Quantum Chevalley groups

QUANTUM CHEVALLEY GROUPS arXiv:1205.7020v2 [math.QA] 11 Jun 2012 ARKADY BERENSTEIN AND JACOB GREENSTEIN Abstract. The goal of this paper is to construct quantum analogues of Chevalley groups inside completions of quantum groups or, more precisely, inside completions of Hall algebras of finitary categories. In particular, we obtain pentagonal and other identities in the quantum Chevalley groups which generalize their classical counterparts and explain Faddeev-Volkov quantum dilogarithmic identities and their recent generalizations due to Keller 1. Introduction It is well-known that a quantum group is not a group, in particular, it has very few elements that can be regarded as “group-like”. To construct group-like elements, it suffices to introduce a collection of “exponentials” in the completed quantum group which are analogues of formal exponentials exp(x), where x runs over a Lie algebra g and the exponential exp(x) is viewed as b (g). Since each quantum group can be realized an element of the completed enveloping algebra U as a (twisted) Hall-Ringel algebra of a (finitary) abelian category, this motivates the following b A of the Hall algebra HA of A with respect to its standard definition. Consider the completion H grading by the Grothendieck group of A . Definition 1.1. Let A be a finitary abelian category. Define the exponential ExpA of A in the b A by: completed Hall algebra H X ExpA = [A] [A]∈Iso A where Iso A is the set of isomorphism classes [A] of objects in A . We can justify this definition, in particular, by the following simple observations. Lemma 1.2. For any finitary category A one has ∆(ExpA ) = ExpA ⊗ ExpA , b A is a coproduct (though, not always co-associative) defined in Appendix A. where ∆ : HA → HA ⊗H Lemma 1.3. If A is the category of finite dimensional vector spaces over a finite field k, then X xn ExpA = expq (x) = [n]q ! n≥0 where x is the isomorphism class of k in A , q = |k|, and [n]q ! = [1]q [2]q · · · [n]q , where [k]q = 1 + q + · · · + q k−1 is the q-number. Date: June 12, 2012. The authors were supported in part by NSF grants DMS-0800247, DMS-1101507 (A.B.) and DMS-0654421 (J. G.) . 1 2 A. BERENSTEIN AND J. GREENSTEIN In general the exponentials behave nicely due to the following extension of Reineke’s inversion formula [7, Lemma 3.4] to more general abelian categories. Theorem 1.4. For any finitary abelian category with the finite length property X c[M ] · [M ], ExpA −1 = [M ] : M semisimple where c[M ] = Y [M :S] (−1)[M :S] | EndA S|( 2 ) [S] : S simple and [M : S] denotes the Jordan-Hölder multiplicity of S in M . These observations suggest the following definition. Definition 1.5. Given a finitary abelian category A , its quantum Chevalley group GA is the b A )× generated by all exponentials ExpC ∈ H bC ⊂ H b A , where C runs over all full subgroup of (H subcategories of A closed under extensions. Now we investigate the adjoint action of some elements of quantum Chevalley groups on their Hall algebras. For each object E of A denote by AddA (E) the minimal additive full subcategory of A containing all direct summands of E. The following result is obvious. Lemma 1.6. If E is an object of A without self-extensions such that EndA E is a field, then AddA (E) is semisimple and the element ExpAddA (E) = expqE ([E]) b A , where qE = | EndA E|, belongs to the quantum Chevalley group. of H Given a full additive subcategory B of A we denote by B (respectively B ) the largest full subcategory C of A whose objects C satisfy Ext1A (B, C) = 0 (respectively Ext1A (C, B) = 0) for all B ∈ Ob B. Clearly, C ⊂ B if and only if B ⊂ C . The following Lemma is immediate. Lemma 1.7. For any full additive subcategory B of A , B and B are closed under extensions. We will show below that the conjugation with exponentials ExpAddA (E) = expqE ([E]) for each E as in Lemma 1.6 almost preserves the entire Hall algebra of the categories AddA (E) and AddA (E) . To make this statement precise, we need more notation. Let R be a ring and let S = {aι }ι∈I be a set of pairwise commuting regular elements (i.e., not zero divisors) in R. The localization R[S −1 ] of R with respect to S is then the quotient of the free product R ∗Z[xι : ι ∈ I] by the ideal generated by aι ∗ xι − 1 and xι ∗ aι − 1, ι ∈ I. Clearly, for each E as in Lemma 1.6 the subset SE ⊂ HAdd (E) ∩ HAdd (E) defined by A SE := {1 + (qE − j 1)qE [E] | j A ∈ Z} consists of pairwise commuting regular elements, therefore the localizations HAdd A (E) HAdd A (E) [SE−1 ] are well-defined. [SE−1 ] and QUANTUM CHEVALLEY GROUPS 3 The following results furthers the analogy between the classical Chevalley group viewed as automorphisms of a semisimple Lie algebra over Z and its quantum counterpart whose subgroup acts by automorphisms of the (slightly extended) Hall subalgebra. Theorem 1.8. Let E be as in Lemma 1.6. Then the assignment