding root systems: 3

4 α 2 Type ( A 1 A 1 ) 2 0 C = 0 2 α 1 α 1 α 1 + α 2 Type ( A C = 1 2 ) α 2 α 2 α 1 + α 2 2α 1 + α 2 Type ( B C = 1 2 ) α 1 4

5 3α 1 + 2α 2 α 2 α 1 + α 2 2α 1 + α 2 3α 1 + α 2 Type ( G C = 1 2 ) α 1 They correspond to tiling of the plane. Theorem An integer n n matrix (c ij ) is a Cartan matrix of a root system iff (1) c ii = 2 for every i (2) c ij 0 if i j, and c ij = 0 c ji = 0 (3) There exists a diagonal matrix D such that DA is symmetric and positive definite (i.e. all eigenvalues > 0) Definition A root system Φ is reducible if Φ = Φ 1 Φ 2 such that α Φ 1, β Φ 2 = (α, β) = 0, i.e. C is block diagonal with > 1 blocks. Otherwise it is irreducible. Definition Cartan matrix of finite type can be encoded by Dynkin diagrams: Theorem 5.18 (Cartan-Killing classification of irreducible root system). Any irreducible root system is isomorphic to the root system corresponding to the Dynkin diagram of A n (n 1), B n (n 2), C n (n 3), D n (n 4), E 6, E 7, E 8, F 4 or G 2. 5

6 Example 5.19 (Root systems of type A n ). The root system is realized on the hyperplane P = {x x n = 0} R n+1. Simple roots are realized as α i := e i+1 e i R n+1 and positive roots are e i e j, 1 j < i n + 1. Example 5.20 (Root systems of type B n ). Simple roots are realized as α 1 = e 1, α i = e i e i 1, i 2 in R n. Positive roots are e i and e i ± e j for 1 j < i n. 6

7 Figure 3: Roots system of type B 3 Example 5.21 (Root systems of type C n ). Simple roots are realized as α 1 = 2e 1, α i = e i e i 1 in R n. Positive roots are 2e i and e i ± e j for 1 j < i n. The reflection group coincide with B n. Figure 4: Roots system of type C 3 In type B n and C n, the action of W on the roots are not transitive, there are 2 orbits, corresponding to short and long roots. Example 5.22 (Root system of type D n ). Simple roots are realized as α 0 = e 1 +e 2 and α i = e i+1 e i. Positive roots are e i ± e j for 1 j < i n. 5.3 Root Systems in Lie Theory Root systems are related to representation theory of Lie algebra. Consider simple Lie algebra g with Cartan subalgebra h (i.e. maximal commutative Lie subalgbera consisting of semisimple elements, i.e. diagonalizable in the adjoint representation). 7

8 Definition Let V be a representation of g. For any homomorphism λ : h C, i.e. λ h, we can define V λ := {v V : ξ h : ξ v = λ(ξ)v} If V λ is nonempty, then V λ is called a weight space of V, and λ is called its weight. We have V = λ h V λ Definition If V is the adjoint representation, i.e. g acts on itself V := g by g v = [g, v], then the set of nonzero weights form a root system Φ. i.e. a root α Φ is an element in h. In this case we have the root space decomposition g = h α Φ g α Classification of root systems then give a classification of simple Lie algebra. In particular, A n = sl n+1, B n = so 2n+1, C n = sp 2n and D n = so 2n. 5.4 Some useful calculations Φ + A 1 = {α} Φ + A 2 = {α 1, α 2, α 1 + α 2 } Φ + A 3 = {α 1, α 2, α 3, α 1 + α 2, α 2 + α 3, α 1 + α 2 + α 3 } Let s i := s αi. Then s i (α i ) = α i if c ij = 1, then s i (α j ) = α i + α j if c ij = 2, then s i (α j ) = 2α i + α j Each element w W can be written as product of simple reflections w = s i1 s i2...s il Shortest factorization of this form is called a reduced word for w, and l is called the length of w. Proposition Any Weyl group has a unique element w 0 of maximal length, called the longest element W A1 = {1, s 1 } W A2 = {1, s 1, s 2, s 1 s 2, s 2 s 1, w 0 }, longest element is w 0 = s 1 s 2 s 1 = s 2 s 1 s 2 W A3 = S 4 the permutation group. w 0 = s 1 s 2 s 1 s 3 s 2 s 1 8

9 W An = S n+1 is generated by transposition s i = (i, i + 1). We have s i s i+1 s i = s i+1 s i s i+1. s 1 s 2 s 1 s 3 s 2 s 3 is non-reduced: it equals s 2 s 1 s 3 s 2. In type A 2, w 0 (α 1 ) = α 2, w 0 (α 2 ) = α 1 In type A 3, w 0 (α 1 ) = α 3, w 0 (α 2 ) = α 2, w 0 (α 3 ) = α 1 In general, w 0 (α i ) = α i. The map i i is called the Dynkin involution. Definition Dynkin diagram has no cycles = bipartite. Let I = I + I where the nodes are marked as + and alternatively. A Coxeter element is defined as c = :=: t + t i I + s i i I s i The order h of c (i.e. c h = 1) is called Coxeter number. Proposition The longest element w 0 can be written as w 0 = t + t t + t...t ± }{{} h In particular if g is not of type A 2n then h is even and w 0 = c h 2. Theorem We have the following table: T ype Φ + h W A n n(n + 1)/2 n + 1 (n + 1)! B n, C n n 2 2n 2 n n! D n n(n 1) 2(n 1) 2 n 1 n! E E E F G