MAT 5330 Algebraic Geometry: Quiver Varieties

Transcription

1 MAT 5330 Algebraic Geometry: Quiver Varieties Joel Lemay 1

2 Abstract Lie algebras have become of central importance in modern mathematics and some of the most important types of Lie algebras are Kac-Moody algebras. In this paper we give a brief overview of varieties defined by Lusztig associated to quivers (directed graphs) and how they are used to give geometric interpretations of the upper half of the universal enveloping algebra of a Kac-Moody algebra. 0 Introduction Quivers are useful tools in several areas of mathematics, most notably in representation theory. In this paper we will be especially interested their connection to Lie algebras. In particular, for every quiver there is an associated Kac-Moody Lie algebra, and vice-versa. Given a Lie algebra L, we have its universal enveloping algebra, U(L). If L is a Kac-Moody Lie algebra, we have a triangular decomposition of U(L): U(L) = U(L) + U 0 (L) U (L). In [6], Lusztig gave a geometric interpretation of U (L) (and hence U + (L)) by introducing a canonical basis of U (L), but only for quivers of type ADE. In [7], he extended his results to arbitrary quivers through the use of quiver varieties, which are now known as Lusztig s quiver varieties. In this paper, we give an overview of these varieties and discuss how we can use them to give a geometric interpretation of U + (L). In Section 1, we introduce the basic definitions and concepts of quivers that we will need to understand Lusztig s quiver varieties. In particular, we discuss the Cartan matrix associated to a quiver, which will be used in Section 2, where we give a basic overview of the necessary theory in Lie algebras. In Section 3, we will define Lusztig s quiver varieties as well as justify the use of the term variety (that is, we will prove that Lusztig s quiver varieties are affine algebraic varieties). Finally, in Section 4, we will show how Lusztig s quiver varieties allow us to interpret the upper half of the universal enveloping algebra of Kac-Moody Lie algebras. Page 2 of 12

3 By convention, we assume throughout that k = C. 1 Quivers The purpose of this section is to introduce quivers and the basic properties we will need to define quiver varieties in the subsequent sections. The material in this section follows [10]. Definition 1.1 (Quiver). A quiver Q is a directed graph. That is Q = (Q 0, Q 1, t, h), where Q 0 and Q 1 are finite sets (called the set of vertices and the set of arrows, respectively), and t and h are set maps Q 1 Q 0. For ρ Q 1, we call t(ρ) and h(ρ) the tail and head of ρ, respectively. By a standard abuse of notation, we often write Q = (Q 0, Q 1 ), leaving the maps t and h implied. Also, for ρ Q 0, we often write ρ : i j to mean t(ρ) = i and h(ρ) = j. Example 1.2. Let Q = (Q 0, Q 1, t, h) be the quiver defined by Q 0 = {1, 2}, Q 1 = {α}, t(α) = 1 and h(α) = 2. As with graphs, it is customary to represent quivers with diagrams. Hence, we would represent Q as follows: α Q: 1 2. We will use this quiver as a running example throughout this paper. Remark 1.3. By a standard abuse of notation, we often write Q = (Q 0, Q 1 ), leaving the maps t and h implied. Also, for ρ Q 0, we often write ρ : i j to mean t(ρ) = i and h(ρ) = j Remark 1.4. For simplicity, we restrict ourselves to quivers that are connected (that is, quivers whose underlying undirected graph is connected) and contain no loops (that is, no arrows with the same tail and head). Definition 1.5 (Double quiver). Let Q = (Q 0, Q 1 ) be a quiver. The double quiver associated to Q is the quiver Q = (Q 0, Q 1 ), where Q 1 = {ρ, ρ} ρ Q 1 and t(ρ) = h(ρ) and h(ρ) = t(ρ) for all ρ Q 1. Example 1.6. Let Q be the quiver defined in Example 1.2. Then the double quiver of Q is Q: α 1 2 α. Our next goal is to define the Cartan matrix associated to a quiver, which will be used in the following section to define Kac-Moody Lie algebras. We begin by recalling the definition of a Cartan matrix. Page 3 of 12

