FINITE-DIMENSIONAL ALGEBRAS AND QUIVERS ALISTAIR SAVAGE Abstract. This is an overview article on finite-dimensional algebras and quivers, written for the Encyclo- pedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima. 1. Introduction a two-sided ideal of A. If I is a two-sided ideal of A, then the factor space A/I is again an algebra. Algebras and their representations are ubiquitous An algebra homomorphism is a linear map f : in mathematics. It turns out that representations A → A between two algebras such that of finite-dimensional algebras are intimately related 1 2 to quivers, which are simply oriented graphs. Quiv- ers arise naturally in many areas of mathematics, in- f(1A1 )= 1A2 , and cluding representation theory, algebraic and differ- f(xy)= f(x)f(y), ∀ x, y ∈ A. ential geometry, Kac-Moody algebras, and quantum groups. In this article, we give a brief overview of some of these topics. We start by giving the basic A representation of an algebra A is an algebra homo- definitions of associative algebras and their represen- morphism ρ : A → Endk(V ) for a k-vector space V . tations. We then introduce quivers and their rep- Here Endk(V ) is the space of endomorphisms of the resentation theory, mentioning the connection to the vector space V with multiplication given by compo- representation theory of associative algebras. We also sition. Given a representation of an algebra A on a discuss in some detail the relationship between quiv- vector space V , we may view V as an A-module with ers and the theory of Lie algebras. the action of A on V given by 2. Associative algebras a · v = ρ(a)v, a ∈ A, v ∈ V. An algebra is a vector space A over a field k equipped with a multiplication which is distributive A morphism ψ : V → W of two A-modules (or and such that equivalently, representations of A), is a linear map a(xy) = (ax)y = x(ay), ∀ a ∈ k, x,y ∈ A. commuting with the action of A. That is, it is a lin- ear map satisfying arXiv:math/0505082v1 [math.RA] 5 May 2005 When we wish to make the field explicit, we call A a k-algebra. An algebra is associative if (xy)z = x(yz) a · ψ(v)= ψ(a · v), ∀ a ∈ A, v ∈ V. for all x,y,z ∈ A. A has a unit, or multiplica- tive identity, if it contains an element 1A such that 1Ax = x1A = x for all x ∈ A. From now on, we Let G be a commutative monoid (a set with an will assume all algebras are associative with unit. A associative multiplication and a unit element). A is said to be commutative if xy = yx for all x, y ∈ A G-graded k-algebra is a k-algebra which can be ex- and finite-dimensional if the underlying vector space pressed as a direct sum A = ⊕g∈GAg such that of A is finite-dimensional. aAg ⊂ Ag for all a ∈ k and Ag1 Ag2 ⊂ Ag1+g2 for A vector subspace I of A is called a left (resp. all g1,g2 ∈ G. A morphism ψ : A → B of G-graded right) ideal if xy ∈ I for all x ∈ A, y ∈ I (resp. x ∈ I, algebras is a k-algebra morphism respecting the grad- y ∈ A). If I is both a right and a left ideal, it is called ing, that is, satisfying ψ(Ag) ⊂ Bg for all g ∈ G. Date: May 4, 2005. 2000 Mathematics Subject Classification. 16G20,16-02. This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. 1 2 ALISTAIR SAVAGE 3. Quivers and path algebras Note that the path algebra kQ is finite-dimensional if and only if Q has no oriented cycles (paths with the A quiver is simply an oriented graph. More pre- same head and tail vertex). cisely, a quiver is a pair Q = (Q0,Q1) where Q0 is a finite set of vertices and Q1 is a finite set of arrows Example 3.3. Let Q be the following quiver. (oriented edges) between them. For a ∈ Q1, we let h(a) denote the head of a and t(a) denote the tail 12 3 n−2 n−1 n of a. A path in Q is a sequence x = ρ1ρ2 ...