x 7→ expqE ([E]) · x · expqE ([E])−1 defines an automorphism of the localized algebras HAdd A (E) [SE−1 ] and HAdd A (E) [SE−1 ]. We refine and prove the theorem in Section 3 (Theorem 3.11). In fact, we show that SE is an Ore set for both HAdd (E) and HAdd (E) . In a special case when E is a projective or an injective A A indecomposable, or, more generally, a preprojective or preinjective indecomposable in a hereditary category, it turns out that we obtain an automorphism of the localized Hall algebra HA [SE−1 ]. Next we discuss factorization of exponentials. Definition 1.9. We say that an ordered pair of full subcategories (A ′ , A ′′ ) of A is a factorizing pair if A ′ , A ′′ are closed under extensions and every object M in A has a unique subobject M ′′ ∈ Ob A ′′ such that M/M ′′ is an object in A ′ . More generally, we define a factorizing sequence (A1 , . . . , Am ), m ≥ 2 for A recursively by: • If m ≥ 3, then (A1 , . . . , Am−1 ) is a factorizing sequence for the category A≤m−1 , which is the minimal additive subcategory of A containing A1 , . . . , Am−1 and closed under extensions; • (A≤m−1 , Am ) is a factorizing pair for A . The following result is immediate. Proposition 1.10. If (A1 , . . . , Am ) is a factorizing sequence for a finitary abelian category A then (a) ExpA = ExpA1 · · · ExpAm . (b) If, additionally, Ai ⊂ Aj for all i < j, then HA ∼ = HA1 ⊗ · · · ⊗ HAm as a vector space. The simplest example of a factorizing pair is provided by the following. Proposition 1.11. Let S be a simple object in a finitary abelian category A with the finite length property. Let A ′ = AddA S and let A ′′ be the maximal full subcategory of A whose objects M satisfy [M : S] = 0. (a) If S is injective, then (A ′ , A ′′ ) is a factorizing pair for A . (b) If S is projective, then (A ′′ , A ′ ) is a factorizing pair for A . Let (A ′ , A ′′ ) be a factorizing pair for A . We say that (A ′ , A ′′ ) is a pentagonal pair if there exists a full additive subcategory A0 closed under extensions such that (A ′′ , A0 , A ′ ) is a factorizing triple. If (A ′ , A ′′ ) is a pentagonal pair then, clearly ExpA ′ ExpA ′′ = ExpA = ExpA ′′ ExpA0 ExpA ′ , which is the generalized pentagonal relation in the quantum Chevalley group. 4 A. BERENSTEIN AND J. GREENSTEIN Proposition 1.12. Let A be a finitary abelian category with the finite length property. Let S ⊂ F Iso A be the set of isomorphism classes of simples in A and suppose that S = S− S+ where S− (respectively S+ ) is the set of isomorphism classes of projective (respectively, injective) simple objects. Then (A+ , A− ), where A± = AddA S± , is a pentagonal pair with A0 being the full subcategory of A whose objects have no simple direct summands. In particular, if A has only two isomorphism classes of simple objects [S0 ], [S1 ] such that Ext1A (S0 , S1 ) = 0, we obtain [expq0 ([S0 ]), expq1 ([S1 ])] = ExpA0 where [x, y] = x−1 yxy −1 and qi = | EndA Si |. Let ai = dimEndA Si Ext1A (S1 , S0 ). If a0 a1 < 4, A has finitely many isomorphism classes of indecomposables and we obtain   a0 a1 = 1,  expq1 (Xα10 ), [expq0 (Xα0 ), expq1 (Xα1 )] = expq1 (Xα10 ) expq0 (Xα01 ), a0 a1 = 2,    expq1 (Xα10 ) expq0 (Xα01 +α0 ) expq1 (Xα1 +α10 ) expq0 (Xα01 ), a0 a1 = 3, where αij = αi + aj αj (the notation Xα is explained in Theorem 1.15 below). Other examples of pentagonal pairs are provided in Section 5. Given [M ] ∈ Iso A , denote by |M | its image in the Grothendieck group K0 (A ) of A . If A is hereditary, we can consider the twisted group algebra PA of K0 (A ) with the multiplication defined by tα tβ = hα, βi−1 tα+β where h|M |, |N |i = | HomA (M, N )|/| Ext1A (M, N )| is the Ringel form. Then we have an algebra homomorphism ([4, 5, 7]) Z t|M | . : HA → PA , [M ] 7→ | AutA (M )| Applying it to pentagonal identities yields the so-called quantum dilogarithm identities (cf. [5]). In this case, PA is generated by x1 , x0 subject to the relation x1 x0 = q a0 a1 x0 x1 bA of PA and we have the following identities in the natural completion P    Eq1 (x0 x1 ), [Eq0 (x0 ), Eq1 (x1 )] = Eq1 (p0 xa00 x1 )Eq0 (p1 x0 xa11 ),    Eq1 (p0 xa00 x1 )Eq0 (q0 p1 x20 xa11 )Eq1 (p0 q1 xa00 x21 )Eq0 (p1 x0 xa11 ), a0 a1 = 1 a0 a1 = 2 a0 a1 = 3, ( ai ) where Eq (x) = expq (x/(q − 1)) is a quantum dilogarithm and pi = qi 2 . Let A be the category of finite length modules over a finite hereditary ring and denote by A− (respectively A+ ) the preprojective (respectively, the preinjective) component of A . Note that if A is hereditary then A− = A+ = A if and only if A is of finite type. Otherwise, A− ⊂ A+ and the triple (A− , A0 , A+ ), where A0 = A− ∩ A+ , is a factorizing sequence for A and satisfies hypotheses of Proposition 1.10(b). This implies the following result. QUANTUM CHEVALLEY GROUPS 5 Corollary 1.13. Suppose that A is hereditary of infinite type. Then (a) The Hall algebra HA admits a triangular decomposition