Valued Graphs and the Representation Theory of Lie Algebras
aa r X i v : . [ m a t h . R T ] S e p VALUED GRAPHS AND THE REPRESENTATION THEORYOF LIE ALGEBRAS
JOEL LEMAY
Abstract.
Quivers (directed graphs) and species (a generalization of quivers) and theirrepresentations play a key role in many areas of mathematics including combinatorics, ge-ometry, and algebra. Their importance is especially apparent in their applications to therepresentation theory of associative algebras, Lie algebras, and quantum groups. In thispaper, we discuss the most important results in the representation theory of species, suchas Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite andtame representation type. We also explain the link between species and K -species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategoryof the category of species. Furthermore, we prove two results about the structure of thetensor ring of a species containing no oriented cycles that do not appear in the literature.Specifically, we prove that two such species have isomorphic tensor rings if and only if theyare isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensoralgebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Moritaequivalent to, the path algebra of a quiver. Contents
Introduction 11. Valued Quivers 32. Species and K -species 73. The Path and Tensor Algebras 114. The Frobenius Morphism 145. A Closer Look at Tensor Rings 176. Representations 237. Ringel-Hall Algebras 31References 35 Introduction
Species and their representations were first introduced in 1973 by Gabriel in [13]. Let K be a field. Let A be a finite-dimensional, associative, unital, basic K -algebra and letrad A denote its Jacobson radical. Then A/ rad A ∼ = Π i ∈I K i , where I is a finite set and K i is a finite-dimensional K -division algebra for each i ∈ I . Moreover, rad A/ (rad A ) ∼ = L i,j ∈I j M i , where j M i is a finite-dimensional ( K j , K i )-bimodule for each i, j ∈ I . We thenassociate to A a valued graph ∆ A with vertex set I and valued arrows i ( d ij ,d ji ) −−−−→ j for Mathematics Subject Classification. each j M i = 0, where d ij = dim K j ( j M i ). The valued graph ∆ A , the division algebras K i ( i ∈ I ) and the bimodules j M i ( i, j ∈ I ) constitute a species and contain a great deal ofinformation about the representation theory of A (in some cases, all the information). Whenworking over an algebraically closed field, a species is simply a quiver (directed graph) inthe sense that all K i ∼ = K and all j M i ∼ = K n so only ∆ A is significant. In this case,Gabriel was able to classify all quivers of finite representation type (that is, quivers withonly finitely many non-isomorphic indecomposable representations); they are precisely thosewhose underlying graph is a (disjoint union of) Dynkin diagram(s) of type A, D or E.Moreover, he discovered that the isomorphism classes of indecomposable representations ofthese quivers are in bijection with the positive roots of the Kac-Moody Lie algebra associatedto the corresponding diagram. Gabriel’s theorem is the starting point of a series of remarkableresults such as the construction of Kac-Moody Lie algebras and quantum groups via Ringel-Hall algebras, the geometry of quiver varieties, and Lusztig’s categorification of quantumgroups via perverse sheaves. Lusztig, for example, was able to give a geometric interpretationof the positive part of quantized enveloping algebras using quiver varieties (see [21]).While quivers are useful tools in representation theory, they have their limitations. Inparticular, their application to the representation theory of associative unital algebras, ingeneral, only holds when working over an algebraically closed field. Moreover, the Lie the-ory that is studied by quiver theoretic methods is naturally that of symmetric Kac-MoodyLie algebras. However, many of the fundamental examples of Lie algebras of interest tomathematicians and physicists are symmetrizable Kac-Moody Lie algebras which are notsymmetric. Species allow us to relax these limitations.In his paper [13], Gabriel outlined how one could classify all species of finite representationtype over non-algebraically closed fields. However, it was Dlab and Ringel in 1976 (see [11])who were ultimately able to generalize Gabriel’s theorem and show that a species is offinite representation type if and only if its underlying valued graph is a Dynkin diagramof finite type. They also showed that, just as for quivers, there is a bijection between theisomorphism classes of the indecomposable representations and the positive roots of thecorresponding Kac-Moody Lie algebra.Despite having been introduced at the same time, the representation theory of quivers ismuch more well-known and well-developed than that of species. In fact, the very definitionof species varies from text to text; some use the more “general” definition of a species (e.g.[11]) while others use the alternate definition of a K -species (e.g. [9]). Yet the relationshipbetween these two definitions is rarely discussed. Moreover, while there are many well-knownresults in the representation theory of quivers, such as Gabriel’s theorem or Kac’s theorem,it is rarely mentioned whether or not these results generalize for species. Indeed, there doesnot appear to be any single comprehensive reference for species in the literature. The maingoal of this paper is to compare the current literature and collect all the major, often hardto find, results in the representation theory of species into one text.This paper is divided into seven sections. In the first, we give all the preliminary materialon quivers and valued quivers that will be needed for the subsequent sections. In particular,we address the fact that two definitions of valued quivers exist in the literature.In Section 2, we define both species and K -species and discuss how the definitions arerelated. Namely, we define the categories of species and K -species and show that the categoryof K -species can be thought of as a subcategory of the category of species. That is, via an ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 3 appropriate functor, all K -species are species. There are, however, species that are not K -species for any field K .The third section deals with the tensor ring (resp. algebra) T ( Q ) associated to a species(resp. K -species) Q . This is a generalization of the path algebra of a quiver. If K is aperfect field, then for any finite-dimensional associative unital K -algebra A , the categoryof A -modules is equivalent to the category of T ( Q ) /I -modules for some K -species Q andsome ideal I . Also, it will be shown in Section 6, that the category of representations of Q is equivalent to the category of T ( Q )-modules. These results show why species are suchimportant tools in representation theory; modulo an ideal, they allow us to understand therepresentation theory of finite-dimensional associative unital algebras.In Section 4, we follow the work of [9] to show that, when working over a finite field, onecan simply deal with quivers (with automorphism) rather than species. That is, we showthat if Q is an F q -species, then the tensor algebra of Q is isomorphic to the fixed pointalgebra of the path algebra of a quiver under the Frobenius morphism.In the fifth section, we further discuss the link between a species and its tensor ring. Inparticular, we prove two results that do not seem to appear in the literature. Theorem 5.5
Let Q and Q ′ be two species with no oriented cycles. Then T ( Q ) ∼ = T ( Q ′ ) ifand only if Q C ∼ = Q ′ C (where Q C and Q ′ C denote the crushed species of Q and Q ′ ). Theorem 5.8 and Corollary 5.13
Let K be a perfect field and Q a K -species containingno oriented cycles. Then K ⊗ K T ( Q ) is isomorphic to, or Morita equivalent to, the pathalgebra of a quiver (where K denotes the algebraic closure of K ). Section 6 deals with representations of species. We discuss many of the most importantresults in the representation theory of quivers, such as the theorems of Gabriel and Kac, andtheir generalizations for species.The seventh and final section deals with the Ringel-Hall algebra of a species. It is well-known that the generic composition algebra of a quiver is isomorphic to the positive part ofthe quantized enveloping algebra of the associated Kac-Moody Lie algebra. Also, Sevenhantand Van Den Bergh have shown that the Ringel-Hall algebra itself is isomorphic to thepositive part of the quantized enveloping algebra of a generalized Kac-Moody Lie algebra(see [28]). We show that these results hold for species as well. While this is not a new result,it does not appear to be explained in detail in the literature.We assume throughout that all algebras (other than Lie algebras) are associative andunital.
Acknowledgements . First and foremost, I would like to thank Prof. Alistair Savage forintroducing me to this topic and for his invaluable guidance and encouragement. Further-more, I would like to thank Prof. Erhard Neher and Prof. Vlastimil Dlab for their helpfulcomments and advice. 1.
Valued Quivers
In this section, we present the preliminary material on quivers and valued quivers thatwill be used throughout this paper. In particular, we begin with the definition of a quiver
JOEL LEMAY and then discuss valued quivers. There are two definitions of valued quivers that can befound in the literature; we present both and give a precise relationship between the two interms of a functor between categories (see Lemma 1.5). We also discuss the idea of “folding”,which allows one to obtain a valued quiver from a quiver with automorphism.
Definition 1.1 (Quiver) . A quiver Q is a directed graph. That is, Q = ( Q , Q , t, h ), where Q and Q are sets and t and h are set maps Q → Q . The elements of Q are called vertices and the elements of Q are called arrows . For every ρ ∈ Q , we call t ( ρ ) the tail of ρ and h ( ρ ) the head of ρ . By an abuse of notation, we often simply write Q = ( Q , Q ) leavingthe maps t and h implied. The sets Q and Q may well be infinite; however we will dealexclusively with quivers having only finitely many vertices and arrows. We will also restrictourselves to quivers whose underlying undirected graphs are connected.A quiver morphism ϕ : Q → Q ′ consists of two set maps, ϕ : Q → Q ′ and ϕ : Q → Q ′ ,such that ϕ ( t ( ρ )) = t ( ϕ ( ρ )) and ϕ ( h ( ρ )) = h ( ϕ ( ρ )) for each ρ ∈ Q .For ρ ∈ Q , we will often use the notation ρ : i → j to mean t ( ρ ) = i and h ( ρ ) = j . Definition 1.2 (Absolute valued quiver) . An absolute valued quiver is a quiver Γ = (Γ , Γ )along with a positive integer d i for each i ∈ Γ and a positive integer m ρ for each ρ ∈ Γ suchthat m ρ is a common multiple of d t ( ρ ) and d h ( ρ ) for each ρ ∈ Γ . We call ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ an (absolute) valuation of Γ. By a slight abuse of notation, we often refer to Γ as an absolutevalued quiver, leaving the valuation implied.An absolute valued quiver morphism is a quiver morphism ϕ : Γ → Γ ′ respecting thevaluations. That is, d ′ ϕ ( i ) = d i for each i ∈ Γ and m ′ ϕ ( ρ ) = m ρ for each ρ ∈ Γ .Let Q abs denote the category of absolute valued quivers.A (non-valued) quiver can be viewed as an absolute valued quiver with trivial values (i.e.all d i = m ρ = 1). Thus, valued quivers are a generalization of quivers.Given a quiver Q and an automorphism σ of Q , we can construct an absolute valuedquiver Γ with valuation ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ by “folding” Q as follows: • Γ = { vertex orbits of σ } , • Γ = { arrow orbits of σ } , • for each i ∈ Γ , d i is the number of vertices in the orbit i , • for each ρ ∈ Γ , m ρ is the number of arrows in the orbit ρ .Given ρ ∈ Γ , let m = m ρ and d = d t ( ρ ) . The orbit ρ consists of m arrows in Q , say { ρ i = σ i − ( ρ ) } mi =1 . Because σ is a quiver morphism, we have that each t ( ρ i ) = t ( σ i − ( ρ )) isin the orbit t ( ρ ) and that t ( ρ i ) = t ( σ i − ( ρ )) = σ i − ( t ( ρ )). The value d is the least positiveinteger such that σ d ( t ( ρ )) = t ( ρ ) and since σ m ( t ( ρ )) = t ( ρ ) (because σ m ( ρ ) = ρ ), then d | m . By the same argument, d h ( ρ ) | m . Thus, this construction does in fact yield anabsolute valued quiver.Conversely, given an absolute valued quiver Γ with valuation ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ , it is possibleto construct a quiver with automorphism ( Q, σ ) that folds into Γ in the following way. Let x y be the unique representative of ( x mod y ) in the set { , , . . . , y } for x, y positive integers.Then define: • Q = { v i ( j ) | i ∈ Γ , ≤ j ≤ d i } , • Q = { a ρ ( k ) | ρ ∈ Γ , ≤ k ≤ m ρ } , ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 5 • t ( a ρ ( k )) = v t ( ρ ) (cid:16) k d t ( ρ ) (cid:17) and h ( a ρ ( k )) = v h ( ρ ) (cid:16) k d h ( ρ ) (cid:17) , • σ ( v i ( j )) = v i (cid:16) ( j + 1) d i (cid:17) , • σ ( a ρ ( k )) = a ρ (cid:16) ( k + 1) m ρ (cid:17) .It is clear that Q is a quiver. It is easily verified that σ is indeed an automorphism of Q .Given the construction, we see that ( Q, σ ) folds into Γ. However, we do not have a one-to-onecorrespondence between absolute valued quivers and quivers with automorphism since, ingeneral, several non-isomorphic quivers with automorphism can fold into the same absolutevalued quiver, as the following example demonstrates.
