Finite-Dimensional Algebras and Quivers 3

arXiv:math/0505082v1 [math.RA] 5 May 2005 qipdwt utpiainwihi distributive is that which such and multiplication a with equipped 1 o all for ieidentity tive of r n h hoyo i algebras. Lie of quiv- theory between the relationship and the ers detail some in also We rep- discuss algebras. their associative the of to and theory representation connection quivers the introduce mentioning basic theory, the then resentation We giving of represen- by their overview and start tations. brief algebras We associative a of give topics. definitions we these article, of this some In quantum and differ- groups. algebras, and Kac-Moody geometry, algebraic in- ential theory, mathematics, of representation Quiv- areas many cluding graphs. in naturally oriented arise simply ers are representations related which that intimately quivers, are to out algebras turns finite-dimensional It of mathematics. in hnw iht aetefil xlct ecall we explicit, field the make to wish we When and ilasm l lersaeascaiewt unit. be with to associative said are is algebras all assume will k y ih)ideal right) agba nagbais algebra An -algebra. A ∈ A 2000 Date An etrsubspace vector A lersadterrpeettosaeubiquitous are representations their and Algebras hsrsac a upre npr yteNtrlSine a Sciences Natural the by part in supported was research This x A finite-dimensional sfinite-dimensional. is = a .If ). ( a ,2005. 4, May : algebra xy ahmtc ujc Classification. Subject Mathematics ,y z y, x, x Abstract. uzi n Nakajima. and Lusztig Physics Mathematical of pedia 1 ( = ) I A if fi otisa element an contains it if , sbt ih n etiel ti called is it ideal, left a and right a both is 2. commutative = xy ax ∈ soitv algebras Associative savco space vector a is x ∈ 1. ) A y o all for hsi noeve ril nfiiedmninlagba a algebras finite-dimensional on article overview an is This I . = Introduction o all for IIEDMNINLAGBA N QUIVERS AND ALGEBRAS FINITE-DIMENSIONAL x A I fteudryn etrspace vector underlying the if ( ay associative of x a a has if ) x , ∈ A xy ∈ ∀ A scle a called is A a = rmnwo,we on, now From . unit , ∈ y yx ecvrpt lers iglHl lersadtequive the and algebras Ringel-Hall algebras, path cover We . A ,x y x, k, ∈ f( if or , o all for vrafield a over I 16G20,16-02. 1 xy (resp. A ) multiplica- et(resp. left ∈ z uhthat such ,y x, = LSARSAVAGEALISTAIR A. x x ( ∈ ∈ A yz I A A k a ) , 1 a etrspace vector End Here hntefco space factor the then iin ie ersnaino nalgebra an of representation a Given sition. h cinof action the etrspace vector A a a satisfying map ear A soitv utpiainadaui lmn) A element). unit a G and multiplication associative omtn ihteato of action the with commuting aA morphism all qiaety ersnain of representations equivalently, rse sadrc sum direct a as pressed n,ta s satisfying is, that ing, lersi a is algebras dEgneigRsac oni NEC fCanada. of (NSERC) Council Research Engineering nd 1 -graded two-sided representation Let An A g g → 1 ⊂ g , morphism A ler homomorphism algebra G 2 A 2 a ∈ k g ewe w lerssc that such algebras two between eacmuaiemni astwt an with set (a monoid commutative a be k · ( ρ -algebra f V G ψ o all for da of ideal k ( : a V V ( morphism A . A stesaeo noopim fthe of endomorphisms of space the is ) xy dqies rte o the for written quivers, nd agbamrhs epcigtegrad- the respecting morphism -algebra A v · emyview may we , = ) v ihmlilcto ie ycompo- by given multiplication with on = ) ψ → f fa algebra an of = ( a 1 V : ψ End ρ f A sa is A V ( ( ∈ ( ie by given 1 a A/I If . a x ψ = ) ) · ) → k k ,a v, ( f v k ( A V ( I ) agbawihcnb ex- be can which -algebra and saana algebra. an again is 1 A y , g W o a for ) satosddielof ideal two-sided a is ) ) A ∀ ψ , ∈ ⊂ V 2 = , a ftwo of ∀ A A A : ,v A, B salna map linear a is san as and hti,i salin- a is it is, That . ∈ A A g ,y x, sa ler homo- algebra an is ⊕ aite of varieties r 1 g ,i iermap linear a is ), A ,v A, → k g o all for ∈ ∈ g vco space -vector ∈ Encyclo- 2 G A V. B A A A. ⊂ mdl with -module ∈ mdls(or -modules g of V. g A uhthat such G g ∈ 1 -graded + A G g . 2 na on f for V A , . : 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 representations 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. −1 2 −1 0 C =   . Then any representation Z of Q is isomorphic to 0 −1 2 −1  0 0 −1 2  ∼ d1−r d2−r r   Z = U ⊕ V ⊕ W ,   The Tits form q of a quiver Q is defined by where d = dim Z , d = dim Z , r = rank Z . 1 1 2 2 ρ 1 q(α)= hα, αi = (α, α). Example 4.3. Let Q be the Jordan quiver. Then 2 representations V of Q are classified up to isomor- It is known that the number of continuous parameters phism by the Jordan normal form of Vρ where ρ is describing representations of dimension α for α 6= 0 the single arrow of the quiver. Indecomposable repre- is greater than or equal to 1 − q(α). sentations correspond to single Jordan blocks. These Let g be the Kac-Moody algebra associated to the are parametrized by a discrete parameter n (the size Cartan matrix of a quiver Q. By forgetting the ori- of the block) and a continuous parameter λ (the eigen- entation of the arrows of Q, we obtain the underlying value of the block). (undirected) graph. This is the Dynkin graph of g. g A quiver is said to be of finite type if it has only Associated to is a root system and a set of simple finitely many indecomposable representations (up to roots {αi | i ∈ Q0} indexed by the vertices of the isomorphism). If a quiver has infinitely many isomor- Dynkin graph. phism classes but they can be split into families, each Theorem 4.5 (Gabriel’s Theorem). parametrized by a single continuous parameter, then (1) A quiver is of finite type it and only if the we say the quiver is of tame (or affine) type. Ifa underlying graph is a union of Dynkin graphs quiver is of neither finite nor tame type, it is of wild of type A, D, or E. type. It turns out that there is a rather remarkable (2) A quiver is of tame type if and only if the un- relationship between the classification of quivers and derlying graph is a union of Dynkin graphs of their representations and the theory of Kac-Moody type A, D, or E and extended Dynkin graphs algebras. of type A, D, or E (with at least one extended The Euler form or Ringel form of a quiver Q is Dynkin graph). defined to be the asymmetric bilinear form on ZQ0 (3) The isomorphism classes of indecomposable given by b b b representations of a quiver Q of finite type are hα, βi = α(i)β(i) − α(t(ρ))β(h(ρ)). in one-to-one correspondence with the posi-

i∈Q0 ρ∈Q1 tive roots of the root system associated to the X X 4 ALISTAIR SAVAGE