ρm of arrows such that h(ρi+1) = t(ρi) for 1 ≤ i ≤ m − 1. Then for every 1 ≤ i ≤ j ≤ n, there is a unique We let t(x)= t(ρm) and h(ρ)= h(ρ1) denote the ini- path from i to j. Let f : kQ → Mn(k) be the lin- tial and final vertices of the path x. For each vertex ear map from the path algebra to the n × n matrices i ∈ Q0, we let ei denote the trivial path which starts with entries in the field k that sends the unique path and ends at the vertex i. from i to j to the matrix Eji with (j, i) entry 1 and Fix a field k. The path algebra kQ associated to all other entries zero. Then one can show that f is a quiver Q is the k-algebra whose underlying vector an isomorphism onto the algebra of lower triangular space has basis the set of paths in Q, and with the matrices. product of paths given by concatenation. Thus, if 4. Representations of quivers x = ρ1 ...ρm and y = σ1 ...σn are two paths, then xy = ρ1 ...ρmσ1 ...σn if h(y) = t(x) and xy = 0 Fix a field k. A representation of a quiver Q is an otherwise. We also have assignment of a vector space to each vertex and to each arrow a linear map between the vector spaces ei if i = j eiej = , assigned to its tail and head. More precisely, a rep- 0 if i 6= j ( resentation V of Q is a collection {V | i ∈ Q } x if h(x)= i i 0 eix = , 0 if h(x) 6= i of finite-dimensional k-vector spaces together with a ( collection x if t(x)= i {Vρ : Vt(ρ) → Vh(ρ) | ρ ∈ Q1} xei = , (0 if t(x) 6= i of k-linear maps. Note that a representation V of a quiver Q is equivalent to a representation of the for x ∈ kQ. This multiplication is associative. Note path algebra kQ. The dimension of V is the map that e A and Ae have bases given by the set of paths i i dV : Q0 → Z≥0 given by dV (i) = dim Vi for i ∈ Q0. ending and starting at i respectively. The path alge- If V and W are two representations of a quiver bra has a unit given by i∈Q0 ei. Q, then a morphism ψ : V → W is a collection of Example 3.1. Let Q beP the following quiver. k-linear maps {ψ : V → W | i ∈ Q } ρ σ λ i i i 0 such that 1 2 3 4 Wρψt(ρ) = ψh(ρ)Vρ, ∀ ρ ∈ Q1. Then kQ has a basis given by the set of paths Proposition 4.1. Let A be a finite-dimensional k- {e1,e2,e3,e4,ρ,σ,λ,σρ}. Some sample products are algebra. Then the category of representations of A ρσ =0, λλ =0, λσ =0, e3σ = σe2 = σ, e2σ =0. is equivalent to the category of representations of the algebra kQ/I for some quiver Q and some two-sided Example 3.2. Let Q be the following quiver (the so- ideal I of kQ. called Jordan quiver). It is for this reason that the study of finite- ρ dimensional associative algebras is intimately related to the study of quivers. We define the direct sum V ⊕ W of two represen- tations V and W of a quiver Q by 1 (V ⊕ W )i = Vi ⊕ Wi, i ∈ Q0 and (V ⊕ W ) : V ⊕ W → V ⊕ W by Then kQ =∼ k[t], the algebra of polynomials in one ρ t(ρ) t(ρ) h(ρ) h(ρ) variable. (V ⊕ W )ρ((v, w)) = (Vρ(v), Wρ(w)) FINITE-DIMENSIONAL ALGEBRAS AND QUIVERS 3 Q0 for v ∈ Vt(ρ), w ∈ Wt(ρ), ρ ∈ Q1. A representation In the standard coordinate basis of Z , the Euler V is trivial if Vi = 0 for all i ∈ Q0 and simple if its form is represented by the matrix E = (aij ) where only subrepresentations are the zero representation a = δ − #{ρ ∈ Q | t(ρ)= i, h(ρ)= j}. and V itself. We say that V is decomposable if it is ij ij 1 isomorphic to W ⊕ U for some nontrivial representa- Here δij is the Kronecker delta symbol. We define tions W and U. Otherwise, we call V indecompos- the Cartan form of the quiver Q to be the symmetric able. Every representation of a quiver has a decom- bilinear form given by position into indecomposable representations that is unique up to isomorphism and permutation of the (α, β)= hα, βi + hβ, αi . components. Thus, to classify all representations of Note that the Cartan form is independent of the ori- a quiver, it suffices to classify the indecomposable entation of the arrows in Q. In the standard coordi- representations. nate basis of ZQ0 , the Cartan form is represented by Example 4.2. Let Q be the following quiver. the Cartan matrix C = (cij ) where cij = aij + aji. ρ Example 4.4. For the quiver in Example 3.1, the Euler matrix is 1 2 1 −1 0 0 0 1 −1 0 Then Q has three indecomposable representations U, E = 00 10 V and W given by: 0 0 −1 1 U = k, U =0, U =0, 1 2 ρ and the Cartan matrix is V =0, V = k, V =0, 1 2 ρ 2 −1 0 0 W1 = k, W2 = k, Wρ =1.