Example 1.3. [9, Example 3.4] Consider the following two quivers. Q : Q ′ : a b cd eα α α α α α β β β β β β Define σ ∈ Aut( Q ) and σ ′ ∈ Aut( Q ′ ) by σ : (cid:18) (cid:19) , (cid:18) α α α α α α α α α α α α (cid:19) σ ′ : (cid:18) a b c d ea d e b c (cid:19) , (cid:18) β β β β β β β β β β β β (cid:19) . Then, both (
Q, σ ) and ( Q ′ , σ ′ ) fold into (1) (2) (2)(2) (2)(2) ,yet Q and Q ′ are not isomorphic as quivers. Definition 1.4 (Relative valued quiver) . A relative valued quiver is a quiver ∆ = (∆ , ∆ )along with positive integers d ρij , d ρji for each arrow ρ : i → j in ∆ such that there existpositive integers f i , i ∈ ∆ , satisfying d ρij f j = d ρji f i for all arrows ρ : i → j in ∆ . We call ( d ρij , d ρji ) ( ρ : i → j ) ∈ ∆ a (relative) valuation of ∆. By aslight abuse of notation, we often refer to ∆ as a relative valued quiver, leaving the valuationimplied.We will use the notation: i jρ ( d ρij , d ρji ) .In the case that ( d ρij , d ρji ) = (1 , relative valued quiver morphism is a quiver morphism ϕ : ∆ → ∆ ′ satisfying:( d ′ ) ϕ ( ρ ) ϕ ( i ) ϕ ( j ) = d ρij and ( d ′ ) ϕ ( ρ ) ϕ ( j ) ϕ ( i ) = d ρji for all arrows ρ : i → j in ∆ . JOEL LEMAY
Let Q rel denote the category of relative valued quivers.Note that the definition of a relative valued quiver closely resembles the definition of asymmetrizable Cartan matrix. We will explore the link between the two in Section 6, whichdeals with representations.As with absolute valued quivers, one can view (non-valued) quivers as relative valuedquivers with trivial values (i.e. all ( d ρij , d ρji ) = (1 , Q abs and Q rel are related. Given Γ ∈ Q abs with valuation ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ , define F (Γ) ∈ Q rel with valuation ( d ρij , d ρji ) ( ρ : i → j ) ∈ F (Γ) asfollows: • the underlying quiver of F (Γ) is equal to that of Γ, • the values ( d ρij , d ρji ) are given by: d ρij = m ρ d j and d ρji = m ρ d i for all arrows ρ : i → j in F (Γ) .It is clear that F (Γ) satisfies the definition of a relative valued quiver (simply set all the f i = d i ). Given a morphism ϕ : Γ → Γ ′ in Q abs , one can simply define F ( ϕ ) : F (Γ) → F (Γ ′ )to be the morphism given by ϕ , since Γ and Γ ′ have the same underlying quivers as F (Γ)and F (Γ ′ ), respectively. By construction of F (Γ) and F (Γ ′ ), it is clear then that F ( ϕ ) is amorphism in Q rel . Thus, F is a functor from Q abs to Q rel . Lemma 1.5.
The functor F : Q abs → Q rel is faithful and surjective.Proof. Suppose F ( ϕ ) = F ( ψ ) for two morphisms ϕ, ψ : Γ → Γ ′ in Q abs . By definition, F ( ϕ ) = ϕ on the underlying quivers of Γ and Γ ′ . Likewise for F ( ψ ) and ψ . Thus, ϕ = ψ and F is faithful.Suppose ∆ is a relative valued quiver. By definition, there exist positive integers f i , i ∈ ∆ , such that d ρij f j = d ρji f i for each arrow ρ : i → j in ∆ . Fix a particular choice ofthese f i . Define Γ ∈ Q abs as follows: • the underlying quiver of Γ is the same as that of ∆, • set d i = f i for each i ∈ Γ = ∆ , • set m ρ = d ρij f j = d ρji f i for each arrow ρ : i → j in Γ = ∆ .Then, Γ is an absolute valued quiver and F (Γ) = ∆. Thus, F is surjective. (cid:3) Note that F is not full, and thus not an equivalence of categories, as the following exampleillustrates. Example 1.6.
Consider the following two non-isomorphic absolute valued quivers.Γ: Γ ′ : (2)(2)(1) (4)(4)(2) Both Γ and Γ ′ are mapped to: F (Γ) = F (Γ ′ ): (2,1) . ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 7
One sees that Hom Q abs (Γ , Γ ′ ) is empty whereas Hom Q rel ( F (Γ) , F (Γ ′ )) is not (it contains theidentity). Thus, F : Hom Q abs (Γ , Γ ′ ) → Hom Q rel ( F (Γ) , F (Γ ′ ))is not surjective, and hence F is not full.It is tempting to think that one could remedy this by restricting F to the full subcategoryof Q abs consisting of objects Γ with valuations ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ such that the greatest commondivisor of all d i is 1. While one can show that F restricted to this subcategory is injectiveon objects, it would still not be full, as the next example illustrates. Example 1.7.
Consider the following two absolute valued quivers.Γ: Γ ′ : (2)(2)(1) ρ (4)(4)(2)(2)(1) α β The values of the vertices of Γ have greatest common divisor 1. The same is true of Γ ′ . Byapplying F we get: F (Γ): F (Γ ′ ): (2,1) ρ (2,1) (2,1) α β One sees that Hom Q abs (Γ , Γ ′ ) contains only one morphism (induced by ρ β ), while on theother hand Hom Q rel ( F (Γ) , F (Γ ′ )) contains two morphisms (induced by ρ α and ρ β ).Thus, F : Hom Q abs (Γ , Γ ′ ) → Hom Q rel ( F (Γ) , F (Γ ′ ))is not surjective, and hence F is not full, even when restricted to the subcategory of objectswith vertex values having greatest common divisor 1.Note that there is no similar functor Q rel → Q abs . Following the proof of Lemma 1.5, onesees that finding a preimage under F of a relative valued quiver ∆ is equivalent to making achoice of f i (from Definition 1.4). One can show that there is a unique such choice satisfyinggcd( f i ) i ∈ ∆ = 1 (so long as ∆ is connected). Thus, there is a natural and well-defined wayto map objects of Q rel to objects of Q abs by mapping a relative valued quiver ∆ to theunique absolute valued quiver Γ with valuation ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ satisfying F (Γ) = ∆ andgcd( d i ) i ∈ Γ = 1. However, there is no such natural mapping on the morphisms of Q rel . Forinstance, under this natural mapping on objects, in Example 1.7, the relative valued quivers F (Γ) and F (Γ ′ ) are mapped to Γ and Γ ′ , respectively. However, there is no natural way tomap the morphism F (Γ) → F (Γ ′ ) induced by ρ α to a morphism Γ → Γ ′ since there isno morphism Γ → Γ ′ such that ρ α . Thus, there does not appear to be a functor similarto F from Q rel to Q abs . 2. Species and K -species The reason for introducing two different definitions of valued quivers in the previoussection, is that there are two different definitions of species in the literature: one for each ofthe two versions of valued quivers. In this section, we introduce both definitions of speciesand discuss how they are related (see Proposition 2.6 as well as Examples 2.7, 2.8 and 2.9).First, we begin with the more general definition of species (see for example [13] or [11]).Recall that if R and S are rings and M is an ( R, S )-bimodule, then Hom R ( M, R ) is an
JOEL LEMAY ( S, R )-bimodule via ( s · ϕ · r )( m ) = ϕ ( m · s ) r and Hom S ( M, S ) is an (
S, R )-bimodule via( s · ψ · r )( m ) = sψ ( r · m ) for all r ∈ R , s ∈ S , m ∈ M , ϕ ∈ Hom R ( M, R ) and ψ ∈ Hom S ( M, S ). Definition 2.1 (Species) . Let ∆ be a relative valued quiver with valuation( d ρij , d ρji ) ( ρ : i → j ) ∈ ∆ . A modulation M of ∆ consists of a division ring K i for each i ∈ ∆ ,and a ( K h ( ρ ) , K t ( ρ ) )-bimodule M ρ for each ρ ∈ ∆ such that the following two conditionshold:(a) Hom K t ( ρ ) ( M ρ , K t ( ρ ) ) ∼ = Hom K h ( ρ ) ( M ρ , K h ( ρ ) ) as ( K t ( ρ ) , K h ( ρ ) )-bimodules, and(b) dim K t ( ρ ) ( M ρ ) = d ρh ( ρ ) t ( ρ ) and dim K h ( ρ ) ( M ρ ) = d ρt ( ρ ) h ( ρ ) .A species (also called a modulated quiver ) Q is a pair (∆ , M ), where ∆ is a relative valuedquiver and M is a modulation of ∆.A species morphism Q → Q ′ consists of a relative valued quiver morphism ϕ : ∆ → ∆ ′ ,a division ring morphism ψ i : K i → K ′ ϕ ( i ) for each i ∈ ∆ , and a compatible abelian grouphomomorphism ψ ρ : M ρ → M ′ ϕ ( ρ ) for each ρ ∈ ∆ . That is, for every ρ ∈ ∆ we have ψ ρ ( a · m ) = ψ h ( ρ ) ( a ) · ψ ρ ( m ) and ψ ρ ( m · b ) = ψ ρ ( m ) · ψ t ( ρ ) ( b ) for all a ∈ K h ( ρ ) , b ∈ K t ( ρ ) and m ∈ M ρ .Let M denote the category of species. Remark 2.2.
Notice that we allow parallel arrows in our definition of valued quivers andthus in our definition of species. However, many texts only allow for single arrows in theirdefinition of species. We will see in Sections 5 and 6 that we can always assume, without lossof generality, that we have no parallel arrows. Thus our definition of species is consistentwith the other definitions in the literature.Another definition of species also appears in the literature (see for example [9]). Thisdefinition depends on a central field K , and so to distinguish between the two definitions,we will call these objects K -species. Definition 2.3 (K-species) . Let Γ be an absolute valued quiver with valuation( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ . A K -modulation M of Γ consists of a K -division algebra K i for each i ∈ Γ ,and a ( K h ( ρ ) , K t ( ρ ) )-bimodule M ρ for each ρ ∈ Γ , such that the following two conditionshold:(a) K acts centrally on M ρ (i.e. k · m = m · k ∀ k ∈ K, m ∈ M ρ ), and(b) dim K ( K i ) = d i and dim K ( M ρ ) = m ρ .A K -species (also called a K -modulated quiver ) Q is a pair (Γ , M ), where Γ is an absolutevalued quiver and M is a K -modulation of Γ.A K -species morphism Q → Q ′ consists of an absolute valued quiver morphism ϕ : Γ → Γ ′ , a K -division algebra morphism ψ i : K i → K ′ ϕ ( i ) for each i ∈ Γ , and a compatible K -linear map ψ ρ : M ρ → M ′ ϕ ( ρ ) for each ρ ∈ Γ . That is, for every ρ ∈ Γ we have ψ ρ ( a · m ) = ψ h ( ρ ) ( a ) · ψ ρ ( m ) and ψ ρ ( m · b ) = ψ ρ ( m ) · ψ t ( ρ ) ( b ) for all a ∈ K h ( ρ ) , b ∈ K t ( ρ ) and m ∈ M ρ .Let M K denote the category of K -species.Note that, given a base field K , not every absolute valued quiver has a K -modulation.For example, it is well-known that the only division algebras over R are R , C and H , whichhave dimension 1, 2 and 4, respectively. Thus, any absolute valued quiver containing a ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 9 vertex with value 3 (or any value not equal to 1, 2 or 4) has no R -modulation. However,given an absolute valued quiver, we can always find a base field K for which there exists a K -modulation. For example, Q admits field extensions (thus division algebras) of arbitrarydimension, thus Q -modulations always exist.It is also worth noting that given a valued quiver, relative or absolute, there may existseveral non-isomorphic species or K -species (depending on the field K ). Example 2.4.
Consider the following absolute valued quiver Γ and its image under F .Γ: F (Γ): (2) (1)(2) (2,1) One can construct the following two Q -species of Γ. Q : Q ′ : Q ( √ Q ( √ Q Q ( √ Q ( √ Q Then Q ≇ Q ′ as Q -species, since Q ( √ ≇ Q ( √
3) as algebras. Also, one can show that Q and Q ′ are species of F (Γ) (indeed, this will follow from Proposition 2.6). But again, Q ≇ Q ′ as species since Q ( √ ≇ Q ( √
3) as rings.It is natural to ask how species and K -species are related, i.e. how the categories M and M K are related. To answer this question, we first need the following lemma. Lemma 2.5.