underlying graph of Q. The correspondence Theorem 4.6 (Kac’s Theorem). Let Q be an arbi- is given by trary quiver. The dimension vectors of indecomposable representations of Q correspond to positive roots V 7→ dV (i)αi. of the root system associated to the underlying graph i∈Q0 X of Q (and are thus independent of the orientation of The Dynkin graphs of type A, D, and E are as the arrows of Q). The correspondence is given by follows. dV 7→ dV (i)αi. i∈Q0 X An Note that in Kac’s Theorem, it is not asserted that the isomorphism classes are in one-to-one correspondence with the roots as in the ﬁnite case considered Dn in Gabriel’s theorem. It turns out that in the general case, dimension vectors for which there is exactly one isomorphism class correspond to real roots while imaginary roots correspond to dimension vectors for which there are families of representations. E6 Example 4.7. Let Q be the quiver of type An, oriented as follows. ρ1 ρ2 ρn−2 ρn−1

E7 12 3 n−2 n−1 n It is known that the set of positive roots of the simple Lie algebra of type An is

E8 l α 1 ≤ j ≤ l ≤ n ⊔{0}.  i  i=j Here the subscript indicates the number of vertices X  in the graph. The zero root corresponds to the trivial representa-  l  The extended Dynkin graphs of type A, D, and E tion. The root i=j αi for some 1 ≤ j ≤ l ≤ n are as follows. corresponds to the unique (up to isomorphism) repre- P b b b sentation V with k if j ≤ i ≤ l A Vi = n (0 otherwise b and 1 if j ≤ i ≤ l − 1 Dn Vρi = . (0 otherwise b Example 4.8. Let Q be the quiver of type An, with all arrows oriented in the same direction (for in- n stance, counter-clockwise). The positive root b i=0 αi (where {0, 1, 2,...,n} are the vertices of the quiver) is E 6 imaginary. There is a one parameter familyP of isomorphism classes of indecomposable representations b where the maps assigned to each arrow are non-zero. E7 The parameter is the composition of the maps around the loop. b If a quiver Q has no oriented cycles, then the only i E8 simple kQ-modules are the modules S for i ∈ Q0 where b i k if i = j Here we have used an open dot to denote the vertex Sj = that was added to the corresponding Dynkin graph (0 if i 6= j i of type A, D, or E. and Sρ = 0 for all ρ ∈ Q1. FINITE-DIMENSIONAL ALGEBRAS AND QUIVERS 5

5. Ringel-Hall Algebras Ei, i ∈ Q0 and relations

1−cij Let k be the finite field Fq with q elements and 1 − c 1−c −p (−1)p ij EpE E ij , i 6= j, let Q be a quiver with no oriented cycles. Let P p i j i p=0 be the set of all isomorphism classes of kQ-modules X which are finite as sets (since k is finite-dimensional, where cij are the entries of the Cartan matrix and these are just the quiver representations we consid- m [m]! = , ered above). Let A be a commutative integral domain p [p]![m − p]! containing Z and elements v, v−1 such that v2 = q. tn − t−n The Ringel-Hall algebra H = HA,v(kQ) is the free A- [n]= − , [n]! = [1][2] . . . [n]. module with basis {[V ]} indexed by the isomorphism t − t 1 classes of representations of the quiver Q, with an Theorem 5.1. There is a Q(t)-algebra isomorphism ∗ + A-bilinear multiplication defined by C → U sending ui 7→ Ei for all i ∈ Q0.

1 2 1 2 hdim V ,dim V i V The proof of Theorem 5.1 is due to Ringel in the [V ] · [V ]= v gV 1,V 2 [V ]. V case that the underlying graph of Q is of finite or X affine type. The more general case presented here is 1 2 V Here dim V , dim V is the Euler form and gV 1,V 2 due to Green. is the number of submodules W of V such that All of the Kac-Moody algebras considered so far ∼ 1 ∼ 2 Q0 V/W = V and W = V . H is an associative Z≥0- have been simply-laced. That is, their Cartan ma- graded algebra, with identity element [0], the isomor- trices are symmetric. There is