4 Definition 1.7 (Cartan matrix). Let I be an indexing set. A (generalized) Cartan matrix C = (c ij ), i, j I is an integer matrix satisfying: c ii = 2, for all i I, c ij 0, for all i j I, c ij = 0 c ji = 0, for all i, j I. Definition 1.8 ((Symmetric) Euler form). Let Q = (Q 0, Q 1 ) be a quiver. bilinear form, : Z Q0 Z Q0 Z defined by The Euler form of Q is the x, y = x i y i x t(ρ) y h(ρ). i Q 0 ρ Q 1 The symmetric Euler form of Q is the symmetric bilinear form (, ) : Z Q0 Z Q0 Z defined by (x, y) = x, y + y, x. Let {ε i } i Q0 denote the standard basis of Z Q0. Then we have ε i, ε j = δ ij {ρ Q 1 ρ : i j}, where δ ij is the Kronecker delta. Thus, (ε i, ε j ) = 2δ ij {ρ Q 1 ρ : i j or ρ : j i}. Since we assume that Q does not contain any loops, we have that (ε i, ε i ) = 2 and (ε i, ε j ) 0 for all i j Q 0. Moreover, the symmetric Euler form is symmetric, thus (ε i, ε j ) = 0 if and only if (ε j, ε i ) = 0. Hence, the matrix (c ij ) = (ε i, ε j ) is a (symmetric) Cartan matrix. This leads to the following definition. Definition 1.9 (Cartan matrix of a quiver). Let Q be a quiver. The Cartan matrix associated to Q, denoted C Q, is the matrix associated to the symmetric Euler form with respect to the standard basis of Z Q0. That is, the (i, j)-th entry of C Q is (ε i, ε j ). Example Let Q be the quiver from Example 1.2. A quick calculation shows that C Q = Lie algebras and the Universal Enveloping Algebra In this section we will discuss Lie algebras (specifically Kac-Moody Lie algebras) and their universal enveloping algebras. These concepts will be useful when discussing Lusztig s quiver varieties in the following Page 4 of 12

5 sections. Unless otherwise specified, the material presented here can be found in [2] or [3]. Recall that a Lie algebra L over k is a k-algebra whose mulitiplication, denoted by [, ], satisfies [x, x] = 0 for all x L, and [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z L. While there are many interesting examples of Lie algebras, we will restrict our attention to Kac-Moody Lie algebras. Definition 2.1 (Kac-Moody Lie algebra). Let C = (c ij ) be an n n Cartan matrix. Then the Kac-Moody Lie algebra of C is the Lie algebra over k generated by e i, f i, h i for 1 i n, subject to the following relations. [h i, h j ] = 0 for all i, j, [h i, e j ] = c ij e j and [h i, f j ] = c ij f j for all i, j, [e i, f i ] = h i for each i and [e i, f j ] = 0 for all i j, (ad e i ) 1 cij (e j ) = 0 and (ad f i ) 1 cij (f j ) for all i j, where (ad x)(y) = [x, y] for x, y L. Remark 2.2. A proof that this defines a Lie algebra for any Cartan matrix can be found in Chapter 1 of [4]. We saw in Section 1 that, given a quiver, we have its associated Cartan matrix. following definition. We thus have the Definition 2.3 (Kac-Moody Lie algebra of a quiver). Let Q be a quiver and C Q its associated Cartan matrix. Then the Kac-Moody Lie algebra associated to Q, denoted g Q, is the Kac-Moody Lie algebra of C Q. Example 2.4. Let Q be the quiver consisting of one vertex and no arrows. Then C Q = (2) and g Q is the Lie algebra generated by e, f, h subject to [e, f] = h, [h, e] = 2e, and [h, f] = 2f. But this is simply the multiplication of sl 2. Hence, g Q = sl 2. Remark 2.5. Let Q be a quiver. It is clear that the Dynkin diagram of g Q is the underlying undirected graph of Q. Hence, we can identify g Q just by looking at Q. Example 2.6. Let Q be the quiver defined in 1.2. Then the underlying undirected graph of Q is the Dynkin diagram A 2, which is the Dynkin diagram of sl 3. Hence, g Q = sl 3. Page 5 of 12