Let F and G be finite-dimensional (nonzero) division algebras over a field K and let M be a finite-dimensional ( F, G ) -bimodule on which K acts centrally. Then Hom F ( M, F ) ∼ = Hom G ( M, G ) as ( G, F ) -bimodules.Proof. A proof can be found in [22, Lemma 3.7], albeit using slightly different terminology.For convenience, we present a brief sketch of the proof.Let τ : F → K be a nonzero K -linear map such that τ ( ab ) = τ ( ba ) for all a, b ∈ F . Sucha map is known to exist; one can take the reduced trace map F → Z ( F ), where Z ( F ) isthe centre of F (see [29, Chapter IX, Section 2, Proposition 6]) and compose it with anynonzero map Z ( F ) → K . Then T : Hom F ( M, F ) → Hom K ( M, K ) defined by ϕ τ ◦ ϕ isa ( G, F )-bimodule isomorphism. By an analogous argument, Hom G ( M, G ) ∼ = Hom K ( M, K )completing the proof. (cid:3)
Given Lemma 2.5, we see that if Q is a K -species with absolute valued quiver Γ, then Q isa species with underlying relative valued quiver F (Γ). Also, a K -species morphism Q → Q ′ is a species morphism when viewing Q and Q ′ as species (because an algebra morphism is aring morphism and a linear map is a group homomorphism). Thus, we may define a forgetfulfunctor U K : M K → M , which forgets the underlying field K and views absolute valuedquivers as relative valued quivers via the functor F . This yields the following result. Proposition 2.6.
The functor U K : M K → M is faithful and injective on objects. Hence,we may view M K as a subcategory of M .Proof. Faithfulness is clear, since F is faithful and U K then simply forgets the underlyingfield K .To see that U K is injective on objects, suppose Q and Q ′ are K -species with modulations( K i , M ρ ) i ∈ Γ ,ρ ∈ Γ and ( K ′ i , M ′ ρ ) i ∈ Γ ′ ,ρ ∈ Γ ′ , respectively, such that U K ( Q ) = U K ( Q ′ ). Then, the underlying (non-valued) quivers of Q and Q ′ are equal. Moreover, K i = K ′ i for all i ∈ Γ = Γ ′ and M ρ = M ′ ρ for all ρ ∈ Γ = Γ ′ . So, Q = Q ′ and thus U K is injective. (cid:3) Note that U K is not full (and hence we cannot view M K as a full subcategory of M ) noris it essentially surjective. In fact, there are objects in M which are not of the form U K ( Q )for Q ∈ M K for any field K . The following examples illustrate these points. Example 2.7.
Consider C as a C -species, that is C is a C -modulation of the trivially valuedquiver consisting of one vertex and no arrows. Then, the only morphism in Hom C ( C , C )is the identity, since any such morphism must send 1 to 1 and be C -linear. However,Hom U C ( C ) ( U C ( C ) , U C ( C )) contains more than just the identity. Indeed, let ϕ : C → C given by z z . Then, ϕ is a ring morphism and thus defines a species morphism. Hence, U C : Hom C ( C , C ) → Hom U C ( C ) ( U C ( C ) , U C ( C ))is not surjective and so U C is not full. Example 2.8.
There exist division rings which are not finite-dimensional over their centres;such division rings are called centrally infinite . Hilbert was the first to construct such aring (see for example [19, Proposition 14.2]). Suppose R is a centrally infinite ring. Then,for any field K contained in R such that R is a K -algebra, K ⊆ Z ( R ) and so R is notfinite-dimensional K -algebra. Thus, any species containing R as part of its modulation isnot isomorphic to any object in the image of U K for any field K .One might think that we could eliminate this problem by restricting ourselves to modu-lations containing only centrally finite rings. In other words, one might believe that if Q isa species whose modulation contains only centrally finite rings, then we can find a field K and a K -species Q ′ such that Q ∼ = U K ( Q ′ ). However, this is not the case as we see in thefollowing example. Example 2.9.
Let p be a prime. Consider: Q : G FM ,where F = G = F p ( F and G are then centrally finite since they are fields) and M = F p isan ( F, G )-bimodule with actions: f · m · g = f mg p for all f ∈ F , g ∈ G and m ∈ M . We claim that Q is a species. The dimension criterion isclear, as dim F M = dim G M = 1. Thus, it remains to show thatHom F ( M, F ) ∼ = Hom G ( M, G )as (
G, F )-bimodules. Recall that in F p , p -th roots exist and are unique. Indeed, for any a ∈ F p , the p -th roots of a are the roots of the polynomial x p − a . Because F p is algebraicallyclosed, this polynomial has a root, say α . Because char F p = p we have( x − α ) p = x p − α p = x p − a. Hence, α is the unique p -th root of a . Therefore, we have a well-defined map:Φ : Hom F ( M, F ) → Hom G ( M, G ) ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 11 ϕ ρ ◦ ϕ, where ρ is the p -th root map. It is straightforward to show that Φ is a ( G, F )-bimoduleisomorphism.Therefore, Q is a species. Yet the field F p does not act centrally on M . Indeed, take anelement a / ∈ F p , then a · a = a p = 1 · a. In fact, the only subfield that does act centrally on M is F p since a p = a if and only if a ∈ F p . But, F and G are infinite-dimensional over F p . Thus, there is no field K for which Q is isomorphic to an object of the form U K ( Q ′ ) with Q ′ ∈ M K .3. The Path and Tensor Algebras
In this section we will define the path and tensor algebras associated to quivers andspecies, respectively. These algebras play an important role in the representation theory offinite-dimensional algebras (see Theorems 3.4 and 3.6, and Corollary 3.13). In subsequentsections, we will give a more in-depth study of these algebras (Sections 4 and 5) and we willshow that modules of path and tensor algebras are equivalent to representations of quiversand species, respectively (Section 6).Recall that a path of length n in a quiver Q is a sequence of n arrows in Q , ρ n ρ n − · · · ρ ,such that h ( ρ i ) = t ( ρ i +1 ) for all i = 1 , , . . . , n −
1. For every vertex, we have a trivial pathof length 0 (beginning and ending at that vertex).
Definition 3.1 (Path algebra) . The path algebra , KQ , of a quiver Q is the K -algebra withbasis the set of all the paths in Q and multiplication given by:( β n β n − · · · β )( α m α m − · · · α ) = ( β n β n − · · · β α m α m − · · · α , if t ( β ) = h ( α m ) , , otherwise . Remark 3.2.
According to the convention used, a path i ρ −→ i ρ −→ · · · ρ n − −−−→ i n ρ n −→ i n +1 is written from “right to left” ρ n ρ n − · · · ρ . However, some texts write paths from “left toright” ρ ρ · · · ρ n . Using the “left to right” convention yields a path algebra that is oppositeto the one defined here.Note that KQ is associative and unital (its identity is P i ∈ Q ε i , where ε i is the pathof length zero at i ). Also, KQ is finite-dimensional precisely when Q contains no orientedcycles. Definition 3.3 (Admissible ideal) . Let Q be a quiver and let P n ( Q ) = span K { all paths in Q of length ≥ n } . An admissible ideal I of the path algebra KQ is a two-sided ideal of KQ satisfying P n ( Q ) ⊆ I ⊆ P ( Q ) , for some positive integer n .If Q has no oriented cycles, then any ideal I ⊆ P ( Q ) of KQ is an admissible ideal, since P n ( Q ) = 0 for sufficiently large n .There is a strong relationship between path algebras and finite-dimensional algebras,touched upon by Brauer [6], Jans [17] and Yoshii [30], but fully explored by Gabriel [13].Let A be a finite-dimensional K -algebra. We recall a few definitions. An element ε ∈ A is called an idempotent if ε = ε . Two idempotents ε and ε are called orthogonal if ε ε = ε ε = 0. An idempotent ε is called primitive if it cannot be written as a sum ε = ε + ε , where ε and ε are orthogonal idempotents. A set of idempotents { ε , . . . , ε n } is called complete if P ni =1 ε i = 1. If { ε , . . . , ε n } is a complete set of primitive (pairwise)orthogonal idempotents of A , then A = Aε ⊕ · · · ⊕ Aε n is a decomposition of A (as a left A -module) into indecomposable modules; this decomposition is unique up to isomorphismand permutation of the terms. We say that A is basic if Aε i ≇ Aε j as (left) A -modulesfor all i = j (or, alternatively, the decomposition of A into indecomposable modules admitsno repeated factors). Finally, A is called hereditary if every A -submodule of a projective A -module is again projective. Theorem 3.4.
Let K be an algebraically closed field and let A be a finite-dimensional K -algebra. (a) If A is basic and hereditary, then A ∼ = KQ (as K -algebras) for some quiver Q . (b) If A is basic, then A ∼ = KQ/I (as K -algebras) for some quiver Q and some admissibleideal I of KQ . For a proof of Theorem 3.4 see [1, Sections II and VII] or [3, Propositions 4.1.7 and 4.2.4](though it also follows from [10, Proposition 10.2]). The above result is powerful, but itdoes not necessarily hold over fields which are not algebraically closed. If we want to workwith algebras over non-algebraically closed fields, we need to generalize the notion of a pathalgebra. We look, then, at the analogue of the path algebra for a K -species.Let Q be a species of a relative valued quiver ∆ with modulation ( K i , M ρ ) i ∈ ∆ ,ρ ∈ ∆ . Let D = Π i ∈ ∆ K i and let M = L ρ ∈ ∆ M ρ . Then D is a ring and M naturally becomes a( D, D )-bimodule. If Q is a K -species, then D is a K -algebra. Definition 3.5 (Tensor ring/algebra) . The tensor ring , T ( Q ), of a species Q is defined by T ( Q ) = ∞ M n =0 T n ( M ) , where T ( M ) = D and T n ( M ) = T n − ( M ) ⊗ D M for n ≥ . Multiplication is determined by the composition T m ( M ) × T n ( M ) ։ T m ( M ) ⊗ D T n ( M ) ∼ = −→ T m + n ( M ) . If Q is a K -species, then T ( Q ) is a K -algebra. In this case we call T ( Q ) the tensor algebra of Q .Admissible ideals for tensor rings/algebras are defined in the same way as admissibleideals for path algebras by setting P n ( Q ) = L ∞ m = n T m ( M ).Suppose that Γ is an absolute valued quiver with trivial valuation (all d i and m ρ areequal to 1) and Q is a K -species of Γ. Then, for each i ∈ Γ , dim K K i = 1, which impliesthat K i ∼ = K (as K -algebras). Likewise, dim K M ρ = 1 implies that M ρ ∼ = K (as ( K, K )-bimodules). Therefore, it follows that T ( Q ) ∼ = KQ where Q = (Γ , Γ ). Thus, when viewingnon-valued quivers as absolute valued quivers with trivial valuation, the tensor algebra ofthe K -species becomes simply the path algebra (over K ) of the quiver. Therefore, the tensoralgebra is indeed a generalization of the path algebra. Additionally, the tensor algebra allowsus to generalize Theorem 3.4. ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 13
Recall that a field K is called perfect if either char( K ) = 0 or, if char( K ) = p >
0, then K p = { a p | a ∈ K } = K . Theorem 3.6.
Let K be a perfect field and let A be a finite-dimensional K -algebra. (a) If A is basic and hereditary, then A ∼ = T ( Q ) (as K -algebras) for some K -species Q . (b) If A is basic, then A ∼ = T ( Q ) /I (as K -algebras) for some K -species Q and someadmissible ideal I of T ( Q ) . For a proof of Theorem 3.6, see [10, Proposition 10.2] or [3, Corollary 4.1.11 and Propo-sition 4.2.5] or [12, Section 8.5]. Note that Theorem 3.6 does not necessarily hold overnon-perfect fields. To see why, we first introduce a useful tool in the study of path andtensor algebras.
Definition 3.7 (Jacobson radical) . The
Jacobson radical of a ring R is the intersection ofall maximal left ideals of R . We denote the Jacobson radical of R by rad R . Remark 3.8.
The intersection of all maximal left ideals coincides with the intersection ofall maximal right ideals (see, for example, [19, Corollary 4.5]), so the Jacobson radical couldalternatively be defined in terms of right ideals.
Lemma 3.9.
Let Q be a species. (a) If Q contains no oriented cycles, then rad T ( Q ) = L ∞ n =1 T n ( M ) . (b) Let I be an admissible ideal of T ( Q ) . Then, rad ( T ( Q ) /I ) = ( L ∞ n =1 T n ( M )) /I .Proof. It is well known that if R is a ring and J is a two-sided nilpotent ideal of R such that R/J is semisimple, then rad R = J (see for example [19, Lemma 4.11 and Proposition 4.6]together with the fact that the radical of a semisimple ring is 0). Let J = L ∞ n =1 T n ( M ). If Q contains no oriented cycles, then T n ( M ) = 0 for some positive integer n . Thus, J n = 0and J is then nilpotent. Then T ( Q ) /J ∼ = T ( M ) = D , which is semisimple. Therefore,rad T ( Q ) = J , proving Part 1.If I is an admissible ideal of T ( Q ), let J = ( L ∞ n =1 T n ( M ) /I ). By definition, P n ( Q ) ⊆ I for some n and so J n = 0. Thus, Part 2 follows by a similar argument. (cid:3) Remark 3.10.