a way to deal with phism class of the trivial representation. The grad- non-simply-laced Kac-Moody algebras using species. ing H = ⊕αHα is given by letting Hα be the A- We will not treat this subject in this article. span of the set of isomorphism classes [V ] such that dim V = α. 6. Quiver Varieties Let C = CA,v(kQ) be the A-subalgebra of H gen- One can use varieties associated to quivers to yield i erated by the isomorphism classes [S ] of the sim- a geometric realization of the upper half of the uni- ple kQ-modules. C is called the composition algebra. versal enveloping algebra of a Kac-Moody algebra g If the underlying graph of Q is of finite type, then and its irreducible highest-weight representations. C = H. Now let K be a set of finite fields k such that the 6.1. Lusztig’s Quiver Varieties. We first intro- set {|k| | k ∈ K} is infinite. Let A be an integral do- duce the quiver varieties, first defined by Lusztig, main containing Q and, for each k ∈ K, an element which yield a geometric realization of the upper half 2 + vk such that vk = |k|. For each k ∈ K we have the U of the universal enveloping algebra of a simply- corresponding composition algebra Ck, generated by laced Kac-Moody algebra g. Let Q = (Q0,Q1) be k i the elements [ S ] (here we make the field k explicit). the quiver whose vertices Q0 are the vertices of the g Now let C be the subring of k∈K Ck generated by Dynkin diagram of and whose set of arrows Q1 con- Q and the elements sists of all the edges of the Dynkin diagram with both Q orientations. By definition, U + is the Q-algebra de- t = (tk)k∈K, tk = vk, fined by generators ei, i ∈ Q0, subject to the Serre −1 −1 −1 k −1 relations t = (tk )k∈K, tk = (v ) , 1−cij i i i k i 1 − c u = (uk)k∈K, uk = [ S ], i ∈ Q0. (−1)p ij epe e1−cij −p =0 p i j p=0 Now, t lies in the center of C and if p(t) = 0 for X some polynomial p, then p must be the zero polyno- for all i 6= j in Q0, where cij are the entries of the Car- mial since the set of vk is infinite. Thus we may tan matrix associated to Q. For any ν = i∈Q0 νii, i + + think of C as the A-algebra generated by the u , νi ∈ N, let Uν be the subspace of U generated −1 P i ∈ Q0, with A = Q[t,t ] and t an indeterminate. by the monomials ei1 ei2 ...ein for various sequences ∗ ∗ Let C = Q(t) ⊗A C. We call C the generic compo- i1,i2,...,in in which i appears νi times for each + + + sition algebra. i ∈ Q0. Thus U = ⊕ν Uν . Let UZ be the subring + p Let g be the Kac-Moody algebra associated to the of U generated by the elements ei /p! for i ∈ Q0, + + + + + Cartan matrix of the quiver Q and let U be the quan- p ∈ N. Then UZ = ⊕ν UZ,ν where UZ,ν = UZ ∩Uν . tum group associated by Drinfeld and Jimbo to g. It We define the involution ¯ : Q1 → Q1 to be the − 0 + has a triangular decomposition U = U ⊗ U ⊗ U . function which takes ρ ∈ Q1 to the element of Q1 Specifically, U + is the Q(t)-algebra with generators consisting of the same edge with opposite orientation. 6 ALISTAIR SAVAGE