HA ∼ = HA− ⊗ HA0 ⊗ HA+ . (b) The exponential ExpA admits a factorization ExpA = ExpA− ExpA0 ExpA+ . Finally, we describe HA± and ExpA± (and, to some extent, HA0 and ExpA0 ) in the case when A is the representation category of a finite hereditary algebra over k. Number the isomorphism classes of simples [Si ], 1 ≤ i ≤ r in such a way that Ext1A (Sj , Si ) = 0 if i < j. The elements αi = |Si | form a basis of K0 (A ). Let K0+ (A ) be the submonoid of K0 (A ) generated by the αi , 1 ≤ i ≤ r. Since the form h·, ·i is non-degenerate for each 1 ≤ i ≤ r there exist unique γi , γ−i ∈ K0 (A ) such that for all 1 ≤ j ≤ r δ hγ−i , αj i = qi ij = hαj , γi i where qi = | EndA Si |. Then {γi }1≤i≤r and {γ−i }1≤i≤r are bases of K0 (A ). Let c be the unique automorphism of K0 (A ) defined by c(γ−i ) = −γi . Lemma 1.14. (a) For all 1 ≤ i ≤ r, γ±i ∈ K0+ (A ) (b) Set Γ± = {γ±i,±k := c±k (γ±i ) : 1 ≤ i ≤ r, k ≥ 0} ∩ K0+ (A ). Then for each γ ∈ Γ+ ∪Γ− there exists a unique, up to an isomorphism, indecomposable object Eγ with |Eγ | = γ. Moreover, an indecomposable M is preprojective (respectively, preinjective) if and only if M ∼ = Eγ with γ ∈ Γ− (respectively, γ ∈ Γ+ ). (c) If A has finitely many isomorphism classes of indecomposables then Γ+ = Γ− . Otherwise, Γ+ ∩ Γ− = ∅ and γ±i,±k ∈ K0+ (A ) for all 1 ≤ i ≤ r, k ≥ 0. Theorem 1.15. In the notation of Corollary 1.13: ← −− − −→ O O Q[Xγ ] Q[Xγ ], HA+ = HA− = γ∈Γ+ γ∈Γ− as vector spaces, while the exponentials ExpA± factor as ← −− −Y −→ Y Expqγ (Xγ ) ExpA+ = Expqγ (Xγ ), ExpA− = γ∈Γ+ γ∈Γ− where qγ±i,±r = qi , Xγ = [Eγ ] and both products are taken in the total order on Γ− (respectively, on Γ+ ) defined by γ−j,−r < γ−i,−s (respectively, γi,s < γj,r ) if either r < s or r = s and i < j. We prove Lemma 1.14, Theorem 1.15 and its Corollary in §4.4. It turns out that the factorization of ExpA allows one to obtain an elementary proof of Ringel’s result (cf. [2]) that the [Eγ ] with γ ∈ Γ+ ∪ Γ− are contained in the composition algebra CA of A . After Ringel ([9]), the composition algebra CA of A , that is, the subalgebra of HA generated by the [Si ], 1 ≤ i ≤ r, has the following presentation (cf. (3.13)) (ad −aij (1,qi ,qi2 ,...,qi ) [Si ])([Sj ]) = 0, (ad∗ −aji (1,qj ,...,qj ) [Sj ])([Si ]), j<i (the repeated quantum commutators (ad(q0 ,...,qm ) x)(y) and (ad∗(q0 ,...,qm ) x)(y) are defined in §3.2). In fact, this algebra is isomorphic to a Sevostyanov’s analogue of the upper half of the quantized universal enveloping algebra corresponding to the symmetrized Cartan matrix DA ([11]). 6 A. BERENSTEIN AND J. GREENSTEIN Corollary 1.16. For all γ ∈ Γ+ ∪ Γ− , [Eγ ] ∈ CA . This statement is proven in §4.5. Acknowledgments. An important part of this work was done while both authors were visiting the Massachusetts Institute of Technology, and it is our pleasure to thank the Department of Mathematics for its hospitality and Pavel Etingof for his support and stimulating discussions. We are grateful to Jiarui Fei for inspiring conversations. 2. Preliminaries We begin by fixing some notation. Let A be an abelian category. We will mostly assume that A is a k-category for some field k, that is, there exists an exact functor A → Vectk , and is essentially small. We denote by Iso A the set of isomorphism classes of objects in A . Given an object M ∈ Ob A , we denote by [M ] ∈ Iso A its isomorphism class. For any S ⊂ Iso A , let Add S be the smallest additively closed full subcategory of A containing all direct summands of all objects M ∈ Ob A with [M ] ∈ S. We set AddA M = AddA {[M ]} for any object M in A . We denote by A (S) the smallest full subcategory of A containing all objects M with [M ] ∈ S and closed under extensions. We denote by K0+ (A ) the submonoid of K0 (A ) generated by the |M |, [M ] ∈ Iso A and define a partial order ≤ on K0+ (A ) by λ ≤ µ if µ − λ ∈ K0+ (A ). If B is a full subcategory of A closed under extensions, we denote by K0 (B) the subgroup of K0 (A ) generated by the |M |, [M ] ∈ Iso B, and we set K0+ (B) = K0+ (A ) ∩ K0 (B). 2.1. Recall that an object M in A is called indecomposable if M is not isomorphic to a direct sum of two non-zero subobjects. The category A is said to be Krull-Schmidt if every object in A can be written, uniquely (up to an isomorphism) as a direct sum of finitely many indecomposable objects. If A is Krull-Schmidt, we denote by Ind A ⊂ Iso A the set of isomorphism classes of all indecomposable objects in A . The following simple Lemma will be used repeatedly in the sequel. F F Lemma 2.1. Suppose that A is Krull-Schmidt and Ind A = I1 · · · Ir such that for all 1 ≤ i < j ≤ r, HomA (Mi , Mj ) = 0 if [Ms ] ∈ Is . Then the categories As = AddA Is , 1 ≤ s ≤ r, are closed under extensions. Proof. Consider a short exact sequence 0→M →X →N →0 with M, N ∈ Ob As . Applying HomA (Y, −) (respectively, HomA (−, Z)) with Y , Z indecomposable such that [Y ] ∈ It (respectively, [Z] ∈ Ik ) and t < s < k, we conclude that HomA (Y, X) = 0 (respectively, HomA (X, Z) = 0), hence X cannot have Y and Z as its indecomposable summands. Thus, the isomorphism classes of all indecomposable summands of X lie in Is and so X ∈ Ob As .  