Part 1 of Lemma 3.9 is false if Q contains oriented cycles. One does notneed to look beyond quivers to see why. For example, following [1, Section II, Chapter 1],we can consider the path algebra of the Jordan quiver over an infinite field K . That is, weconsider KQ , where: Q : .Then it is clear that KQ ∼ = K [ t ], the polynomial ring in one variable. For each α ∈ K , let I α be the ideal generated by t + α . Each I α is a maximal ideal and T α ∈ K I α = 0 since K is infinite. Thus rad KQ = 0 whereas L ∞ n =1 T n ( Q ) ∼ = ( t ) (the ideal generated by the lonearrow of Q ).With the concept of the Jacobson radical and Lemma 3.9, we are ready to see whyTheorem 3.6 fails over non-perfect fields. Recall that a K -algebra epimorphism ϕ : A ։ B is said to split if there exists a K -algebra morphism µ : B → A such that ϕ ◦ µ = id B . We seethat if A = T ( Q ) /I for a K -species Q and admissible ideal I , then the canonical projection A ։ A/ rad A splits (since A ∼ = D ⊕ rad A ). Thus, to construct an example where Theorem3.6 fails, it suffices to find an algebra where this canonical projection does not split. This ispossible over a non-perfect field. Example 3.11. [3, Remark (ii) following Corollary 4.1.11] Let K be a field of characteristic p = 0 and let K = K ( t ), which is not a perfect field. Let A = K [ x, y ] / ( x p , y p − x − t ). Aquick calculation shows that rad A = ( x ) and thus A/ rad A ∼ = K [ y ] / ( y p − t ). One can easilyverify that the projection A ։ A/ rad A does not split. Hence, A is not isomorphic to thequotient of the tensor algebra of a species by some admissible ideal.Theorems 3.4 and 3.6 require our algebras to be basic. There is a slightly weaker propertythat holds in the case of non-basic algebras, that of Morita equivalence . Definition 3.12 (Morita equivalence) . Two rings R and S are said to be Morita equivalent if their categories of (left) modules, R -Mod and S -Mod, are equivalent. Corollary 3.13.
Let K be a field and let A be a finite-dimensional K -algebra. (a) If K is algebraically closed and A is hereditary, then A is Morita equivalent to KQ for some quiver Q . (b) If K is algebraically closed, then A is Morita equivalent to KQ/I for some quiver Q and some admissible ideal I of KQ . (c) If K is perfect and A is hereditary, then A is Morita equivalent to T ( Q ) for some K -species Q . (d) If K is perfect, then A is Morita equivalent to T ( Q ) /I for some K -species Q andsome admissible ideal I of T ( Q ) .Proof. Every algebra is Morita equivalent to a basic algebra (see [3, Section 2.2]) and Moritaequivalence preserves the property of being hereditary (indeed, an equivalence of categoriespreserves projective modules). Thus, the result follows as a consequence of Theorems 3.4and 3.6. (cid:3) The Frobenius Morphism
When working over the finite field of q elements, F q , it is possible to avoid dealing withspecies altogether and deal only with quivers with automorphism. This is achieved by usingthe Frobenius morphism (described below). Definition 4.1 (Frobenius morphism) . Let K = F q (the algebraic closure of F q ). Given aquiver with automorphism ( Q, σ ), the
Frobenius morphism F = F Q,σ,q is defined as F : KQ → KQ X i λ i p i X i λ qi σ ( p i )for all λ i ∈ K and paths p i in Q . The F -fixed point algebra is( KQ ) F = { x ∈ KQ | F ( x ) = x } . Note that while KQ is an algebra over K , the fixed point algebra ( KQ ) F is an algebraover F q . Indeed, suppose 0 = x ∈ ( KQ ) F , then F ( λx ) = λ q F ( x ) = λ q x . Thus, λx ∈ ( KQ ) F if and only if λ q = λ , which occurs if and only if λ ∈ F q . ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 15
Suppose Γ is the absolute valued quiver obtained by folding (
Q, σ ). For each i ∈ Γ andeach ρ ∈ Γ define A i = M a ∈ i Kε a and A ρ = M τ ∈ ρ Kτ. where ε a is the trivial path at vertex a . Then as an F q -algebra, A Fi ∼ = F q di . Indeed, fix some a ∈ i , then, A Fi = ( x = d i − X j =0 λ j ε σ j ( a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ j ∈ K and F ( x ) = x ) Applying F to an arbitrary x = P d i − j =0 λ j ε σ j ( a ) ∈ A Fi , we obtain: F ( x ) = d i − X j =0 λ qj σ ( ε σ j ( a ) ) = d i − X j =0 λ qj ε σ j +1 ( a ) . The equality F ( x ) = x yields λ qj = λ j +1 for j = 0 , , . . . , d i − λ qd i − = λ . By successivesubstitution, we get λ q di = λ , which occurs if and only if λ ∈ F q di , and λ j = λ q j . Thus, A Fi can be rewritten as: A Fi = ( d i − X j =0 λ q j ε σ j ( a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ∈ F q di ) ∼ = F q di (as fields).It is easy to see that A ρ is an ( A h ( ρ ) , A t ( ρ ) )-bimodule via multiplication, thus A Fρ is an( A Fh ( ρ ) , A Ft ( ρ ) )-bimodule (on which F q acts centrally). Since A Fi ∼ = F q di for each i ∈ Γ , A Fρ isthen an ( F q dh ( ρ ) , F q dt ( ρ ) )-bimodule. Over fields, we make no distinction between left and rightmodules because of commutativity. Thus, A Fρ is an F q dh ( ρ ) -module and an F q dt ( ρ ) -module,and hence A Fρ is a module of the composite field of F q dh ( ρ ) and F q dt ( ρ ) , which in this case issimply the bigger of the two fields (recall that the composite of two fields is the smallest fieldcontaining both fields). Over fields, all modules are free and thus A Fρ is a free module of thecomposite field (this fact will be useful later on). Also, dim F q A Fi = d i and dim F q A Fρ = m ρ (the dimensions are the number of vertices/arrows in the corresponding orbits). Therefore, M = ( A Fi , A Fρ ) i ∈ Γ ,ρ ∈ Γ defines an F q -modulation of Γ. We will denote the F q -species (Γ, M )by Q Q,σ,q . This leads to the following result.
Theorem 4.2. [9, Theorem 3.25]
Let ( Q, σ ) be a quiver with automorphism. Then ( KQ ) F ∼ = T ( Q Q,σ,q ) as F q -algebras. In light of Theorem 4.2, the natural question to ask is: given an arbitrary F q -species, is itstensor algebra isomorphic to the fixed point algebra of a quiver with automorphism? And ifso, to which one?Suppose Q is an F q -species with underlying absolute valued quiver Γ and F q -modulation( K i , M ρ ) i ∈ Γ ,ρ ∈ Γ . Each K i is, by definition, a division algebra containing q d i elements. Ac-cording to the well-known Wedderburn’s little theorem, all finite division algebras are fields.Thus, K i ∼ = F q di . Similar to the above discussion, M ρ is then a free module of the composite field of F q dh ( ρ ) and F q dt ( ρ ) . Therefore, by unfolding Γ (as in Section 1, say) to a quiver withautomorphism ( Q, σ ), we get
Q ∼ = Q Q,σ,q as F q -species. This leads to the following result. Proposition 4.3. [9, Proposition 3.37]
For any F q -species Q , there exists a quiver withautomorphism ( Q, σ ) such that T ( Q ) ∼ = ( KQ ) F as F q -algebras. Note that, given an F q -species Q = (Γ , M ) and a quiver with automorphism ( Q, σ ) suchthat T ( Q ) ∼ = ( KQ ) F , we cannot to conclude that ( Q, σ ) folds into Γ as the following exampleillustrates.
Example 4.4.
Consider the following quiver. Q :There are two possible automorphisms of Q : σ = id Q and σ ′ , the automorphism definedby interchanging the two arrows of Q . By folding Q with respect to σ and σ ′ , we obtain thefollowing two absolute valued quivers.Γ: Γ ′ : (1) (1)(1) (1) (1) (2) (1) It is clear that Γ and Γ ′ are not isomorphic. Now, construct F q -species of Γ and Γ ′ withthe following F q -modulations. Q : Q ′ : F q F q F q F q F q F q F q Then we have that T ( Q ) ∼ = T ( Q ′ ). Thus, ( KQ ) F Q,σ,q ∼ = T ( Q ′ ), yet ( Q, σ ) does not foldinto Γ ′ .The above example raises an interesting question. Notice that the two F q -species Q and Q ′ are not isomorphic, but their tensor algebras T ( Q ) and T ( Q ′ ) are isomorphic. Thisphenomenon is not restricted to finite fields either; if we replaced F q with some arbitraryfield K , we still get Q ≇ Q ′ as K -species, but T ( Q ) ∼ = T ( Q ′ ) as K -algebras. So we mayask: under what conditions are the tensor algebras (or rings) of two K -species (or species)isomorphic? We answer this question in the following section.It is worth noting that over infinite fields, there are no (known) methods to extend theresults of this section. It is tempting to think that, given an infinite field K and a quiverwith automorphism ( Q, σ ) that folds into an absolute valued quiver Γ, Theorem 4.2 mightbe extended by saying that the fixed point algebra ( KQ ) σ ( σ extends to an automorphismof KQ ) is isomorphic to the tensor algebra of a K -species of Γ. This is, however, not thecase as the following example illustrates. Example 4.5.