An orientation of our graph/quiver is a choice of a Let V, V′, V′′ ∈ V such that dim V = dim V′ + ′′ subset Ω ⊂ Q1 such that Ω ∪ Ω=¯ Q1 and Ω ∩ Ω=¯ ∅. dim V . Now, suppose that S is an I-graded subspace Let V be the category of finite-dimensional Q0- of V. For x ∈ ΛV we say that S is x-stable if x(S) ⊂ ′ ′′ graded vector spaces V = ⊕i∈Q0 Vi over C with S. LetΛV;V ,V be the variety consisting of all pairs morphisms being linear maps respecting the grad- (x, S) where x ∈ ΛV and S is an I-graded x-stable ing. Then V ∈ V shall denote that V is an ob- subspace of V such that dim S = dim V′′. Now, if we ject of V. The dimension of V ∈ V is given by fix some isomorphisms V/S =∼ V′, S =∼ V′′, then x ′ ′′ v = dim V = (dim V0,..., dim Vn). induces elements x ∈ ΛV′ and x ∈ ΛV′′ . We then Given V ∈ V, let EV be the space of represen- have the maps tations of Q with underlying vector space V. That p1 p2 ΛV′ × ΛV′′ ←− ΛV V′ V′′ −→ ΛV is, ; , ′ ′′ where p1(x, S) = (x , x ), p2(x, S)= x. EV = Hom(Vt(ρ), Vh(ρ)). ρ∈Q1 For a holomorphic map π between complex vari- ′ M eties A and B, let π denote the map between the For any subset Q of Q , let EV ′ be the subspace ! 1 1 ,Q1 spaces of constructible functions on A and B given of EV consisting of all vectors x = (x ) such that ρ by x = 0 whenever ρ 6∈ Q′ . The algebraic group ρ 1 −1 −1 V V V ′ (π!f)(y)= aχ(π (y) ∩ f (a)). G = i Aut(Vi) acts on E and E ,Q1 by a∈Q ′ − 1 ∗ X (g, x)Q = ((gi), (xρ)) 7→ gx = (xρ) = (gh(ρ)xρgt(ρ)). Let π be the pullback map from functions on B to functions on A acting as π∗f(y)= f(π(y)). We then Define the function ε : Q1 → {−1, 1} by ε(ρ)=1 for all ρ ∈ Ω and ε(ρ) = −1 for all ρ ∈ Ω.¯ Let h·, ·i define a map be the nondegenerate, GV-invariant, symplectic form (6.1) M(ΛV′ ) × M(ΛV′′ ) → M(ΛV) on EV with values in C defined by by (f ′,f ′′) 7→ f ′ ∗ f ′′ where f f f hx, yi = ε(ρ) tr(xρyρ¯). ′ ′′ ∗ ′ ′′ f ∗ f = (p2)!p1(f × f ). ρ∈Q1 X ′ ′′ ′ ′′ ′ Note that EV can be considered as the cotangent Here f × f ∈ M(ΛV × ΛV ) is defined by (f × ′′ ′ ′′ ′ ′ ′′ ′′ space of EV,Ω under this form. f )(x , x ) = f (x )f (x ). The map (6.1) is bilin- The moment map associated to the GV-action on ear and defines anf associatve Q-algebra structure on ν the symplectic vector space EV is the map ψ : EV → ⊕ν M(ΛVν ) where V is the object of V defined by Vν = Cνi . glV = i End Vi, the Lie algebra of GLV, with i- i component ψi : EV → End Vi given by Theref is a unique algebra homomorphism κ : Q + U → ⊕ν M(ΛVν ) such that κ(ei) is the function ψi(x)= ε(ρ)xρxρ¯. on the point ΛVi with value 1. Then κ restricts to ρ∈Q1,h(ρ)=i + X a map κν :fUν → M(ΛVν ). It can be shown that p Definition 6.1. An element x ∈ EV is said to be κpi(ei /p!) is the function 1 on the point ΛVpi for nilpotent if there exists an N ≥ 1 such that for any i ∈ Q0, p ∈ Z≥0. f sequence ρ1,ρ2,...,ρN in H satisfying t(ρ1)= h(ρ2), Let MZ(ΛV) be the set of all functions in M(ΛV) t(ρ2)= h(ρ3),..., t(ρN−1)= h(ρN ), the composition that take on only integer values. One can show V V ′ ′′ xρ1 xρ2 ...xρN : t(ρN ) → h(ρ1) is zero. that if ff ∈ MZ(ΛV′ ) and f ∈ MZ(ΛV′′f) then ′ ′′ Definition 6.2. Let ΛV be the set of all nilpotent f ∗ f ∈ MZ(ΛV) in the setup of (6.1). Thus + f f elements x ∈ EV such that ψi(x)=0 for all i ∈ I. κν (UZ,ν ) ⊆ MZ(ΛVν ). Let IrrΛVfdenote the set of irreducible compo- A subset of an algebraic variety is said to be con- nents of ΛVf. The following proposition was conjec- structible if it is obtained from subvarieties from a tured by Lusztig and proved by him in the affine (and finite number of the usual set-theoretic operations. finite) case. The general case was proved by Kashi- A function f : A → Q on an algebraic variety A wara and Saito. is said to be a constructible function if f −1(a) is Q0 a constructible set for all a ∈ Q and is empty for Proposition 6.3. For any ν ∈ (Z≥0) , we have + all but finitely many a. Let M(ΛV) denote the Q- dim Uν = #IrrΛVν . vector space of all constructible functions on ΛV. Let We then have the following important result due M(ΛV) denote the Q-subspace of M(ΛV) consisting to Lusztig. of those functions that are constant on any GV-orbit Q0 inf ΛV. Theorem 6.4. Let ν ∈ (Z≥0) . Then FINITE-DIMENSIONAL ALGEBRAS AND QUIVERS 7