If Ai , i ∈ I are full additive subcategories of A such that the additive monoid Iso A is the W direct sum of its submonoids Iso Ai , we write A = i∈I Ai . A typical example is provided by a partition of the set Ind A if A is Krull-Schmidt (and so Iso A is a free monoid generated by Ind A ) F W if Ind A = i∈I Si then A = i∈I AddA Si . Also, we denote by B1 ∩ B2 the full subcategory of A whose objects are in B1 and B2 . QUANTUM CHEVALLEY GROUPS 2.2. 7 Given M, N, K ∈ Ob A , set K PM N = {(f, g) ∈ HomA (N, K) × HomA (K, M ) : ker g = Im f }. K The group AutA (M ) × AutA (N ) acts freely on PM N and the set of orbits identifies with K ∼ ∼ FM N = {U ⊂ K : U = N, K/U = M }. It is easy to see that isomorphisms of objects M → M ′ , N → N ′ and K → K ′ induce bijections K K′ between the sets FM N and FM ′ K ′ . Suppose that A is finitary, that is, HomA (M, N ) and Ext1A (M, N ) are finite sets for all M, N ∈ Ob A . Definition 2.2 (Ringel [9]). The Hall algebra HA of A is the Q-vector space with the basis [M ] ∈ Iso A and with the multiplication defined by X K [M ][N ] = FM N [K]. [K]∈Iso A K := #F K The number FM N M N is independent of representatives of isomorphism classes. A Hall algebra can be defined for an exact category. In particular, if B is a full subcategory of A and is closed under extensions, then we can consider its Hall algebra HB which identifies with the subspace of HA spanned by the [M ] ∈ Iso B. Note that HB is graded by K0 (B) M HB = HB,γ , γ∈K0 (B) b B of HB is defined where HB,γ is spanned by the [M ] ∈ Iso B with |M | = γ. Then the completion H by Y bB = H HB,γ . γ∈K0 (B) 1 1 K Clearly, there is a natural surjective map PM N → ExtA (M, N )K , where ExtA (M, N )K denotes the set of equivalence classes of short exact sequences 0→N →E→M →0 ∼ K. The fiber of that map is given by AutA (K)/ StabAut K (f, g) and it can be shown with E = A that the map HomA (M, N ) → StabAutA K (f, g), ψ 7→ 1 + f ψg is a bijection. This yields Proposition 2.3 (Riedtmann’s formula [8]). Let A be a finitary category. Then for all M, N, K ∈ Ob A | Ext1A (M, N )K || AutA (K)| K FM = (2.1) N | AutA (M ) × AutA (N ) × HomA (M, N )| We note the following simple corollary which will be used repeatedly. Corollary 2.4. (a) Suppose that HomA (N, M ) = 0 = Ext1A (M, N ). Then [M ][N ] = [M ⊕ N ]. [E]a where qE = (b) Let E be such that EndA E is a field and Ext1A (E, E) = 0. Then [E ⊕a ] = [a]qE ! | EndA E|. 8 A. BERENSTEIN AND J. GREENSTEIN Proof. To prove (a), observe that | Ext1A (M, N )M ⊕N | = 1. Then HomA (N, M ) = 0 implies that AutA (M ⊕ N ) ∼ = AutA (M ) × AutA (N ) × HomA (M, N ), and it remains to apply (2.1). For part (b), note that AutA E ⊕a ∼ = GL(a, EndA E), hence Qa j a+1 − qE ) | AutA (E ⊕(a+1) )| j=0 (qE a = = qE [a + 1]qE . Q a − qj ) | AutA (E ⊕a )|| AutA E| (qE − 1) a−1 (q j=0 E E a , (2.1) implies that Since | HomA (E ⊕a , E)| = qE [E ⊕a ][E] = [a + 1]qE [E ⊕(a+1) ], and the second identity follows by an obvious induction.  W W Proposition 2.5. Let A be a finitary abelian category and suppose that A = A1 · · · Ar where the Aj , 1 ≤ j ≤ r are closed under extensions and for all 1 ≤ i < j ≤ r, Aj ⊂ Ai . Then (a) the multiplication map m : HA1 ⊗ · · · ⊗ HAr → HA is an isomorphism of vector spaces. In particular, if A = A1 ⊕ · · · ⊕ Ar then HA ∼ = HA1 ⊗ · · · ⊗ HAr as an algebra; (b) If HomA (Aj , Ai ) = 0 for all 1 ≤ i < j ≤ r, where As ∈ Ob As , then (A1 , . . . , Ar ) is a factorizing sequence for A . Proof. Clearly, it is sufficient to prove this statement for r = 2. Let B = A1 , C = A2 . By (2.1), m([B] ⊗ [C]) = [B][C] = c[B],[C] [B ⊕ C] where c[B],[C] = | AutA (B ⊕ C)| 6= 0. | HomA (B, C)|| AutA (B) × AutA (C)| On the other hand, since for any [M ] ∈ Iso A , there exist a unique ([BM ], [CM ]) ∈ Iso B × Iso C with [M ] = [BM ⊕ CM ], the assignment [M ] 7→ c−1 [BM ],[CM ] [BM ] ⊗ [CM ] provides a well-defined linear map ψ : HA → HB ⊗ HC . Clearly, ψ ◦ m = 1HB ⊗HA and m ◦ ψ = 1HA . If A = B ⊕ C then HomA (B, C) = HomA (C, B) = 0 = Ext1A (B, C) = Ext1A (C, B) for all B ∈ Ob B, C ∈ Ob C . In particular, [B][C] = [C][B] = [B ⊕ C] and so HB HC = HC HB . Thus, in this case HA ∼ = HB ⊗ HC as an algebra. To prove (b), note that by Corollary 2.4(a), [B][C] = [B ⊕ C] for all B ∈ Ob B, C ∈ Ob C . Thus, X X X X ExpB ExpC = [B] [C] = [B][C] = [A] = ExpA .  [B]∈Iso B [C]∈Iso C ([B],[C])∈Iso B×Iso C [A]∈Iso A 2.3. Given an object X in A without self-extensions, let ExpA ([X]) = ExpAddA X . For example, if A is the category of finite dimensional vector spaces over a finite field k, ExpA = ExpA ([k]). Proposition 2.6. Let S be a simple object in a finitary abelian category A with the finite length property. Let A ′ = AddA S and let A ′′ be the maximal full subcategory of A whose objects M satisfy [M : S] = 0. (a) If S is injective, then for any M ∈ Ob A , there is a unique subobject M ′′ ∈ Ob A of M such that M/M ′′ ∼ = S ⊕[M :S] . In particular, (A ′ , A ′′ ) is a factorizing pair. QUANTUM CHEVALLEY GROUPS 9 (b) If S is projective, then for any M ∈ Ob A , there is a unique subobject M ′ ∼ = S ⊕[M :S] such that M/M ′ ∈ Ob A ′′ . In particular, (A ′′ , A ′ ) is a factorizing pair for A . Proof. We prove part (a), the proof for the other part being similar. Let M ∈ Ob A . Then M has a unique maximal subobject M ′′ which is an object in A ′′ . Indeed, if X, Y ∈ Ob A ′′ are subobjects of M , then their sum X + Y which is the image of X ⊕ Y in M under the map canonically induced by the natural inclusions X ֒→ M and Y ֒→ M , is also an object in A ′′ . We claim that M/M ′′ ∼ = S ⊕[M :S]. Indeed, let a ≥ 0 be maximal such that M/M ′′ has a subobject isomorphic to S ⊕a . Since S is injective, M/M ′′ ∼ = R⊕S ⊕a for some subobject R of M/M ′′ . Suppose that R 6= 0. Then R has a simple subobject S ′ , which by the choice of a is not isomorphic to S. Let K be the subobject of M such that M ′′ is a subobject of K and K/M ′′ = S ′ . Then K is an object in A ′′ and since K 6= M ′′ , we obtain a contradiction by the maximality of M ′′ . Thus, R = 0 and so M/M ′′ ∼ = S ⊕a . Since [M ′′ : S] = 0, the short exact sequence 0 → M ′′ → M → S ⊕a → 0 yields a = [M : S].  2.4. From now on, we only consider abelian k-categories for a field k. In this case, if S is a simple object in A , EndA S is a division algebra over k. Note also that Ext1A (S, S ′ ) is a right EndA Smodule and a left EndA S ′ -module. Consider the Ext-quiver EA of A , which is the valued quiver (a,b) with vertices corresponding to isomorphism classes of simple objects and arrows [S] −−−→ [S ′ ] if Ext1A (S, S ′ ) 6= 0 where a = dimEndA S ′ Ext1A (S, S ′ ) and b = dim Ext1A (S, S ′ )EndA S . We say that A is acyclic if EA is acyclic. If A has the finite length property, denote by [M : S] the Jordan-Hölder multiplicity of a simple object S in an object M . We say that A admits a source order if the set of vertices of EA is countable and there exists a numbering of vertices such that [Si ] is a source in the quiver obtained by removing all vertices [Sj ], j < i and their adjacent arrows. Equivalently, there exists a numbering of the isomorphism classes of simples such that Si is injective in the Serre subcategory whose objects satisfy [M : Sj ] = 0 if j < i. Lemma 2.7. Let A be an acyclic abelian category with the finite length property. Suppose that EA has finitely many vertices. Then A admits a source order. Proof. Since EA is finite and acyclic, it contains at least one source. Let [S1 ] be the corresponding class and consider the Serre subcategory whose objects satisfy [M : S1 ] = 0. Its Ext quiver is obtained from that of A by removing the vertex [S1 ] and its adjacent arrows. Continuing this way, we obtain the desired total order.  The following is an immediate consequence of Proposition 2.6. Corollary 2.8. Let A be a finitary category and suppose that A admits a source order [S1 ] ≺ [S2 ] ≺ · · · . Then − → Y ExpA ([Si ]) ExpA = i Now we can prove Proposition 1.12 from the introduction. Proof of Proposition 1.12. It is immediate from Proposition 2.6 that (A+ , A− ) is a factorizing pair W W for A . Clearly, A = A− A0 A+ , A− ⊂ A0 , A+ and A0 ⊂ A+ . Suppose that M is an object 10 A. BERENSTEIN AND J. GREENSTEIN in A0 . Then HomA (M, S− ) = 0 for any [S− ] ∈ S− , for otherwise S− is a direct summand of M , and similarly, HomA (S+ , M ) = 0 for all [S+ ] ∈ S+ . Therefore, (A− , A0 , A+ ) is a factorizing triple by Proposition 2.5.  2.5. Let B be a full subcategory of A closed under extensions. Given γ ∈ K0 (B) and an element b B , denote by X|γ its canonical projection onto HB,γ . Clearly, X = P X of HB or H γ∈K0 (B) X|γ . Define EB as the subalgebra of HB generated by the ExpB |γ , γ ∈ K0 (B). Lemma 2.9. Suppose that (A+ , A− ) is a factorizing pair for a finitary category A with the finite length property. Assume that K0+ (A+ ) ∩ K0+ (A− ) = 0. (2.2) Then EA± ⊂ EA . Proof. Since (A+ , A− ) is a factorizing pair, ExpA = ExpA+ ExpA− which is equivalent to X ExpA |γ = (ExpA+ |γ+ )(ExpA− |γ− ), γ ∈ K0+ (A ). (γ+ ,γ− )∈K0+ (A+ )×K0+ (A− ) : γ+ +γ− =γ Then by (2.2) we have for all γ± ∈ K0 (A± )+ ExpA± |γ± = ExpA |γ± − X (ExpA+ |µ+ )(ExpA− |µ− ). (µ+ ,µ− )∈(K0+ (A+ )\{0})×(K0+ (A− )\{0}) : µ+ +µ− =γ± We can now finish the proof by a simultaneous induction on partially ordered sets K0+ (A+ ) and F K0+ (A− ) (note that the induction begins since |S| ∈ K0+ (A+ ) K0+ (A− ) for every simple object S in A ).  