Take K = R to be our (infinite) base field. Consider the following quiver. Q : ε ε ε α β Let σ be the automorphism of Q given by: σ : (cid:18) ε ε ε α βε ε ε β α (cid:19) . ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 17
Then (
Q, σ ) folds into the following absolute valued quiver.Γ : (2)(2)(1)So, we would like for ( KQ ) σ to be isomorphic to the tensor algebra of a K -species of Γ.However, this does not happen.An element x = a ε + a ε + a ε + a α + a β ∈ KQ is fixed by σ if and only if σ ( x ) = x ,that is, if and only if a ε + a ε + a ε + a α + a β = a ε + a ε + a ε + a α + a β, which occurs if and only if a = a and a = a . Hence, ( KQ ) σ has basis { ε + ε , ε , α + β } and we see that it is isomorphic to the path algebra (over K ) of the following quiver. Q ′ : ε + ε α + β ε The algebra KQ ′ is certainly not isomorphic to the tensor algebra of any K -species of Γ.Indeed, over K , KQ ′ has dimension 3 whereas the tensor algebra of any K -species of Γ hasdimension 5. 5. A Closer Look at Tensor Rings
In this section, we find necessary and sufficient conditions for two tensorrings/algebras to be isomorphic. We show that the isomorphism of tensorrings/algebras corresponds to an equivalence on the level of species (see Theorem 5.5). Fur-thermore, we show that if Q is a K -species, where K is a perfect field, then K ⊗ K T ( Q )is either isomorphic to, or Morita equivalent to, the path algebra of a quiver (see Theorem5.8 and Corollary 5.13). This serves as a partial generalization to [15, Lemma 21] in whichHubery proved a similar result when K is a finite field. We begin by introducing the notionof “crushing”. Definition 5.1 (Crushed absolute valued quiver) . Let Γ be an absolute valued quiver. Definea new absolute valued quiver, which we will denote Γ C , as follows: • Γ C = Γ , • i → j = ( , if ∃ ρ : i → j ∈ Γ , , otherwise , • d Ci = d i for all i ∈ Γ C = Γ , • m Cρ = P ( α : t ( ρ ) → h ( ρ )) ∈ Γ m α for all ρ ∈ Γ C .Intuitively, one “crushes” all parallel arrows of Γ into a single arrow and sums up thevalues. ( d i ) ( d j )( m ρ )( m ρ )( m ρ r )... ( d i ) ( d j )( m ρ + m ρ + · · · + m ρ r ) The absolute valued quiver Γ C will be called the crushed (absolute valued) quiver of Γ. Note that Γ C does indeed satisfy the definition of an absolute valued quiver. Take any ρ : i → j ∈ Γ C . Then d Ci = d i | m α for all α : i → j in Γ . Thus, d Ci | (cid:16)P ( α : i → j ) ∈ Γ m α (cid:17) = m Cρ . The same is true for d Cj . Therefore, Γ C is an absolute valued quiver.The notion of crushing can be extended to relative valued quivers via the functor F . Recallthat if Γ is an absolute valued quiver with valuation ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ , then F (Γ) is a relativevalued quiver with valuation ( d ρij , d ρji ) ( ρ : i → j ) ∈ F (Γ) given by d ρij = m ρ /d j and d ρji = m ρ /d i . So,the valuation of F (Γ C ) is given by( d C ) ρij = X ( α : i → j ) ∈ Γ m α /d j = X ( α : i → j ) ∈ Γ d αij and likewise ( d C ) ρji = X ( α : i → j ) ∈ Γ m α /d i = X ( α : i → j ) ∈ Γ d αji for each ρ : i → j in F (Γ C ) . We take this to be the definition of the crushed (relativevalued) quiver of a relative valued quiver. Definition 5.2 (Crushed relative valued quiver) . Let ∆ be a relative valued quiver. Definea new relative valued quiver, which we will denote ∆ C , as follows: • ∆ C = ∆ , • i → j = ( , if ∃ ρ : i → j ∈ ∆ , , otherwise , • ( d C ) ρij = P ( α : i → j ) ∈ ∆ d αij and ( d C ) ρji = P ( α : i → j ) ∈ ∆ d αji for all ρ : i → j in ∆ C .Again, the intuition is to “crush” all parallel arrows in ∆ into a single arrow and sum thevalues. i j ( d ρ ij , d ρ ji )( d ρ ij , d ρ ji )( d ρ r ij , d ρ r ji )... i j ( d ρ ij + · · · + d ρ r ij , d ρ ji + · · · + d ρ r ji ) The relative valued quiver ∆ C will be called the crushed (relative valued) quiver of ∆. Definition 5.3 (Crushed species) . Let Q be a species with underlying relative valued quiver∆. Define a new species, which we will denote Q C , as follows: • the underlying valued quiver of Q C is ∆ C , • K Ci = K i for all i ∈ ∆ C = ∆ , • M Cρ = L ( α : i → j ) ∈ ∆ M α for all ρ : i → j in ∆ C .The intuition here is similar to that of the previous definitions; one “crushes” all bimodulesalong parallel arrows into a single bimodule by taking their direct sum. The species Q C willbe called the crushed species of Q . Remark 5.4. A crushed K -species is defined in exactly the same way, only using the crushedquiver of an absolute valued quiver instead of a relative valued quiver. ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 19
Note that in the above definition Q C is indeed a species of ∆ C . Clearly, M Cρ is a ( K Cj , K Ci )-bimodule for all ρ : i → j in ∆ C . Moreover,Hom K Cj ( M Cρ , K Cj ) = Hom K j M ( α : i → j ) ∈ ∆ M α , K j ∼ = M ( α : i → j ) ∈ ∆ Hom K j ( M α , K j ) ∼ = M ( α : i → j ) ∈ ∆ Hom K i ( M α , K i ) ∼ = Hom K i M ( α : i → j ) ∈ ∆ M α , K i = Hom K i ( M Cρ , K Ci ) , where all isomorphisms are ( K Cj , K Ci )-bimodule isomorphisms (in the second isomorphismwe use the fact that Q is a species). Thus the duality condition for species holds. As for thedimension condition:dim K Cj ( M Cρ ) = dim K j M ( α : i → j ) ∈ ∆ M α = X ( α : i → j ) ∈ ∆ dim K j ( M α ) = X ( α : i → j ) ∈ ∆ d αij = ( d C ) ρij , where in the third equality we use the fact that Q is a species. Likewise, dim K Ci ( M Cρ ) =( d C ) ρji . Thus, Q C is a species of ∆ C .Note also that if Q is a K -species of an absolute valued quiver Γ, then Q C is a K -speciesof Γ C . Indeed, it is clear that M Cρ is a ( K Cj , K Ci )-bimodule on which K acts centrally forall ρ : i → j in Γ C (since each summand satisfies this condition). Moreover, dim K ( K Ci ) =dim K ( K i ) = d i = d Ci for all i ∈ Γ C , thus the dimension criterion for the vertices is satisfied.Also, dim K ( M Cρ ) = m Cρ by a computation similar to the above.With the concept of crushed species/quivers, we obtain the following result, which givesa necessary and sufficient condition for two tensor rings/algebras to be isomorphic. Theorem 5.5.
Let Q and Q ′ be two species containing no oriented cycles. Then T ( Q ) ∼ = T ( Q ′ ) as rings if and only if Q C ∼ = Q ′ C as species. Moreover, if Q and Q ′ are K -species,then T ( Q ) ∼ = T ( Q ) as K -algebras if and only if Q C ∼ = Q ′ C as K -species.Proof. Suppose Q and Q ′ are species with underlying relative valued quivers ∆ and ∆ ′ ,respectively. Throughout the proof we will use the familiar notation D := Π i ∈ ∆ K i and M := L ρ ∈ ∆ M ρ (add primes for Q ′ ).The proof of the reverse implication is straightforward and so we leave the details to thereader.For the forward implication, assume T ( Q ) ∼ = T ( Q ′ ). Let A = T ( Q ) and B = T ( Q ′ ) andlet ϕ : A → B be a ring isomorphism. Then, there exists an induced ring isomorphism ϕ : A/ rad A → B/ rad B . By Lemma 3.9, A/ rad A ∼ = D and B/ rad B ∼ = D ′ . Thus, wehave an isomorphism e ϕ D : D → D ′ . It is not difficult to show that { K i } i ∈ ∆ is the onlycomplete set of primitive orthogonal idempotents in D and that, likewise, { K ′ i } i ∈ ∆ ′ is theonly complete set of primitive orthogonal idempotents of D ′ . Since any isomorphism mustbijectively map a complete set of primitive orthogonal idempotents to a complete set ofprimitive orthogonal idempotents, we may identify ∆ and ∆ ′ and assume, without loss of generality, that e ϕ D (1 K i ) = 1 K ′ i for each i ∈ ∆ = ∆ ′ . Since 1 K i · D = K i and 1 K ′ i · D ′ = K ′ i ,we have that e ϕ D | K i is a ring isomorphism K i → K ′ i for each i ∈ ∆ = ∆ ′ .Now, ϕ | rad A is a ring isomorphism from rad A to rad B . Thus, as before, we have an in-duced ring isomorphism (and hence an abelian group isomorphism) ϕ | rad A : rad A/ (rad A ) → rad B/ (rad B ) . Since rad A/ (rad A ) ∼ = M and rad B/ (rad B ) ∼ = M ′ , we have an isomor-phism e ϕ M : M → M ′ . For any i, j ∈ ∆ , 1 K j · M · K i = L ( ρ : i → j ) ∈ ∆ M ρ =: j M i and1 K ′ j · M ′ · K ′ i = L ( ρ : i → j ) ∈ ∆ ′ M ′ ρ =: j M ′ i . Therefore, e ϕ M | j M i is an abelian group isomorphism j M i → j M ′ i .Hence, { e ϕ D | K i , e ϕ M | j M i } i ∈ ∆ C =∆ , ( i → j ) ∈ ∆ C defines an isomorphism of species from Q C to Q ′ C .In the case of K -species, one simply has to replace the terms “ring” with “ K -algebra”,“ring morphism” with “ K -algebra morphism” and “abelian group homomorphism” with“ K -linear map” and the proof is the same. (cid:3) If Q (and Q ′ ) contain oriented cycles, the arguments in the proof of Theorem 5.5 fail since,in general, it is not true that rad T ( Q ) = L ∞ n =1 T n ( M ). However, it seems likely that onecould modify the proof to avoid using the radical. Hence, we offer the following conjecture. Conjecture 5.6.
Theorem 5.5 holds even if Q and Q ′ contain oriented cycles. Remark 5.7.
Theorem 5.5 serves as a first step in justifying Remark 2.2 (i.e. that we canalways assume, without loss of generality, that we have no parallel arrows in our valuedquivers) since a species with parallel arrows can always be crushed to one with only singlearrows and its tensor algebra remains the same.Theorem 5.5 shows that there does not exist an equivalence on the level of valued quivers(relative or absolute) such that T ( Q ) ∼ = T ( Q ′ ) ⇐⇒ ∆ is equivalent to ∆ ′ since there are species (respectively K -species), with identical underlying valued quivers,that are not isomorphic as crushed species (respectively K -species) and hence have non-isomorphic tensor rings (respectively algebras) (see Example 2.4).In the case of K -species, one may wonder what happens when we tensor T ( Q ) with thealgebraic closure of K . Indeed, maybe we can find an equivalence on the level of absolutevalued quivers such that K ⊗ K T ( Q ) ∼ = K ⊗ K T ( Q ′ ) ⇐⇒ Γ is equivalent to Γ ′ . The answer, unfortunately, is no. However, this idea does yield an interesting result. In[15], Hubery showed that if K is a finite field, then there is a field extension F/K such that F ⊗ K T ( Q ) is isomorphic to the path algebra of a quiver. Our strategy of tensoring with thealgebraic closure allows us to generalize this result for an arbitrary perfect field. Theorem 5.8.
Let K be a perfect field and Q be a K -species with underlying absolutevalued quiver Γ containing no oriented cycles such that K i is a field for each i ∈ Γ . Then K ⊗ K T ( Q ) is isomorphic to the path algebra of a quiver.Proof. Let A = K ⊗ K T ( Q ). Take any i ∈ Γ . It is a well-known fact that K ⊗ K K i = K d i (see for example [5, Chapter V, Section 6, Proposition 2] or the proof of [16, Theorem 8.46]). ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 21
Let I = K ⊗ ( P ∞ n =1 T n ( M )). It is clear that I is a two-sided ideal of A . Moreover, sinceΓ has no cycles, I is also nilpotent. Considering A/I , we see that
A/I ∼ = K ⊗ K (Π i ∈ Γ K i ) ∼ = K × · · · × K | {z } P i ∈ Γ0 d i times . So, as in Lemma 3.9, rad A = I and by [1, Section I, Proposition 6.2], A is a basic finite-dimensional K -algebra.We claim that A is also hereditary. It is well-known that a ring is hereditary if and only ifit is of global dimension at most 1. According to [2, Theorem 16], if Λ and Λ are K -algebrassuch that Λ and Λ are semiprimary (recall that a K -algebra Λ is semiprimary if there is atwo-sided nilpotent ideal I such that Λ /I is semisimple) and (Λ / rad Λ ) ⊗ K (Λ / rad Λ ) issemisimple, then gl . dim(Λ ⊗ K Λ ) = gl . dim Λ + gl . dim Λ . The K -algebras K and T ( Q )satisfy these conditions. Indeed, K is simple and thus semiprimary. We know also that T ( Q ) / rad T ( Q ) is semisimple and rad T ( Q ) is nilpotent since Γ has no oriented cycles; thus T ( Q ) is semiprimary. Moreover,( K/ rad K ) ⊗ K ( T ( Q ) / rad T ( Q )) ∼ = K ⊗ K (Π i ∈ Γ K i ) ∼ = K × · · · × K | {z } P i ∈ Γ0 d i times , which is semisimple.Therefore we have that:gl . dim A = gl . dim( K ⊗ K T ( Q )) = gl . dim K + gl . dim T ( Q ) . However, gl . dim K = 0 (since all K -modules are free) and gl . dim T ( Q ) ≤ T ( Q ) ishereditary). Hence, gl . dim A ≤ A is hereditary. By Theorem 3.4, A is isomorphicto the path algebra of a quiver. (cid:3) Remark 5.9.
Hubery goes further in [15], constructing an automorphism σ of the quiver Q whose path algebra is isomorphic to K ⊗ K T ( Q ) such that ( Q, σ ) folds into Γ. It seemslikely that this is possible here as well.
Conjecture 5.10.
Let K be a perfect field, let Q be a K -species with underlying absolutevalued quiver Γ containing no oriented cycles such that K i is a field for each i ∈ Γ andlet Q be a quiver such that K ⊗ K T ( Q ) ∼ = KQ (as in Theorem 5.8). Then there exists anautomorphism σ of Q such that ( Q, σ ) folds into Γ . With Theorem 5.8, we are able to use the methods of [1, Chapter II, Section 3] to constructthe quiver, Q , whose path algebra is isomorphic to A = K ⊗ K T ( Q ). That is, the vertices of Q are in one-to-one correspondence with { ε , . . . , ε n } , a complete set of primitive orthogonalidempotents of A , and the number of arrows from the vertex corresponding to ε i to thevertex corresponding to ε j is given by dim K ( ε j · (rad A/ rad A ) · ε i ). We illustrate this inthe next example. Example 5.11.