6 Our next aim is to define the Universal enveloping algebra of a Lie algebra. Again, we will only be interested in Kac-Moody Lie algebras, but the definition holds for arbitrary Lie algebras. Definition 2.7 (Universal enveloping algebra). Let L be a Lie algebra. The universal enveloping algebra of L, denoted U(L), is the quotient of the tensor algebra of (the underlying vector space) L modulo the ideal generated by elements of the form x y y x [x, y]. That is, U(L) = T (L)/I, where T (L) is the tensor algebra of L and I = x y y x [x, y] x,y L. Remark 2.8. We will write x 1 x 2 x n for all x 1 x 2 x n T (L). Definition 2.9 (Upper half of the universal enveloping algebra). Let g be a Kac-Moody Lie algebra with n n Cartan matrix (c ij ) and U = U(g) its universal enveloping algebra. The upper half of U, denoted U + = U + (g) is the Q-algebra generated e i, for i = 1,..., n, subject to the Serre relations for all i j. 1 c ij p=0 ( 1) p 1 c ij p e p i e je 1 cij p i = 0 Remark The upper half of the universal enveloping algebra comes from the triangular decomposition of U(g) = U + (g) U 0 (g) U (g) when g is a Kac-Moody Lie algebra (see [9, Chapter 3]). However, since we will focus only on the upper half, we will not present all the details of this triangular decomposition. Example Let Q be the quiver defined in Example 1.2. We have seen in Examples 1.10 and 2.6 that C Q = 2 1 and g Q = sl Thus, U + (g Q ) = U + (sl 3 ) is generated by e 1 and e 2 and subject to e 2 e 2 1 2e 1 e 2 e 1 + e 2 1e 2 = 0 and e 1 e 2 2 2e 2 e 1 e 2 + e 2 2e 1 = 0. 3 Lusztig s Quiver Varieties In this section, we define Lusztig s quiver varieties and prove that they are affine algebraic varieties. The definitions presented here come from Lusztig s own paper, [7]. The proof that Lusztig s quiver varieties are indeed varieties (Proposition 3.7), while by no means an original result, is the author s own work, as is Example 3.8. Page 6 of 12

7 Let Q = (Q 0, Q 1 ) be a quiver and let V = i Q 0 V i be a finite-dimensional Q 0 -graded k-vector space. Define E V = Hom k (V t(ρ), V h(ρ) ). ρ Q 1 Let ε : Q 1 {±1} be the map defined by 1, if ρ Q 1, ε(ρ) = 1 if ρ / Q 1. Finally, let ψ : E V gl V = Π i Q0 gl(v i ) (where gl(v i ) is the general linear algebra on V i ) whose i-th component ψ i : E V gl(v i ) is given by ψ i (x) = ε(ρ)x ρ x ρ. ρ Q 1,h(ρ)=i Remark 3.1. For our purposes, it will be sufficient to simply know the definition of the map ψ, however it is worth noting that it originates from the use of methods in symplectic geometry. Let G V = Π i Q0 Aut(V i ) (this is an algebraic group). Then G V acts on E V by g x = (g h(ρ) x ρ g 1 t(ρ) ) ρ Q 1, for all g = (g i ) i Q0 G V and x = (x ρ ) ρ E Q1 V. Then we have a nondegenerate, G V -invariant, symplectic form, : E V E V k given by x, y = ε(ρ) tr(x ρ y ρ ), ρ Q 1 where tr denotes the trace map. Thus, (E V,, ) is a symplectic vector space. The map ψ is the then moment map associated to the G V action on E V. These arguments come from [7] while the necessary background theory in symplectic geometry can be found in [1]. Definition 3.2 (Nilpotent). An element x = (x ρ ) ρ E Q1 V is called nilpotent if there exists an N N + such that for all paths ρ N ρ 1 of length N in Q, the composition x ρn x ρ1 : V t(ρ1) V h(ρn ) is zero. With these definitions in mind, we are ready to define Lusztig s quiver varieties. Definition 3.3 (Lusztig s (nilpotent) quiver variety). The Lusztig (nilpotent) quiver variety associated to V, denoted Λ V, is the set of all nilpotent elements x E V such that ψ i (x) = 0 for all i Q 0. The use of the term variety in this definition is worth justifying. Firstly, note that E V is a finitedimensional vector space. Hence, E V = k n = A n for some n N. Hence, we may simply identify E V with Page 7 of 12