Lemma 2.10. Let A be a finitary abelian category with the finite length property. Let B be a full subcategory of A closed under extensions and direct summands and satisfying 1◦ If M, N are indecomposable objects in B with |M | = |N | then M ∼ = N; 2◦ HB is generated by the [M ] ∈ Ind A ∩ Iso B. Then HB = EB . Proof. If M ∈ Ob B is indecomposable then 1◦ implies that X [M ] = ExpB ||M | − [N ]. (2.3) [N ] decomposable : |N |=|M | Given γ ∈ K0+ (B), define HB,<γ = M HB,µ , HB,≤γ = HB,<γ ⊕ HB,γ . µ∈K0+ (B) : µ<γ Clearly, the condition 2◦ implies that if M is decomposable then [M ] ∈ (HB,<|M | )2 . We prove by induction on the partially ordered set K0+ (B) that HB,≤γ ⊂ EB for all γ ∈ K0+ (B). If γ ∈ K0+ (B) is minimal then by (2.3) [M ] = ExpB |γ with [M ] indecomposable and so the induction begins. Suppose that the claim is proven for all γ ′ < γ. Thus, in particular, HB,<γ ⊂ EB . If γ is such that all objects M with |M | = γ are decomposable, we are done by the induction hypothesis. QUANTUM CHEVALLEY GROUPS 11 Otherwise, for the unique [M ] ∈ Ind A ∩ Iso B with |M | = γ we have [M ] ∈ EB by (2.3) and the induction hypothesis, hence HB,γ ⊂ EB and so HB,≤γ ⊂ EB .  3. q-commutators and proof of Theorem 1.8 3.1. Given an integral domain R and q ∈ R, q 6= 0, define an R-linear operator Dq on R[[x]] (the q-derivative) by f (qx) − f (x) (Dq f )(x) = (q − 1)x and set 1 Dj . Dq(j) := [j]q ! q Since, clearly   a (j) a Dq x = xa−j , (3.1) j q where   [a]q ! a = [j]q ![a − j]q ! j q (j) is the Guassian q-binomial which is well-known to be a polynomial in q, the Dq are well-defined R-endomorphisms of R[[x]] and of R[x] regarded as a subring of R[[x]]. We gather some elementary properties of Dq in the following Lemma. Lemma 3.1. (a) For all f, g ∈ R[[x]], Dq(j) (f g) = j X (Dq(j−s) f )(q s x)Dq(s) g. s=0 (b) If f ∈ R[x] is invertible in R[[x]] then j Y f (q s x)Dq(j) (f −1 ) ∈ R[x], j ≥ 0. s=0 Proof. For j = 1 it is easy to check that Dq (f g) = f (qx)Dq (g) + g(Dq f ). (3.2) For the inductive step, we have Dq(r+1) (f g) = r X 1 Dq ((Dq(r−s) f )(q s x)Dq(s) g). [r + 1]q s=0 Since Dq (P (q a x)) = q a (Dq P )(q a x), we obtain Dq(r+1) (f g) = r X 1 q s [r + 1 − s]q (Dq(r+1−s) f )(q s x)Dq(s) g + [s + 1]q (Dq(r−s) f )(q s+1 x)Dq(s+1) g. [r + 1]q s=0 To complete part (a), it remains to observe that q s [r + 1 − s]q + [s]q = [r + 1]q . For part (b), we use induction on j. Note that the assertion trivially holds for j = 0 and (a) implies r X (Dq(r−j) f )(q j x)Dq(j) (f −1 ) = 0. j=0 12 A. BERENSTEIN AND J. GREENSTEIN Therefore,  r+1 Y j=0 r r  Y  X (r−j) j (r+1) −1 f (q t x) Dq(j) (f −1 ). (Dq f )(q x) (f ) = − f (q x) Dq j t=0 j=0 Q  j t x) D (j) (f −1 ) ∈ R[x], hence the right hand side of the By the induction hypothesis, f (q q t=0 above expression is contained in R[x].  3.2. Let F be a field and let A be a unital associative F -algebra. Given x ∈ A and q ∈ F × , let adq x, ad∗q x be the F -linear endomorphisms of A defined by adq x(y) = xy − qyx and ad∗q x(y) = yx − qxy Q for all y ∈ A. Since, clearly adq x ◦ adq′ x = adq′ x ◦ adq x, we can define ad(q0 ,...,qr ) x = ri=0 adqi x Q and ad∗(q0 ,...,qr ) x = ri=0 ad∗qi x. Lemma 3.2. Let x, y ∈ A and (q0 , . . . , qr ) ∈ (F × )r+1 . Then (ad(q0 ,...,qr ) x)(y) = r+1 X (−1)j ej (q0 , . . . , qr )xr+1−j yxj , j=0 r+1 X (−1)j ej (q0 , . . . , qr )xj yxr+1−j , (ad∗(q0 ,...,qr ) x)(y) = (3.3) j=0 and r x y= r X hr−j (q0 , . . . , qj )(ad(q0 ,...,qj−1 ) x)(y)xr−j , j=0 r yx = r X (3.4) hr−j (q0 , . . . , qj )x r−j (ad∗(q0 ,...,qj−1 ) x)(y), j=0 where es (respectively, hs ) denotes the elementary (respectively, the complete) symmetric polynomial of degree s. In particular, if q0 , q ∈ F ×   r X j r+1 (−1)j q0j q (2) xr+1−j yxj , j q j=0   r X j j (2j ) r + 1 ∗ (−1) q0 q (adq0 (1,...,qr ) x)(y) = xj yxr+1−j j q (adq0 (1,...,qr ) x)(y) = (3.5) j=0 and r x y= r X j=0 q0r−j   r (adq0 (1,...,qj−1 ) x)(y)xr−j , j q r yx = r X j=0 q0r−j   r xr−j (ad∗q0 (1,...,qj−1 ) x)(y). (3.6) j q Proof. We prove the identities involving ad, the proof of the ones involving ad∗ being similar. The argument is by induction on r, the case r = 0 being obvious in both (3.3) and (3.4). For the QUANTUM CHEVALLEY GROUPS 13 inductive step in (3.3), we have r r X X j r+1−j j (−1)j ej (q0 , . . . , qr−1 )qr xr−j yxj+1 (−1) ej (q0 , . . . , qr−1 )x yx − (ad(q0 ,...,qr ) x)(y) = j=0 j=0 = r+1 X (−1)j (ej (q0 , . . . , qr−1 ) + ej−1 (q0 , . . . , qr−1 )qr )xr+1−j yxj j=0 where we use the usual convention that es (x1 , . . . , xk ) = 0 if s < 0 or s > k. To prove (3.4), let y0 = y and define yj+1 = (adqj x)(yj ), j ≥ 0. Then xr+1 y = r X hr−j (q0 , . . . , qj )(qj yj xr+1−j + yj+1 xr−j ) j=0 = r+1 X (hr−j (q0 , . . . , qj )qj + hr+1−j (q0 , . . . , qj−1 ))yj xr+1−j . j=0 To complete both inductive steps, it remains to observe that ej (q0 , . . . , qr−1 ) + ej−1 (q0 , . . . , qr−1 )qr = ej (q0 , . . . , qr ) and hr−j (q0 , . . . , qj )qj + hr+1−j (q0 , . . . , qj−1 ) = hr+1−j (q0 , . . . , qj ) which follow from the formulae k+1 X es (x0 , . . . , xk )ts = s=0 Since k Y (1 + xi t), i=0 n Y r (1 + q t) = r=0 X s≥0   s n+1 s ( ) q 2 t , s q X hs (x0 , . . . , xk )ts = j=0 s≥0 n Y r=0 k Y 1 . 