st st (1) For any Z ∈ IrrΛVν , there exists a unique Definition 6.6. Let Λ = Λ(v, w) be the set of all + fZ ∈ κν (UZ,ν ) such that fZ is equal to one (x, t) ∈ Λ(v, w) satisfying the following condition: If on an open dense subset of Z and equal to S = (Si) with Si ⊂ Vi is x-stable and ti(Si)=0 for ′ zero on an open dense subset of Z ∈ IrrΛVν i ∈ I, then Si =0 for i ∈ I. for all Z′ 6= Z. + The group GV acts on Λ(v, w) via (2) {fZ | Z ∈ IrrΛVν } is a Q-basis of κν (Uν ). (3) κ : U + → κ (U +) is an isomorphism. −1 −1 ν ν ν ν (g, (x, t)) 7→ ((g xρg ), (tig )). + h(ρ) t(ρ) i (4) Define [Z] ∈ Uν by κν ([Z]) = fZ . Then st Bν = {[Z] | Z ∈ IrrΛVν } is a Q-basis of and the stabilizer of any point of Λ(v, w) in GV is + Uν . trivial. We then make the following definition. + + (5) κν (UZ,ν )= κν (Uν ) ∩ MZ(ΛVν ). st + Definition 6.7. Let L ≡ L(v, w) =Λ(v, w) /GV. (6) Bν is a Z-basis of UZ,ν. f Let w, v, v′, v′′ ∈ ZI be such that v = v′ + v′′. From this theorem, we see that B = ⊔ B is a ≥0 ν ν Consider the maps Q-basis of U +, which is called the semicanonical ba- (6.2) sis. This basis has many remarkable properties. One ′′ ′ p1 ˜ ′′ p2 ′′ p3 of these properties is as follows. Via the algebra in- Λ(v , 0)×Λ(v , w) ← F(v, w; v ) → F(v, w; v ) → Λ(v, w), volution of the entire universal enveloping algebra U where the notation is as follows. A point of of g given on the Chevalley generators by ei 7→ fi, F(v, w; v′′) is a point (x, t) ∈ Λ(v, w) together with fi 7→ ei and h 7→ −h for h in the Cartan subalgebra an I-graded, x-stable subspace S of V such that of g, one obtains from the results of this section a sem- dim S = v′ = v − v′′. A point of F˜(v, w; v′′) is − icanonical basis of U , the lower half of the universal a point (x, t, S) of F(v, w; v′′) together with a col- enveloping algebra of g. For any irreducible highest ′ ′ ∼ ′′ lection of isomorphisms Ri : Vi = Si and Ri : weight integrable representation V of U (or, equiv- ′′ ∼ Vi = Vi/Si for each i ∈ I. Then we define alently, g), let v ∈ V be a non-zero highest weight ′ ′′ p2(x, t, S, R , R ) = (x, t, S), p3(x, t, S) = (x, t) and vector. Then the set ′ ′′ ′′ ′ ′ ′′ ′ ′ p1(x, t, S, R , R ) = (x , x ,t ) where x , x ,t are de- {bv | b ∈ B, bv 6=0} termined by ′ ′ ′ ′ is a Q-basis of V , called the semicanonical basis of Rh(ρ)xρ = xρRt(ρ) : Vt(ρ) → Sh(ρ), V . Thus the semicanonical basis of U − is simulta- ′ ′ ′ ti = tiRi : Vi → Wi neously compatible will all irreducible highest weight R′′ x′′ = x R′′ : V′′ → V /S . integrable modules. There is also a way to define the h(ρ) ρ ρ t(ρ) t(ρ) h(ρ) h(ρ) semicanonical basis of a representation directly in a It follows that x′ and x′′ are nilpotent. geometric way. This is the subject of the next sub- section. Lemma 6.8. One has One can also obtain a geometric realization of st st (p ◦ p )−1(Λ(v, w) ) ⊂ p−1(Λ(v′′, 0) × Λ(v′, w) ). the upper part U + of the quantum group in a sim- 3 2 1 ilar manner using perverse sheaves instead of con- Thus, we can restrict (6.2) to Λst, forget the ′′ structible functions. This construction yields the Λ(v , 0)-factor and consider the quotient by GV and canonical basis which also has many remarkable prop- GV′ . This yields the diagram erties and is closely related to the theory of crystal π1 π2 bases. (6.3) L(v′, w) ←F(v, w; v − v′) → L(v, w),