8 A n. So, E V identification). is an affine algebraic variety (it inherits the Zariski topology and structure sheaf of A n via the We claim that Λ V is a closed subvariety of E V. To prove this, however, we need a few additional concepts. Definition 3.4 (x-stable flag). Let Q = (Q 0, Q 1 ) be a quiver and V = i Q 0 be a Q 0 -graded vector space. A flag of V is a sequence V = V 0 V 1 V m = 0 of Q 0 -graded vector spaces. Let x = (x ρ ) ρ Q1 Lemma 3.5. [7, Lemma 1.8] E V. A flag is called x-stable if x ρ (V l t(ρ) ) V l h(ρ) for all ρ Q 1 and l = 0, 1,..., m. 1. If x E V and V = V 0 V 1 V m = 0 is an x-stable flag, then x ρ (V l 1 t(ρ) ) V l h(ρ) for all ρ Q 1 and l = 0, 1,..., m. In particular, x is nilpotent. 2. Conversely, if x E V is nilpotent, then there exists an x-stable flag of V. Corollary 3.6. Let dim E V = n. Then x E V is nilpotent if and only if x ρn x ρ1 = 0 for all paths ρ n ρ 1 of length n in Q. Proof. Given that dim E V = n, the longest flag is of length n, since dim V l 1 > dim V l for any flag V = V 0 V 1 V m = 0. Now, if x E V, then by Part 1 of Lemma 3.5, there is an x-stable flag V = V 0 V 1 V m = 0 with x ρ (V l 1 t(ρ) ) V h(ρ) l. Thus, x ρ m x ρ1 = 0 for all paths ρ m ρ 1 of length m in Q. Since m n, we have x ρn x ρ1 = 0 for all paths ρ n ρ 1 of length n in Q. The reverse implication is trivial. Proposition 3.7. The set Λ V is a closed subvariety of E V. In particular, Λ V is an affine algebraic variety. Proof. Suppose dim E V = n. Let x = (x ρ ) ρ Q1 E V. For each ρ Q 1, x ρ is a linear map from V t(ρ) to V h(ρ) and hence, by fixing bases for V t(ρ) and V h(ρ), we may view x ρ as a dim V h(ρ) dim V t(ρ) matrix. By matrix multiplication, the entries of x ρ x ρ may be viewed as (quadratic) polynomials in the entries of x ρ and x ρ. Thus, for each i Q 0, the entries of ψ i (x) = ρ Q 1,h(ρ)=i ε(ρ)x ρx ρ are polynomials in the entries of x ρ and x ρ for all ρ Q 1 with h(ρ) = i. Therefore, under the identification E V = A n, the condition x E V such that ψ i (x) = 0 is equivalent to x A n such that x is in the vanishing set of the polynomials defined by the entries of ψ i. By Corollary 3.6, x is nilpotent if and only if x ρn x ρ1 = 0 for all paths ρ n ρ 1 of length n in Q. By viewing each x ρ as a dim V h(ρ) dim V t(ρ) matrix, the entries of x ρn x ρ1 may be viewed as polynomials in the entries of x ρn,..., x ρ1. Thus, under the identification E V = A n, the condition x E V is nilpotent is equivalent to x A n is in the vanishing set of the polynomials defined by the entries of x ρn x ρ1 paths ρ n ρ 1 of length n in Q. for all Therefore, under the identification E V = A n, Λ V is the vanishing set of the polynomials defined by the entries of ψ i, for each i Q 0, and those defined by the entries of x ρn x ρ1, for all paths ρ n ρ 1 of length n in Q. Thus, Λ V is a closed subset of E V, and hence a closed subvariety of E V. Since E V is affine, Λ V is an affine algebraic variety. Page 8 of 12