1 − xj t X s + n  1 = ts , 1 − qr t s q (3.7) s≥0 (3.5) and (3.6) follow, respectively, from (3.3) and (3.4) b Suppose that A admits a completion A.  Corollary 3.3. Let x, y ∈ A and suppose that (adq0 (1,q,...,qr ) x)(y) = 0 for some q0 , q ∈ F × and r ≥ 0 b Then for any and that the assignment t 7→ x extends to an algebra homomorphism F [[t]] → A. P ∈ F [[t]] we have in Ab r X (adq0 (1,q,...,qj−1 ) x)(y)(Dq(j) P )(q0 x). P (x)y = j=0 Similarly, if (ad∗q0 (1,q,...,qs ) x)(y) = 0, yP (x) = s X (Dq(j) P )(q0 x)(ad∗q0 (1,q,...,qj−1 ) x)(y) j=0 Given q ∈ F × , define expq (t) = X tj ∈ F [[t]]. [j]q ! j≥0 14 A. BERENSTEIN AND J. GREENSTEIN Lemma 3.4. In F [[t]], we have Dq (expq (t)) = expq (t) and for ν ∈ Z ν−1 Y   r    (1 + q (q − 1)t), expq (t)−1 expq (q ν t) =: Φν (t, q) = r=0 −ν Y   −r −1    (1 + q (q − 1)t) , ν≥0 ν < 0. r=1 Proof. The first identity is obvious. Since −1 expq (t)    r r X (−1)a q νb ta+b X  X t a (a2)+ν(r−a) r expq (q t) = (−1) q = a q [r]q ! [a]q ![b]q ! ν r≥0 a,b≥0 a=0 and by (3.7) the inner sum equals r−1 Y q νr (1 − q a−ν ). a=0 Suppose first that ν ≥ 0. Then we can rewrite it as   r r ν ( ) 2 (q − q ) = [r]q !q (q − 1) r q a=0 r−1 Y ν a hence −1 expq (t) which by (3.7) equals If ν < 0, we write q νr r−1 Y (1 − q −ν+a ν expq (q t) = X r≥0 Qν−1 r=0 (1 + q r (q   r ν ( ) q 2 ((q − 1)t)r r q − 1)t). r νr r ) = (−1) q (q − 1) r−1 Y a=0 a=0  −ν + r − 1 [−ν + r − 1 − a]q = (−1) q [r]q !(q − 1) r r νr r  q hence −1 expq (t) ν expq (q t) = X −ν + r − 1 r r≥0 (−(q − 1)q ν t)r = q −ν−1 Y (1 + (q − 1)q ν+s t)−1 s=0 where we used the second identity in (3.7).  Corollary 3.5. Let x, y ∈ A and assume that (ad(qν ,qν+1 ,...,qν+r ) x)(y) = 0 for some q ∈ F × , ν ∈ Z and r ≥ 0. Then in Ab r  X 1 (ad(qν ,...,qν+j−1 ) x)(y) Φν (x, q) expq (x)y expq (x)−1 = [j]q ! j=0 and −1 expq (x) y expq (x) = j r X (−1)j q (2) j=0 [j]q ! (ad(qν ,...,qν+j−1 ) x)(y)Φ−j−ν (q ν+j x, q). exp(x)y exp(x)−1 In particular, ∈ A (respectively, exp(x)−1 y exp(x) ∈ A) if and only if ν ≥ 0 (respectively, −ν ≥ r). Similarly, if (ad∗(qµ ,qµ+1 ,...,qµ+s ) x)(y) = 0 for some q ∈ R, µ ∈ Z and s ≥ 0 −1 expq (x) s  X 1 y expq (x) = Φµ (x, q) (ad∗(qµ ,...,qµ+j−1 ) x)(y) [j]q ! j=0 QUANTUM CHEVALLEY GROUPS 15 and −1 expq (x)y expq (x) = j s X (−1)j q (2) [j]q ! j=0 Φ−j−µ(q j+µ x, q)(ad∗(qµ ,...,qµ+j−1 ) x)(y). In particular, expq (x)−1 y expq (x) ∈ A (respectively, expq (x)y expq (x)−1 ∈ A) if and only if µ ≥ 0 (respectively, −µ ≥ s). Recall that a multiplicatively closed subset S of an algebra A is said to be an Ore set if S does not contain zero divisors and for all a ∈ A, s ∈ S there exist u, u′ ∈ A and t, t′ ∈ S such that at = su and t′ a = u′ s. Proposition 3.6. Let Y be a generating set for A. Let x ∈ A be such that the assignment t 7→ x extends to an algebra homomorphism F [[t]] 7→ Ab and suppose that there exists q ∈ F × such that for all y ∈ Y either (ad(qν ,qν+1 ,...,qν+r ) x)(y) = 0 or (ad∗(qν ,qν+1 ,...,qν+r ) x)(y) = 0 for some ν = ν(y) ∈ Z, r = r(y) ≥ 0. Let Sx be the minimal multiplicatively closed subset of A containing 1 + q s (q − 1)x for all s ∈ Z. Then Sx is an Ore set in A and the assignment z 7→ expq (x)z expq (x)−1 , z ∈ A defines an automorphism of A[Sx−1 ]. b Since Proof. The elements of Sx are not zero divisors since they are invertible in the completion A. the elements of Sx commute, it is enough to prove that the Ore condition holds for all y ∈ A and s ∈ Sx . We need the following simple
(s) Qr −j Lemma 3.7. Let P ∈ F [t]. Then P divides Dq j=0 P (q t) for all 0 ≤ s ≤ r. Q Proof. Let Pr = rj=0 P (q −j t). The argument is by induction on r. For r = 0 there is nothing to do. For the inductive step, note that by Lemma 3.1(a) Dq(s) Pr+1 = s X q −(r+1)(s−a) (Dq(s−a) P )(q a−r−1 t)Dq(a) Pr . a=0 (a) If s < r + 1 then by the induction hypothesis P divides Dq Pr , 0 ≤ a ≤ s. If s = r + 1, the only term in the above sum to which the induction hypothesis does not apply is that with a = r + 1 = s, (r+1) which equals P (t)Dq Pr .  Suppose that y ∈ A satisfies (ad∗(qν ,qν+1 ,...,qν+r ) x)(y) = 0 for some q ∈ R× and r, ν ≥ 0 (the case of (ad(qν ,...,qµ+r ) x)(y) = 0 is similar). Let P ∈ Sx and set ′ P = r Y P (q −j−ν x) j=0 Clearly, P ′ ∈ Sx . Then by Corollary 3.3 ′ yP = r X j=0 q −jν Dq(j) r Y t=0 (j) since, by the above Lemma, P divides Dq  P (q −t x) Yj = P Y, Q  k −t t=0 P (q x) Yj , Y ∈ A, for all 0 ≤ j ≤ k. 16 A. BERENSTEIN AND J. GREENSTEIN b we further have Since P is invertible in A, yP −1 r X (Dq(j) P −1 )(q ν x)Yj . = j=0 Qr (j) Let P ′′ = s=0 P (q ν+s x). By Lemma 3.1(b), P ′′ (Dq P −1 ) is a polynomial in x for all 0 ≤ j ≤ k. Therefore, P ′′ yP −1 ∈ A. Thus, Sx is an Ore set in A. It remains to apply Corollary 3.5.  