Let Γ be the following absolute valued quiver.Γ: (2) (2)(4)
We can construct two Q -species of Γ: Q : Q ′ : Q ( √ Q ( √ Q ( √ Q ( √ Q ( √ , √ Q ( √ .Let F = Q , A = F ⊗ Q T ( Q ) and B = F ⊗ Q T ( Q ′ ). We would like to find quivers Q and Q ′ with A ∼ = F Q and B ∼ = F Q ′ .By direct computation, we see that ε = ((1 ⊗
1) + ( √ ⊗ √ ε = ((1 ⊗ − ( √ ⊗ √ F ⊗ Q Q ( √ Q must have 4 vertices. To find the arrows, note that ε j · (rad A/ rad A ) · ε i = ε j · ( F ⊗ Q Q ( √ ) · ε i ∼ = ε j · ( F ⊗ Q Q ( √ · ε i = F ( δ ij ε i , ⊕ F (0 , δ ij ε i ) , which has dimension 2 if i = j and 0 otherwise. Hence, A is isomorphic to the path algebra(over F ) of Q : .For Q ′ , again { ε , ε } is a complete set of primitive orthogonal idempotents of F ⊗ Q Q ( √ ζ = ((1 ⊗
1) + ( √ ⊗ √ ζ = ((1 ⊗ − ( √ ⊗ √ F ⊗ Q Q ( √ Q ′ has 4 vertices. To find thearrows, note that ζ j · (rad B/ rad B ) · ε i = ζ j · ( F ⊗ Q Q ( √ , √ · ε i and, since ζ j · ε i = 0 (as elements in F ⊗ Q Q ( √ , √ i, j ∈ { , } .Hence, B is isomorphic to the path algebra (over F ) of Q ′ : .Notice that in Example 5.11, F Q and
F Q ′ are not isomorphic; this illustrates our earlierpoint; namely that there is no equivalence on the level of absolute valued quivers such that K ⊗ K T ( Q ) ∼ = K ⊗ K T ( Q ′ ) ⇐⇒ Γ is equivalent to Γ ′ . Therefore, it seems likely that Theorem 5.8 is the best that we can hope to achieve.Note, however, that Theorem 5.8 fails if all the division rings in our K -modulation arenot fields. Consider the following simple example. Example 5.12.
View H , the quaternions, as an R -species (that is, H is an R -modulation ofthe absolute valued quiver with one vertex of value 4 and no arrows). Consider the C -algebra C ⊗ R H . It is easy to see that C ⊗ R H ∼ = M ( C ), the algebra of 2 by 2 matrices with entriesin C . This algebra is not basic. Indeed, one can check (by direct computation) that (cid:26) ε = (cid:18) (cid:19) , ε = (cid:18) (cid:19)(cid:27) is a complete set of primitive orthogonal idempotents and that M ( C ) ε ∼ = M ( C ) ε ∼ = C as M ( C )-modules. Thus, C ⊗ R H is not isomorphic to the path algebra of a quiver (sinceall path algebras are basic). ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 23
While we cannot use Theorem 5.8 for arbitrary K -species, we do have the following. Corollary 5.13.
Let K be a perfect field and Q be a K -species with underlying absolutevalued quiver Γ containing no oriented cycles. Then K ⊗ K T ( Q ) is Morita equivalent to thepath algebra of a quiver.Proof. It suffices to show that K ⊗ K T ( Q ) is hereditary (and then invoke Part 1 of Corollary3.13). In the proof of Theorem 5.8, all the arguments proving that K ⊗ K T ( Q ) is hereditary gothrough as before, save for the proof that ( K/ rad K ) ⊗ K ( T ( Q ) / rad T ( Q )) ∼ = K ⊗ K (Π i ∈ Γ K i )is semisimple.To show this in the case that the K i are not necessarily all fields, pick some i ∈ Γ andlet Z be the centre of K i . Then K ⊗ K K i ∼ = K ⊗ K Z ⊗ Z K i . The field Z is a field extension of K and so we may use the same arguments as in the proofof Theorem 5.8 to show K ⊗ K Z ∼ = K × · · · × K . So K ⊗ K K i ∼ = ( K × · · · × K ) ⊗ Z K i ∼ = ( K ⊗ Z K i ) × · · · × ( K ⊗ Z K i ) . One can show that K ⊗ Z K i ∼ = M n ( K ) for some n , which is a simple ring. Thus, K ⊗ K K i is semisimple, meaning that K ⊗ K (Π i ∈ Γ K i ) is semisimple, completing the proof. (cid:3) Representations
In this section, we begin by defining representations of quivers and species. We willthen see (Proposition 6.3) that representations of species (resp. quivers) are equivalent tomodules of the corresponding tensor ring (resp. path algebra). This fact together withSection 3 (specifically Theorems 3.4 and 3.6, and Corollary 3.13) shows why representationsof quivers/species are worth studying; they allow us to understand the representations of anyfinite-dimensional algebra over a perfect field. We then discuss the root system associatedto a valued quiver, which encodes a surprisingly large amount of information about therepresentation theory of species (see Theorems 6.17 and 6.20, and Proposition 6.21). FromSection 1, we know that every valued quiver can be obtained by folding a quiver withautomorphism. Thus, we end the section with a discussion on how much of the data of therepresentation theory of a species is contained in a corresponding quiver with automorphism.Throughout this section, we make the assumption (unless otherwise specified) that allquivers/species are connected and contain no oriented cycles. Also, whenever there is noneed to distinguish between relative or absolute valued quivers, we will simply use the term“valued quiver” and denote it by Ω. We let { e i } i ∈ Ω be the standard basis of Z Ω for a valuedquiver Ω. Definition 6.1 (Representation of a quiver) . A representation V = ( V i , f ρ ) i ∈ Q ,ρ ∈ Q of aquiver Q over the field K consists of a K -vector space V i for each i ∈ Q and a K -linear map f ρ : V t ( ρ ) → V h ( ρ ) , for each ρ ∈ Q . If each V i is finite-dimensional, we call dim V = (dim K V i ) i ∈ Q ∈ N Q the graded dimension of V . A morphism of Q representations ϕ : V = ( V i , f ρ ) i ∈ Q ,ρ ∈ Q → W = ( W i , g ρ ) i ∈ Q ,ρ ∈ Q consists of a K -linear map ϕ i : V i → W i for each i ∈ Q such that ϕ h ( ρ ) ◦ f ρ = g ρ ◦ ϕ t ( ρ ) forall ρ ∈ Q . That is, the following diagram commutes for all ρ ∈ Q . V t ( ρ ) f ρ V h ( ρ ) ϕ t ( ρ ) W t ( ρ ) g ρ W h ( ρ ) ϕ h ( ρ ) We let R K ( Q ) denote the category of finite-dimensional representations of Q over K . Definition 6.2 (Representation of a species) . A representation V = ( V i , f ρ ) i ∈ ∆ ,ρ ∈ ∆ of aspecies (or K -species) Q consists of a K i -vector space V i for each i ∈ ∆ and a K h ( ρ ) -linearmap f ρ : M ρ ⊗ K t ( ρ ) V t ( ρ ) → V h ( ρ ) , for each ρ ∈ ∆ . If all V i are finite-dimensional (over their respective rings), we call dim V =(dim K i V i ) i ∈ ∆ ∈ N ∆ the graded dimension of V .A morphism of Q representations ϕ : V = ( V i , f ρ ) i ∈ ∆ ,ρ ∈ ∆ → W = ( W i , g ρ ) i ∈ ∆ ,ρ ∈ ∆ consists of a K i -linear map ϕ i : V i → W i for each i ∈ ∆ such that ϕ h ( ρ ) ◦ f ρ = g ρ ◦ (id M ρ ⊗ ϕ t ( ρ ) )for all ρ ∈ ∆ . That is, the following diagram commutes for all ρ ∈ ∆ . M ρ ⊗ K t ( ρ ) V t ( ρ ) f ρ V h ( ρ ) id M ρ ⊗ ϕ t ( ρ ) M ρ ⊗ K t ( ρ ) W t ( ρ ) g ρ W h ( ρ ) ϕ h ( ρ ) We let R ( Q ) denote the category of finite-dimensional representations of Q . If Q is a K -species, we use the notation R K ( Q ).Note that if Q is a K -species of a trivially valued absolute valued quiver Γ, then, as before,all K i ∼ = K (as K -algebras) and all M ρ ∼ = K (as bimodules). Thus, a representation of Q isa representation of the underlying (non-valued) quiver of Γ. Therefore, by viewing quiversas trivially valued absolute valued quivers, representations of species are a generalization ofrepresentations of quivers.It is well-known that, for a quiver Q , the category R K ( Q ) is equivalent to KQ -mod, thecategory of finitely-generated (left) KQ -modules. This fact generalizes nicely for species. Proposition 6.3.
Let Q be a species (possibly with oriented cycles). Then R ( Q ) is equivalentto T ( Q ) - mod .Proof. See [10, Proposition 10.1]. While the proof there is given only for K -species, the samearguments hold for species in general. (cid:3) ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 25
Remark 6.4.
Proposition 6.3, together with Theorem 5.5, justifies Remark 2.2 (i.e. thatwe can always assume, without loss of generality, that our valued quivers contain no parallelarrows) since a species with parallel arrows can always be crushed to one with only singlearrows and its tensor algebra remains the same. Since T ( Q )-mod is equivalent to R ( Q )-mod,the representation theory of any species is equivalent to the representation of a species withonly single arrows (its crushed species). While allowing parallel arrows in our definitionof species is not necessary, there are situations where it may be advantageous as the nextexample demonstrates. Example 6.5.
Let ∆ be the following valued quiver.∆ : i j ( d αij , d αji )( d βij , d βji ) Then ∆ C : i j ( d αij + d βij , d αji + d βji ) .Any modulation of ∆, K i K j M α M β ,yields a modulation of ∆ C , K i K j M α ⊕ M β ,and the representation theory of both these species is identical. However, the converse is nottrue. That is, not every modulation of ∆ C yields a modulation of ∆. For example, one canchoose a modulation K i K j M such that M is indecomposable, and thus cannot be written as M = M ⊕ M (with M , M =0) to yield a modulation of ∆. Thus, we can think of modulations of ∆ as being “special”modulations of ∆ C where the bimodule attached to its arrow can be written (nontrivially)as the direct sum of two bimodules.Example 6.5 illustrates why one may wish to allow parallel arrows in the definition ofspecies; they may be used as a way of ensuring that the bimodules in our modulationdecompose into a direct sum of proper sub-bimodules. Definition 6.6 (Indecomposable representation) . Let V = ( V i , f ρ ) i ∈ ∆ ,ρ ∈ ∆ and W = ( W i , g ρ ) i ∈ ∆ ,ρ ∈ ∆ be representations of a species (or a quiver). The direct sum of V and W is V ⊕ W = ( V i ⊕ W i , f ρ ⊕ g ρ ) i ∈ ∆ ,ρ ∈ ∆ . A representation U is said to be indecomposable if U = V ⊕ W implies V = U or W = U .Because we restrict ourselves to finite-dimensional representations, the Krull-Schmidttheorem holds. That is, every representation can be written uniquely as a direct sum ofindecomposable representations (up to isomorphism and permutation of the components).Thus, the study of all representations of a species (or quiver) reduces to the study of itsindecomposable representations.We say that a species/quiver is of finite representation type if it has only finitely manynon-isomorphic indecomposable representations. It is of tame (or affine ) representationtype if it has infinitely many non-isomorphic indecomposable representations, but they canbe divided into finitely many one parameter families. Otherwise, it is of wild representationtype.Thus the natural question to ask is: can we classify all species/quivers of finite type,tame type and wild type? The answer, as it turns out, is yes. However, we first need a fewadditional concepts. Definition 6.7 (Euler, symmetric Euler and Tits forms) . The
Euler form of an absolutevalued quiver Γ with valuation ( d i , m ρ ) i ∈ Γ ,ρ ∈ Γ is the bilinear form h− , −i : Z Γ × Z Γ → Z given by: h x, y i = X i ∈ Γ d i x i y i − X ρ ∈ Γ m ρ x t ( ρ ) y h ( ρ ) . The symmetric Euler form ( − , − ) : Z Γ × Z Γ → Z is given by:( x, y ) = h x, y i + h y, x i . The
Tits form q : Z Γ → Z is given by: q ( x ) = h x, x i . Remark 6.8.
If we take Γ to be trivially valued (i.e. all d i = m ρ = 1), we recover the usualdefinitions of these forms for quivers (see, for example [7, Definitions 3.6.7, 3.6.8, 3.6.9]). Remark 6.9.
Notice that the symmetric Euler form and the Tits form do not depend onthe orientation of our quiver.
Remark 6.10.