9 Example 3.8. Let Q be the quiver defined in Example 1.2 and let V = V 1 V 2 be a Q 0 -graded vector space. Then E V = Hom k (V t(α), V h(α) ) Hom k (V t(α), V h(α) ) = Hom k (V 1, V 2 ) Hom k (V 2, V 1 ), and ψ 1 (x) = x α x α, ψ 2 (x) = x α x α. Notice that ψ 1 (x) = 0 and ψ 2 (x) = 0 implies x is nilpotent, since the only paths of length two are αα and αα. Thus, Λ V = {x E V x α x α = 0 and x α x α = 0}. To illustrate that Λ V is indeed a closed subset of E V, suppose V 1 = k 2 and V 2 = k. Then for x = (x α, x α ) E V, x α : k 2 k and x α : k k 2 may be viewed as 1 2 and 2 1 matrices, respectively. Write ) x α = (x 1 x 2 and x α = x 3. So, x 4 ψ 1 (x) = 0 = x 3 ( ) x 1 x 2 = 0 x 4 x 1x 3 x 2 x 3 = 0, x 1 x 4 x 2 x 4 and ψ 2 (x) = 0 = ) (x 1 x 2 x 3 = 0 x 4 x 1 x 3 + x 2 x 4 = 0. Thus, under the identification E V A 4 given by ( ) x 1 x 2, x 3 (x 1, x 2, x 3, x 4 ), we have that x 4 Λ V = V (x 1 x 3, x 2 x 3, x 1 x 4, x 2 x 4, x 1 x 3 + x 2 x 4 ) = V (x 1 x 3, x 2 x 3, x 1 x 4, x 2 x 4 ). Hence, Λ V is indeed a closed set. Page 9 of 12

10 4 Application of Lusztig s Quiver Varieties In this section we show how Lusztig s quiver varieties are used to obtain a geometric representation of the upper half of the enveloping algebra of the Kac-Moody Lie algebras. The exposition in this section roughly follows [10], however, all definitions can be found in [8]. Let Q = (Q 0, Q 1 ) be a quiver, U + = U + (g Q ) the upper half of the universal enveloping algebra of the associated Kac-Moody Lie algebra (recall Definition 2.9). For any ν = (ν i ) i Q0 N Q0, define U ν + to be the subspace of U + given by U + ν = span{e i1 e i2 e in i 1, i 2,..., i n is a sequence in which i appears ν i times for each i Q 0 }. Thus, U + = ν N Q 0 U + ν. Also, since it will be useful later in the section, let U + Z generated by the elements e p i /p! for all i Q 0 and p N. Moreover, we let U + Z,ν = U + Z U + For any Q 0 -graded vector space V, we write Irr Λ V be the subring of U + for the set of irreducible components of Λ V. For ν = (ν i ) i Q0 N Q0, define V ν = i Q 0 V ν i, where V ν i = k νi for each i Q 0. We have the following proposition. Proposition 4.1. For any ν N Q0, we have dim U + ν = Irr Λ V ν. Remark 4.2. Proposition was proved by Lusztig (see [7, 12.14]) in the case that Q is a quiver whose underlying underlying graph is a Dynkin diagram of type A, D, E, Ã, D or D. The general case was proved by Kashiwara and Saito (see [5]). Example 4.3. Let Q be the quiver defined in Example 1.2 and let ν = (2, 1). Thus, V ν = k 2 k. By Example 3.8, Λ V ν = V (x 1 x 3, x 2 x 3, x 1 x 4, x 2 x 4 ). We claim that Λ V ν = V (x 1, x 2 ) V (x 3, x 4 ). Indeed, suppose (x 1, x 2, x 3, x 4 ) Λ V ν. Then x 1 x 3 = 0, x 2 x 3 = 0, x 1 x 4 = 0, and x 2 x 4 = 0. If x 1 = 0 = x 2 = 0, then (x 1, x 2, x 3, x 4 ) = (0, 0, x 3, x 4 ) V (x 1, x 2 ). If x 1 or x 2 is not zero, then x 3 = x 4 = 0 and so (x 1, x 2, x 3, x 4 ) = (x 1, x 2, 0, 0) V (x 3, x 4 ). Thus, Λ V ν V (x 1, x 2 ) V (x 3, x 4 ). The reverse inclusion is obvious, hence Λ V ν = V (x 1, x 2 ) V (x 3, x 4 ). Also, I(V (x 1, x 2 )) = rad x 1, x 2 = x 1, x 2 (since x 1, x 2 is radical). Thus, Γ(V (x 1, x 2 )) = k[x 1, x 2, x 3, x 4 ]/ x 1, x 2 = k[x 3, x 4 ] is an integral domain. So, V (x 1, x 2 ) is irreducible. Likewise, V (x 3, x 4 ) is irreducible. Moreover, it is clear Page 10 of 12