6.2. Nakajima’s Quiver Varieties. We introduce where here a description of the quiver varieties first pre- F(v, w, v − v′) sented by Nakajima. They yield a geometric realiza- def ′ st tion of the irreducible highest weight representations = {(x, t, S) ∈ F(v, w; v − v ) | (x, t) ∈ Λ(v, w) }/GV. of a simply-laced Kac-Moody algebra. Let M(L(v, w)) be the vector space of all con- I Definition 6.5. For v, w ∈ Z≥0, choose I-graded structible functions on L(v, w). Then define maps vector spaces V and W of graded dimensions v and w respectively. Then define hi : M(L(v, w)) → M(L(v, w)) i ei : M(L(v, w)) → M(L(v − e , w)) Λ ≡ Λ(v, w)=ΛV × Hom(Vi, Wi). i i∈I fi : M(L(v − e , w)) → M(L(v, w)). X 8 ALISTAIR SAVAGE

by and the references cited therein. The reader inter- ested in species, which extend many of these results h f = u f, i i to non-simply-laced Lie algebras, should consult [1]. ∗ eif = (π1)!(π2 f), The book [6] covers the quiver varieties of Lusztig ∗ fig = (π2)!(π1 g). and canonical bases. Canonical bases are closely related to crystal bases and crystal graphs (see [3] for Here t an overview of these topics). In fact, the set of irre- u = (u0,...,un)= w − Cv ducible components of the quiver varieties of Lusztig where C is the Cartan matrix of g and we are using and Nakajima can be endowed with the structure of a diagram (6.3) with v′ = v − ei where ei is the vector crystal graph in a purely geometric way (see [4, 19]). i whose components are given by ej = δij . Many results on Nakajima’s quiver varieties can be Now let ϕ be the constant function on L(0, w) found in the original papers [9, 11]. The overview with value 1. Let L(w) be the vector space of func- article [10] is also useful. tions generated by acting on ϕ with all possible com- Quiver varieties can also be used to give geomet- binations of the operators fi. Then let L(v, w) = ric realizations of tensor products of representations M(L(v, w)) ∩ L(w). (see [7, 8, 13, 20]) and finite-dimensional representations of quantum affine Lie algebras (see [12]). This Proposition 6.9. w The operators ei, fi, hi on L( ) is just a select few of the many applications of quiver provide it with the structure of the irreducible high- varieties. Much more can be found in the literature. est weight integrable representation of g with highest weight i∈Q0 wiωi. Each summand of the decomposition L(w) = L(v, w) is a weight space with P v References weight i∈Q0 wiωi − viαi. Here the ωi and αi are the fundamental weightsL and simple roots of g respec- [1] V. Dlab and C. M. Ringel. Indecomposable representa- tively. P tions of graphs and algebras. Mem. Amer. Math. Soc., 6(173):v+57, 1976. [2] J. A. Green. Hall algebras, hereditary algebras and quan- Let Z ∈ Irr L(v, w) and define a linear map TZ : L(v, w) → C that associates to a constructible func- tum groups. Invent. Math., 120(2):361–377, 1995. [3] J. Hong and S.