Given a relative valued quiver ∆, we have seen in Lemma 1.5 that we canchoose an absolute valued quiver Γ such that F (Γ) = ∆ (this is equivalent to making a choiceof positive integers f i in Definition 1.4) with d ρij = m ρ /d j and d ρji = m ρ /d i for all ρ : i → j in∆ . It is easy to see that (as long as the quiver is connected) for any other absolute valuedquiver Γ ′ with F (Γ ′ ) = ∆, there is a λ ∈ Q + such that d ′ i = λd i for all i ∈ ∆ . Thus, wedefine the Euler, symmetric Euler and Tits forms on ∆ to be the corresponding forms on Γ,which are well-defined up to positive rational multiple. Definition 6.11 (Generalized Cartan matrix) . Let I be an indexing set. A generalizedCartan 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 .A generalized Cartan matrix C is symmetrizable if there exists a diagonal matrix D (calledthe symmetrizer ) such that DC is symmetric. ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 27
Note that, for any valued quiver Ω, c ij = 2 ( e i , e j )( e i , e i ) defines a generalized Cartan matrix,since ( e i , e i ) = 2 d i and ( e i , e j ) = − P ρ m ρ for i = j , where the sum is taken over all arrowsbetween i and j (regardless of orientation). So, c ij = ( , if i = j, − P ρ m ρ /d i = − P ρ d ρij , if i = j. From this we see that two valued quivers Ω and Ω ′ have the same generalized Cartan matrix(up to ordering of the rows and columns) if and only if Ω C ∼ = Ω ′ C as relative valued quivers(by this we mean that if Ω and Ω ′ are relative valued quivers, then Ω C ∼ = Ω ′ C and if they areabsolute valued quivers, then F (Ω) C ∼ = F (Ω ′ ) C ). If all d i are equal (or alternatively, P ρ d ρij = P ρ d ρji for all adjacent i and j ), then the matrix is symmetric, otherwise it is symmetrizablewith symmetrizer D = diag( d i ) i ∈ Ω . Moreover, every symmetrizable Cartan matrix can beobtained in this way. This is one of the motivations for working with species. When workingwith species we can obtain non-symmetric Cartan matrices, but when restricted to quivers,only symmetric Cartan matrices arise. For every generalized Cartan matrix, we have itsassociated Kac-Moody Lie algebra. Definition 6.12 (Kac-Moody Lie algebra) . Let C = ( c ij ) be an n × n generalized Cartanmatrix. Then the Kac-Moody Lie algebra of C is the complex Lie algebra 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 ) − c ij ( e j ) = 0 and (ad f i ) − c ij ( f j ) for all i = j .Therefore, to every valued quiver, we can associate a generalized Cartan matrix and itscorresponding Kac-Moody Lie algebra. It is only fitting then, that we discuss root systems. Definition 6.13 (Root system of a valued quiver) . Let Ω be a valued quiver. • For each i ∈ Ω , define the simple reflection through i to be the linear transformation r i : Z Ω → Z Ω given by: r i ( x ) = x − x, e i )( e i , e i ) e i . • The
Weyl group , which we denote by W , is the subgroup of Aut( Z Ω ) generated bythe simple reflections r i , i ∈ Ω . • An element x ∈ Z Ω is called a real root if ∃ w ∈ W such that x = w ( e i ) for some i ∈ Ω . • The support of an element x ∈ Z Ω is defined as supp( x ) = { i ∈ Ω | x i = 0 } andwe say supp( x ) is connected if the full subquiver of Ω with vertex set supp( x ) isconnected. Then the fundamental set is defined as F = { = x ∈ N Ω | ( x, e i ) ≤ i ∈ Ω and supp( x ) is connected } . • An element x ∈ Z Ω is called an imaginary root if x ∈ S w ∈W w ( F ) ∪ w ( −F ). • The root system of Ω, denoted Φ(Ω) is the set of all real and imaginary roots. • We call a root x positive (resp. negative ) if x i ≥ x i ≤ ∀ i ∈ Ω . We writeΦ + (Ω) for the set of positive roots and Φ − (Ω) for the set of negative roots. Definition 6.14 (Stable element) . An element x ∈ Z Ω is called stable if w ( x ) = x for all w ∈ W . Remark 6.15.
It is worth noting that, while a stable element need not be an imaginaryroot (see [20, Example 6.15]), it is always the sum of imaginary roots (see [20, Lemma 6.16]).
Definition 6.16 (Discrete and continuous dimension types) . An indecomposable represen-tation V of a species (or quiver) is of discrete dimension type if it is the unique indecompos-able representation (up to isomorphism) with graded dimension dim V . Otherwise, it is of continuous dimension type .With all these concepts in mind, we can neatly classify all species of finite and tamerepresentation type. Note that in the case of quivers, this was originally done by Gabriel(see [13]). It was later generalized to species by Dlab and Ringel. Theorem 6.17. [11, Main Theorem]
Let Q be species of a connected relative valued quiver ∆ . Then: (a) Q is of finite representation type if and only if the underlying undirected valued graphof ∆ is a Dynkin diagram of finite type (see [11] for a list of the Dynkin diagrams).Moreover, dim : R ( Q ) → Z ∆ induces a bijection between the isomorphism classes ofthe indecomposable representations of Q and the positive real roots of its root system. (b) If the underlying undirected valued graph of ∆ is an extended Dynkin diagram (see [11] for a list of the extended Dynkin diagrams), then dim : R ( Q ) → Z ∆ inducesa bijection between the isomorphism classes of the indecomposable representations of Q of discrete dimension type and the positive real roots of its root system. More-over, there exists a unique stable element (up to rational multiple) n ∈ Φ( Q ) and theindecomposable representations of continuous dimension type are those whose gradeddimension is a positive multiple of n . If Q is a K -species, then Q is of tame represen-tation type if and only if the underlying undirected valued graph of ∆ is an extendedDynkin diagram. Remark 6.18.
See [11, p. 57] (and [23]) for a proof that a K -species Q is tame if and onlyif ∆ is an extended diagram. Remark 6.19.
In the case that the underlying undirected valued graph of ∆ is an extendedDynkin diagram, the indecomposable representations of continuous dimension type of Q can be derived from the indecomposable representations of continuous dimension type of asuitable species with underlying undirected valued graph e A or e A (see [11, Theorem 5.1]).Theorem 6.17 shows a remarkable connection between the representation theory of speciesand the theory of root systems of Lie algebras. In the case of quivers, Kac was able to showthat this connection is stronger still. Theorem 6.20. [18, Theorems 2 and 3]
Let Q be a quiver with no loops (though possiblywith oriented cycles) and K an algebraically closed field. Then there is an indecomposablerepresentation of Q of graded dimension α if and only if α ∈ Φ + ( Q ) . Moreover, if α is a realpositive root, then there is a unique indecomposable representation of Q (up to isomorphism) ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 29 of graded dimension α . If α is an imaginary positive root, then there are infinitely manynon-isomorphic indecomposable representations of Q of graded dimension α . It is not known whether Kac’s theorem generalizes fully for species, however, it does forcertain classes of species. Indeed, in the case of a species of finite or tame representationtype, one can apply Theorem 6.17. In the case of K -species when K is a finite field, we havethe following result by Deng and Xiao. Proposition 6.21. [8, Proposition 3.3]
Let Q be a K -species ( K a finite field) containing nooriented cycles. Then there exists an indecomposable representation of Q of graded dimension α if and only if α ∈ Φ + ( Q ) . Moreover, if α is a real positive root, then there is a uniqueindecomposable representation of Q (up to isomorphism) of graded dimension α . Based on these results, we see that much of the information about the representationtheory of a species is encoded in its underlying valued quiver/graph. Recall from Section1 that any valued quiver can be obtained by folding a quiver with automorphism. So, onemay ask: how much information is encoded in this quiver with automorphism?We continue our assumption that Q contains no oriented cycles; however for what followsthis is more restrictive than we need. It would be enough to assume that Q contains noloops and that no arrow connects two vertices in the same σ -orbit (see [20, Lemma 6.24] fora proof that this is indeed a weaker condition).Suppose ( Q, σ ) is a quiver with automorphism and let V = ( V i , f ρ ) i ∈ Q ,ρ ∈ Q be a repre-sentation of Q . Define a new representation V σ = ( V σi , f σi ) i ∈ Q ,ρ ∈ Q by V σi = V σ − ( i ) and f σρ = f σ − ( ρ ) . Definition 6.22 (Isomorphically invariant representation) . Let (
Q, σ ) be a quiver withautomorphism. A representation V = ( V i , f ρ ) i ∈ Q ,ρ ∈ Q is called isomorphically invariant (orsimply invariant ) if V σ ∼ = V as representations of Q .We say an invariant representation V is invariant-indecomposable if V = W ⊕ W suchthat W and W are invariant representations implies W = V or W = V .It is not hard to see that the invariant-indecomposable representations are precisely thoseof the form V = W ⊕ W σ ⊕ · · · ⊕ W σ r − where W is an indecomposable representation and r is the least positive integer such that W σ r ∼ = W .Let ( Z Q ) σ = { α ∈ Z Q | α i = α j for all i and j in the same orbit } . Suppose ( Q, σ ) foldsinto Ω and write i ∈ Ω for the orbit of i ∈ Q . We then have a well-defined function f : ( Z Q ) σ → Z Ω defined by f ( α ) i = α i for any i ∈ Q . Notice that if V is an invariant representation of Q ,then dim V i = dim V σ − ( i ) for all i ∈ Q . As such, dim V i = dim V j for all i and j in the sameorbit. Thus, dim V ∈ ( Z Q ) σ . We have the following result due to Hubery. Theorem 6.23. [15, Theorem 1]
Let ( Q, σ ) be a quiver with automorphism, Ω a valuedquiver such that ( Q, σ ) folds into Ω , and K an algebraically closed field of characteristic notdividing the order of σ . (a) The images under f of the graded dimensions of the invariant-indecomposable repre-sentations of Q are the positive roots of Φ(Ω) . (b) If f ( α ) is a real positive root, then there is a unique invariant-indecomposable repre-sentation of Q with graded dimension α (up to isomorphism). Theorem 6.23 tells us that if the indecomposables of Q are determined by the positiveroots of Φ(Ω) (such as in the case of species of Dynkin or extended Dynkin type or K -species over finite fields), then finding all the indecomposables of Q reduces to finding theindecomposables of Q , which, in general, is an easier task.One may wonder if there is a subcategory of R K ( Q ), say R σK ( Q ), whose objects are theinvariant representations of Q , that is equivalent to R ( Q ). One needs to determine what themorphisms of this category should be. The most obvious choice is to let R σK ( Q ) be the fullsubcategory of R K ( Q ) whose objects are the invariant representations. This, however, doesnot work. The category R ( Q ) is an abelian category (this follows from Proposition 6.3), but R σK ( Q ), as we have defined it, is not. As the following example demonstrates, this categorydoes not, in general, have kernels. Example 6.24.
Let (
Q, σ ) be the following quiver with automorphism (where the dottedarrows represent the action of σ ).Let V be the following invariant-indecomposable representation of Q . K K Let ϕ : V → V be the morphism defined by VVϕ K KK K ϕ is a morphism of representations since each of the squares in the diagram commutes.However, by a straightforward exercise in category theory, one can show that ϕ does nothave a kernel (in the category R σK ( Q )).Therefore, if we define R σK ( Q ) as a full subcategory of R K ( Q ), it is not equivalent to R ( Q ). It is possible that one could cleverly define the morphisms of R σK ( Q ) to avoid thisproblem, however there are other obstacles to overcome. If R σK ( Q ) and R ( Q ) were equiva-lent, then there should be a bijective correspondence between the (isomorphism classes of)indecomposables in each category. Using the idea of folding, an invariant representation of Q with graded dimension α should be mapped to a representation of Q with graded dimension f ( α ). The following example illustrates the problem with this idea. Note that this example issimilar to the example following Proposition 15 in [15], however we approach it in a differentfashion. Example 6.25.