11 that V (x 1, x 2 ) V (x 3, x 4 ) and V (x 3, x 4 ) V (x 1, x 2 ). Therefore, Irr V ν = {V (x 1, x 2 ), V (x 3, x 4 )} = Irr V ν = 2. By Proposition 4.1, we expect dim U + ν = 2. We can verify this directly. By definition, U + ν = span{e 1 e 2 2, e 2 e 1 e 2, e 2 2e 1 } = span{e 1 e 2 2, e 2 2e 1 } (recall Example 2.11). Since {e 1 e 2 2, e 2 2e 1 } is linearly independent, we have dim U + ν = 2, as wanted. In light of Proposition 4.1, we see there is a connection between the irreducible components of Lusztig s quiver varieties and the dimension of U ν +. In fact, our next goal is to show that we can construct a basis of U + from the irreducible components of Lusztig s quiver varieties. The material in the rest of this section is based on [8]. Definition 4.4 (Constructible). Let X be a variety and Y X. Then Y is called constructible if it can be obtained by from subvarieties of X from a finite number of set-theoretic operations. If X is an algebraic variety, then a function f : X Q is called constructible if f 1 (a) is constructible for all a Q and is empty for all but finitely many a. For a Q 0 -graded vector space, define M(Λ V ) to be the Q-vector space of all constructible functions on Λ V. Define M(Λ V ) to be the Q-subspace of M(Λ V ) consisting of all functions that are constant on the orbits of G V on Λ V (recall G V from Remark 3.1). Let M Z (Λ V ) = {f M(Λ V ) f(a) Z for all a Q}. We have the following theorem due to Lusztig. Theorem 4.5. [8, Theorem 2.7] Let ν N Q0. Then there exists a Q-linear map κ ν : U + ν that the following hold. M(Λ V ν ) such 1. For any Z Irr Λ V ν, there exists a unique f Z κ ν (U + Z,ν ) such that f Z is equal to 1 on an open dense subset of Z and equal to zero on an open dense subset of Z Irr Λ V ν for all Z Z. 2. {f Z Z Irr Λ V ν } is a Q-basis of κ ν (U ν + ). 3. κ ν : U ν + κ ν (U ν + ) is an isomorphism. 4. Let [Z] U ν + be defined by κ ν ([Z]) = f Z. Then B ν := {[Z] Z Irr Λ V ν } is a Q-basis of U ν κ ν (U + Z,ν ) = κ ν(u ν + ) M Z (Λ V ν ). 6. B ν is a Z-basis of U + Z,ν. Page 11 of 12

12 From Theorem 4.5, it follows that B = ν N Q 0 B ν is a Q-basis of U +. The basis B is called the semicanonical basis of U +. This basis has several interesting properties. Among them, by the involution of U(g Q ) given by e i f i, f i e i and h i h i, we can obtain a semicanonical basis of U (g Q ). For any irreducible highest-weight integrable representation V of U, if v V is a nonzero highest-weight vector, then {bv b B, bv 0} is a Q-basis of V. One can obtain a similar basis for U q + (g Q ), the upper half of the quantized enveloping algebra, using perverse sheaves instead of constructible functions. This basis, called the canonical basis of U q + (g Q ), has similarly interesting properties and is closely related to crystal bases. References [1] Ana Cannas da Silva. Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [2] Karin Erdmann and Mark J. Wildon. Introduction to Lie algebras. Springer Undergraduate Mathematics Series. Springer-Verlag London Ltd., London, [3] James E. Humphreys. Introduction to Lie algebras and representation theory. Springer-Verlag, New York, Graduate Texts in Mathematics, Vol. 9. [4] Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, [5] Masaki Kashiwara and Yoshihisa Saito. Geometric construction of crystal bases. Duke Math. J., 89(1):9 36, [6] G. Lusztig. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc., 3(2): , [7] G. Lusztig. Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc., 4(2): , [8] G. Lusztig. Semicanonical bases arising from enveloping algebras. Adv. Math., 151(2): , [9] George Lusztig. Introduction to quantum groups, volume 110 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, [10] Alistair Savage. Finite-dimensional algebras and quivers. Encyclopedia or Mathematical Physics, 2: , Page 12 of 12