-J. Kang. Introduction to quantum groups tion f ∈ L(v, w) the (constant) value of f on a suit- and crystal bases, volume 42 of Graduate Studies in Math- able open dense subset of Z. The fact that L(v, w) is ematics. American Mathematical Society, Providence, RI, finite-dimensional allows us to take such an open set 2002. on which any f ∈ L(v, w) is constant. So we have a [4] M. Kashiwara and Y. Saito. Geometric construction of linear map crystal bases. Duke Math. J., 89(1):9–36, 1997. [5] S. Lang. Algebra, volume 211 of Graduate Texts in Math- v w Φ : L(v, w) → CIrr L( , ). ematics. Springer-Verlag, New York, 2002. [6] G. Lusztig. Introduction to quantum groups, volume 110 of Then we have the following proposition. Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1993. Proposition 6.10. The map Φ is an isomorphism; [7] A. Malkin. Tensor product varieties and crystals: GL case. for any Z ∈ Irr L(v, w), there is a unique func- Trans. Amer. Math. Soc., 354(2):675–704 (electronic), 2002. tion gZ ∈ L(v, w) such that for some open dense subset O of Z we have g | = 1 and for some [8] A. Malkin. Tensor product varieties and crystals: the Z O ADE case. Duke Math. J., 116(3):477–524, 2003. closed GV-invariant subset K ⊂ L(v, w) of dimen- [9] H. Nakajima. Instantons on ALE spaces, quiver varieties, sion < dim L(v, w) we have gZ = 0 outside Z ∪ K. and Kac-Moody algebras. Duke Math. J., 76(2):365–416, The functions gZ for Z ∈ IrrΛ(v, w) form a basis of 1994. L(v, w). [10] H. Nakajima. Varieties associated with quivers. In Rep- resentation theory of algebras and related topics (Mexico City, 1994), volume 19 of CMS Conf. Proc., pages 139– 7. Further Reading 157. Amer. Math. Soc., Providence, RI, 1996. [11] H. Nakajima. Quiver varieties and Kac-Moody algebras. We have given here a brief overview of some top- Duke Math. J., 91(3):515–560, 1998. ics related to finite-dimensional algebras and quivers. [12] H. Nakajima. Quiver varieties and finite-dimensional rep- There is much more to be found in the literature. resentations of quantum affine algebras. J. Amer. Math. For basics on associative algebras and their represen- Soc., 14(1):145–238 (electronic), 2001. tations, the reader may consult introductory texts on [13] H. Nakajima. Quiver varieties and tensor products. In- vent. Math., 146(2):399–449, 2001. abstract algebra such as [5]. For further results (and [14] C. M. Ringel. Hall algebras. In Topics in algebra, Part their proofs) on Ringel-Hall algebras see the papers 1 (Warsaw, 1988), volume 26 of Banach Center Publ., [14, 15, 16, 17, 18] of Ringel and Green’s paper [2] pages 433–447. PWN, Warsaw, 1990. FINITE-DIMENSIONAL ALGEBRAS AND QUIVERS 9

[15] C. M. Ringel. Hall algebras and quantum groups. Invent. Mat. Comun., pages 85–114. Soc. Mat. Mexicana, M´exico, Math., 101(3):583–591, 1990. 1995. [16] C. M. Ringel. Hall algebras revisited. In Quantum de- [18] C. M. Ringel. Green’s theorem on Hall algebras. In Rep- formations of algebras and their representations (Ramat- resentation theory of algebras and related topics (Mexico Gan, 1991/1992; Rehovot, 1991/1992), volume 7 of Israel City, 1994), volume 19 of CMS Conf. Proc., pages 185– Math. Conf. Proc., pages 171–176. Bar-Ilan Univ., Ramat 245. Amer. Math. Soc., Providence, RI, 1996. Gan, 1993. [19] Y. Saito. Crystal bases and quiver varieties. Math. Ann., [17] C. M. Ringel. The Hall algebra approach to quantum 324(4):675–688, 2002. groups. In XI Latin American School of Mathematics [20] A. Savage. The tensor product of representations of (Spanish) (Mexico City, 1993), volume 15 of Aportaciones Uq(sl2) via quivers. Adv. Math., 177(2):297–340, 2003.

Fields Institute and University of Toronto, Toronto, Ontario, Canada E-mail address: [email protected]