Let (
Q, σ ) be the following quiver with automorphism (again, the dottedarrows represent the action of σ ). ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 31
Then (
Q, σ ) folds into the following absolute valued quiver.Γ: (3) (2)(6)One can easily check that β = (1 ,
1) is an imaginary root of Φ(Γ). The only α ∈ ( Z Q ) σ suchthat f ( α ) = β is α = (1 , , . . . , Q with graded dimension α (after all, α is an imaginary root of Φ( Q )), all such invariant representations are isomorphcto KKK KK where every arrow represents the identity map id K . Thus, we have a single isomorphismclass of invariant-indecomposables with graded dimension α .Now, construct a species of Γ. Let γ = 2 / and let Q be the Q -species given by Q ( γ ) Q ( γ ) −−→ Q ( γ ). Thus the underlying valued quiver of Q is Γ. There exists an indecomposablerepresentation of Q with graded dimension β – in fact, there exists more than one.Let V be the representation Q ( γ ) f −→ Q ( γ ) where f : Q ( γ ) ⊗ Q ( γ ) Q ( γ ) ∼ = Q ( γ ) → Q ( γ ) is the Q ( γ )-linear map defined by 1 γ γ V be the representation Q ( γ ) f −→ Q ( γ ) where f : Q ( γ ) ⊗ Q ( γ ) Q ( γ ) ∼ = Q ( γ ) → Q ( γ ) is the Q ( γ )-linear map defined by 1 γ γ V and V are indecomposable and dim V = dim V = β . One can alsoshow that they are not isomorphic as representations of Q . Hence, there are at least twoisomorphism classes of indecomposable representations of Q with graded dimension β .Therefore, any functor R σK ( Q ) → R ( Q ) mapping invariant representations with gradeddimension α to representations with graded dimension f ( α ) cannot be essentially surjective,and thus cannot be an equivalence of categories.While the above example is not enough to conclude that the categories R σK ( Q ) and R ( Q )are not equivalent, it is enough to deduce that one cannot obtain an equivalence via folding.7. Ringel-Hall Algebras
In this section we define the Ringel-Hall algebra of a species (or quiver). We will constructthe generic composition algebra of a species, which is obtained from a subalgebra of theRingel-Hall algebra, and see that it is isomorphic to the positive part of the quantizedenveloping algebra of the corresponding Kac-Moody Lie algebra (see Theorem 7.5). We then give a similar interpretation of the whole Ringel-Hall algebra (see Theorem 7.10). For furtherdetails, see the expository paper by Schiffmann, [27].We continue our assumption that all quivers/species have no oriented cycles. Also, wehave seen in the last section (Proposition 6.3) that R ( Q ) is equivalent to T ( Q )-mod, and sowe will simply identify representations of Q with modules of T ( Q ). Definition 7.1 (Ringel-Hall algebra) . Let Q be an F q -species. Let v = q / and let A bean integral domain containing Z and v, v − . The Ringel-Hall algebra , which we will denote H ( Q ), is the free A -module with basis the set of all isomorphism classes of finite-dimensionalrepresentations of Q . Multiplication is given by[ A ][ B ] = v h dim A, dim B i X [ C ] g CAB [ C ] , where g CAB is the number of subrepresentations (submodules) X of C such that C/X ∼ = A and X ∼ = B (as representations/modules) and h− , −i is the Euler form (see Definition 6.7). Remark 7.2.
It is well-known (see, for example, [24, Lemma 2.2]) that h dim A, dim B i = dim F q Hom T ( Q ) ( A, B ) − dim F q Ext T ( Q ) ( A, B ) . In many texts (for example [14] or [28]) this is the way the form h− , −i is defined. Also,there does not appear to be a single agreed-upon name for this algebra; depending on thetext, it may be called the twisted Hall algebra , the Ringel algebra , the twisted Ringel-Hallalgebra , etc. Regardless of the name one prefers, it is important not to confuse this algebrawith the (untwisted) Hall algebra whose multiplication is given by [ A ][ B ] = P [ C ] g CAB [ C ]. Definition 7.3 (Composition algebra) . Let Q be an F q -species with underlying absolutevalued quiver Γ and F q -modulation ( K i , M ρ ) i ∈ Γ ,ρ ∈ Γ . The composition algebra , C = C ( Q ),of Q is the A -subalgebra of H ( Q ) generated by the isomorphism classes of the simple rep-resentations of Q . Since we assume Γ has no oriented cycles, this means C is generated bythe [ S i ] for i ∈ Γ where S i = (( S i ) j , ( S i ) ρ ) j ∈ Γ ,ρ ∈ Γ is given by( S i ) j = ( K i , if i = j, , if i = j, and ( S i ) ρ = 0 for all ρ ∈ Γ . Let S be a set of finite fields K such that {| K | | K ∈ S} is infinite. Let v K = | K | / foreach K ∈ S . Write C K for the composition algebra of Q for each finite field K in S and[ S ( K ) i ] for the corresponding generators. Let C be the subring of Π K ∈S C K generated by Q and the elements t = ( t K ) K ∈S , t K = v K ,t − = ( t − K ) K ∈S , t − K = v − K ,u i = ( u ( K ) i ) K ∈S , u ( K ) i = [ S ( K ) i ] . So t lies in the centre of C and, because there are infinitely many v K , t does not satisfy p ( t ) = 0 for any nonzero polynomial p ( T ) in Q [ T ]. Thus, we may view C as the A -algebaragenerated by the u i , where A = Q [ t, t − ] with t viewed as an indeterminate. ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 33
Definition 7.4 (Generic composition algebra) . Using the notation above, the Q ( t )-algebra C ∗ = Q ( t ) ⊗ A C is called the generic composition algebra of Q . We write u ∗ i = 1 ⊗ u i for i ∈ Γ .Let Q be an F q -species with underlying absolute valued quiver Γ. Let ( c ij ) be the gen-eralized Cartan matrix associated to Γ and let g be its associated Kac-Moody Lie algebra(recall Definitions 6.11 and 6.12). Let U t ( g ) be the quantized enveloping algebra of g and let U t ( g ) = U + t ( g ) ⊗ U t ( g ) ⊗ U − t ( g ) be its triangular decomposition (see [21, Chapter 3]). Wecall U + t ( g ) the positive part of U t ( g ); it is the Q ( t )-algebra generated by elements E i , i ∈ Γ ,modulo the quantum Serre relations − c ij X p =0 ( − p (cid:20) − c ij p (cid:21) E pi E j E − c ij − pi for all i = j, where (cid:20) mp (cid:21) = [ m ]![ p ]![ m − p ]! , [ n ] = t n − t − n t − t − , [ n ]! = [1][2] · · · [ n ] . In [14], Green was able to show that C ∗ and U + t ( g ) are canonically isomorphic. Of course, inhis paper, Green speaks of modules of hereditary algebras over a finite field K rather thanrepresentations of K -species, but as we have seen, these two notions are equivalent. Theorem 7.5. [14, Theorem 3]
Let Q be an F q -species with underlying absolute valuedquiver Γ and let g be its associated Kac-Moody Lie algebra. Then, there exists a Q ( t ) -algebraisomorphism U + t ( g ) → C ∗ which takes E i u ∗ i for all i ∈ Γ . Remark 7.6.
This result by Green is actually a generalization of an earlier result by Ringelin [25, p. 400] and [26, Theorem 7] who proved Theorem 7.5 in the case that Q is of finiterepresentation type.Theorem 7.5 gives us an interpretation of the composition algebra in terms of the quantizedenveloping algebra of the corresponding Kac-Moody Lie algebra. Later, Sevenhant and VanDen Bergh were able to give a similar interpretation of the whole Ringel-Hall algebra. Forthis, however, we need the concept of a generalized Kac-Moody Lie algebra , which was firstdefined by Borcherds in [4]. Though some authors have used slightly modified definitionsof generalized Kac-Moody Lie algebras over the years, we use here Borcherds’s originaldefinition (in accordance with Sevenhant and Van Den Bergh in [28]). Definition 7.7 (Generalized Kac-Moody Lie algebra) . Let H be a real vector space withsymmetric bilinear product ( − , − ) : H × H → R . Let I be a countable (but possibly infinite)set and { h i } i ∈I be a subset of H such that ( h i , h j ) ≤ i = j and c ij = 2( h i , h j ) / ( h i , h i )is an integer if ( h i , h i ) >
0. Then, the generalized Kac-Moody Lie algebra associated to H , { h i } i ∈I and ( − , − ) is the Lie algebra (over a field of characteristic 0 containing an isomorphiccopy of R ) generated by H and elements e i and f i for i ∈ I whose product is defined by: • [ h, h ′ ] = 0 for all h and h ′ in H , • [ h, e i ] = ( h, h i ) e i and [ h, f i ] = − ( h, h i ) f i for all h ∈ H and i ∈ I , • [ e i , f i ] = h i for each i and [ e i , f j ] = 0 for all i = j , • if ( h i , h i ) >
0, then (ad e i ) − c ij ( e j ) = 0 and (ad f i ) − c ij ( f j ) = 0 for all i = j , • if ( h i , h i ) = 0, then [ e i , e j ] = [ f i , f j ] = 0. Remark 7.8.
Generalized Kac-Moody Lie algebras are similar to Kac-Moody Lie algebras.The main difference is that generalized Kac-Moody Lie algebras (may) contain simple imag-inary roots (corresponding to the h i with ( h i , h i ) ≤ g be a generalized Kac-Moody Lie algebra (with the notation of Definition 7.7) and let v = 0 be an element of the base field such that v is not a root of unity. Write d i = ( h i , h i ) / i ∈ I such that ( h i , h i ) >
0. The quantized enveloping algebra U v ( g ) can be defined inthe same way as the quantized enveloping algebra of a Kac-Moody Lie algebra (see Section2 of [28]). The positive part of U + v ( g ) is the A -algebra generated by elements E i , i ∈ I ,modulo the quantum Serre relations − c ij X p =0 ( − p (cid:20) − c ij p (cid:21) d i E pi E j E − c ij − pi for all i = j, with ( h i , h i ) > E i E j − E j E i if ( h i , h j ) = 0 , where (cid:20) mp (cid:21) d i = [ m ] d i ![ p ] d i ![ m − p ] d i ! , [ n ] d i = ( v d i ) n − ( v d i ) − n v d i − v − d i , [ n ] d i ! = [1] d i [2] d i · · · [ n ] d i . Let n be a positive integer and let { e i } ni =1 be the standard basis of Z n . Let v ∈ R suchthat v > A be as in Definition 7.1. Suppose we have the following:(a) An N n -graded A -algebra A such that:(a) A = A ,(b) dim A A α < ∞ for all α ∈ N n ,(c) A e i = 0 for all 1 ≤ i ≤ n .(b) A symmetric positive definite bilinear form [ − , − ] : A × A → A such that [ A α , A β ] = 0if α = β and [1 ,
1] = 1 (here we assume [ a, a ] ∈ R for all a ∈ A ).(c) A symmetric bilinear form ( − , − ) : R n × R n → R such that ( e i , e i ) > ≤ i ≤ n and c ij = 2( e i , e j ) / ( e i , e i ) is a generalized Cartan matrix as in Definition6.9.(d) The tensor product A ⊗ A A can be made into an algebra via the rule( a ⊗ b )( c ⊗ d ) = v (deg( b ) , deg( c )) ( ac ⊗ bd ) , for homogeneous a, b, c, d . (here deg( x ) = α if x ∈ A α ). We assume that there isan A -algebra homomorphism δ : A → A ⊗ A A which is adjoint under [ − , − ] to themultiplication (that is, [ δ ( a ) , b ⊗ c ] A ⊗ A = [ a, bc ] A where [ a ⊗ b, c ⊗ d ] A ⊗ A = [ a, c ] A [ b, d ] A ). Proposition 7.9. [28, Proposition 3.2]
Under the above conditions, A is isomorphic (as analgebra) to the positive part of the quantized enveloping algebra of a generalized Kac-MoodyLie algebra. ALUED GRAPHS AND THE REPRESENTATION THEORY OF LIE ALGEBRAS 35
The Ringel-Hall algebra, H ( Q ), of a species Q is N n -graded by associating to each rep-resentation its graded dimension, hence H ( Q ) satisfies Condition 1 above. Moreover, thesymmetric Euler form satisfies Condition 3 (if we extend it to all R Γ ). Following Green in[14], we define δ ([ A ]) = X [ B ] , [ C ] v h dim B, dim C i g ABC | Aut( B ) || Aut( C ) || Aut( A ) | ([ B ] ⊗ [ C ])and ([ A ] , [ B ]) H ( Q ) = δ [ A ] , [ B ] | Aut( A ) | . In [14, Theorem 1], Green shows that ( − , − ) H ( Q ) satisfies Condition 2 and that δ satisfiesCondition 4. Hence, we have the following. Theorem 7.10. [28, Theorem 1.1]
Let Q be an F q -species. Then, H ( Q ) is the positive partof the quantized enveloping algebra of a generalized Kac-Moody algebra. Remark 7.11.
In their paper, Sevenhant and Van Den Bergh state Theorem 7.10 only forthe Ringel-Hall algebra of a quiver, but none of their arguments depend on having a quiverrather than a species. Indeed, many of their arguments are based on those of Green in [14],which are valid for hereditary algebras. Moreover, Sevenhant and Van Den Bergh define theRingel-Hall algebra to be an algebra opposite to the one we defined in Definition 7.1 (ourdefinition, which is the one used by Green, seems to be the more standard definition). Thisdoes not affect any of the arguments presented.
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