3.3. We will now use the above identities to study the action of certain elements of the quantum Chevalley group on the Hall algebra. Recall that an object E in an abelian category A is called a brick if EndA E is a division ring. In that case, ExtiA (E, M ) is a right EndA E-vector space while ExtiA (M, E) is a left EndA E-vector space for all i ≥ 0. Set νE (M ) = dim HomA (E, M )EndA E − dimEndA E HomA (M, E). We say that a brick E is exceptional if Ext1A (E, E) = 0. Lemma 3.8. Let E be an exceptional brick in an abelian category A and let M be an object in A . If Ext1A (E, M ) = 0 then for a non-split short exact sequence 0→E→U →M →0 we have HomA (U, E) ∼ = HomA (M, E), Ext1A (E, U ) = 0, Ext1A (U, E) ∼ = Ext1A (M, E)/ EndA E and HomA (E, M ) ∼ = HomA (E, U )/ EndA E. Similarly, if Ext1A (M, E) = 0 then for a non-split short exact sequence 0→M →U →E→0 we have HomA (E, U ) ∼ = HomA (E, M ), Ext1A (U, E) = 0, Ext1A (E, U ) ∼ = Ext1A (E, M )/ EndA E and HomA (M, E) ∼ = HomA (U, E)/ EndA E. Proof. We prove the first statement only, the argument for the second one being similar. Consider a non-split short exact sequence ι 0→E− → U → M → 0. (3.8) Applying HomA (−, E) we obtain 0 → HomA (M, E) → HomA (U, E) → EndA E → Ext1A (M, E) → Ext1A (U, E) → 0. We claim that the natural morphism HomA (U, E) → EndA E, f 7→ f ◦ ι is identically zero. Otherwise, since it is a morphism of left EndA E-vector spaces and EndA E is one dimensional as such, it is surjective and so there exists f ∈ HomA (U, E) such that f ◦ ι = 1E hence (3.8) splits. Thus, HomA (U, E) ∼ = HomA (M, E) and we have a short exact sequence 0 → EndA E → Ext1A (M, E) → Ext1A (U, E) → 0. On the other hand, applying HomA (E, −) to (3.8) we obtain a long exact sequence 0 → EndA E → HomA (E, U ) → HomA (E, M ) → 0 → Ext1A (E, U ) → Ext1A (E, M ) → 0 which yields the remaining isomorphisms.  QUANTUM CHEVALLEY GROUPS 17 3.4. Let A be a finitary abelian category. Let E be an exceptional brick in A and denote kE = EndA E and qE = |kE |. Proposition 3.9. If Ext1A (E, M ) = 0, then (ad∗ νE (M ) (qE ν ,...,qEE (M )+r ) [E])([M ]) = 0, bA where r = dimkE Ext1A (M, E), and for any P ∈ Q[[x]] we have in H [M ]P ([E]) = r X ν (M ) (Dq(j) P )(qEE E [E])(ad∗ νE (M ) (qE j=0 ν ,...,qEE (M )+j−1 ) [E])([M ]) Similarly, if Ext1A (M, E) = 0, (ad −νE (M ) (qE −νE (M )+s ,...,qE ) [E])([M ]) = 0, bA where s = dimkE Ext1A (E, M ), and for any P ∈ Q[[x]] we have in H X −ν (M ) [E]). P )(qE E P ([E])[M ] = (ad −νE (M ) −νE (M )+j−1 [E])([M ])(Dq(j) E (qE j≥0 ,...,qE ) Proof. We prove only the first statement, the prove of the second one being similar. We will need the following Lemma Lemma 3.10. If Ext1A (E, M ) = 0, (ad∗ νE (M ) (qE ν (M )+j−1 ) ,...,qEE X [E])([M ]) = c[U ] [U ] [U ]∈Iso A where c[U ] 6= 0 implies that dimkE Ext1A (U, E) = dimkE Ext1 (M, E) − j, νE (U ) = νE (M ) + j. (3.9) In particular, (ad∗ νE (M ) (qE ν ,...,qEE (M )+j ) [E])([M ]) = 0, j ≥ dimkE Ext1 (M, E). Similarly, if Ext1A (M, E) = 0, (ad −ν (M ) −ν (M )+j−1 (qE E ,...,qE E ) [E])([M ]) = X c[U ′ ] [U ′ ] [U ′ ]∈Iso A where c[U ′ ] = 0 unless dimkE Ext1A (E, U ′ ) = dimkE Ext1 (E, U ′ ) − j, νE (U ′ ) = νE (M ) − j. In particular, (ad −νE (M ) (qE −νE (M )+j ,...,qE ) [E])([M ]) = 0, j ≥ dimkE Ext1 (E, M ). Proof. Consider U such that Ext1A (E, U ) = 0. We have by (2.1) [E][U ] = | AutA (U ⊕ E)| [U ⊕ E]. | HomA (E, U )|| AutA U ||k× E| (3.10) 18 A. BERENSTEIN AND J. GREENSTEIN On the other hand, by Lemma 3.8 and (2.1) [U ][E] = | AutA (U ⊕ E)| [U ⊕ E] + | HomA (U, E)|| AutA U ||k× E| X ν (U ) c[U ′ ] [U ′ ] = qEE [E][U ] + X c[U ′ ] [U ′ ] [U ′ ]∈Iso A [U ′ ]∈Iso A where c[U ′ ] 6= 0 implies that we have a non-split short exact sequence 0 → E → U ′ → U → 0. By Lemma 3.8, νE (U ′ ) = νE (U ) + 1 and dimkE Ext1A (U ′ , E) = dimkE Ext1A (U, E) − 1. An obvious induction now completes the proof of the Lemma.  To complete the proof of the proposition, it remains to apply the Lemma, together with Corolν (M ) , q = qE and r = dimkE Ext1A (M, E) (respectively, lary 3.3 with x = [E], y = [M ], q0 = qEE s = dimkE Ext1A (E, M )).  Thus, if E is an exceptional brick and Ext1A (E, M ) = 0 r+1 X j=0   (2j )+jνE (M ) r + 1 (−1) qE [E]j [M ][E]r+1−j = 0, j qE j r = dimkE Ext1A (M, E) (3.11) r = dimkE Ext1A (E, M ). (3.12) while if Ext1A (M, E) = 0, r+1 X j=0   (2j )−jνE (M ) r + 1 (−1) qE [E]r+1−j [M ][E]j = 0, j qE j In the special case when (E, E ′ ) is an orthogonal exceptional pair of bricks (that is, HomA (E, E ′ ) = HomA (E ′ , E) = Ext1A (E, E ′ ) = 0), we obtain the so-called fundamental relations in the Hall algebra of the subcategory A (E, E ′ ) of A which is defined to be the smallest full subcategory of A containing E, E ′ and closed under extensions. In particular, this yields Serre relations in CA . Namely, S and S ′ are non-isomorphic simples with Ext1A (S ′ , S) = 0, dimkS Ext1A (S, S ′ ) = r and dimkS ′ Ext1A (S, S ′ ) = r ′ then     r+1 j X ′ (2j ) r ′ + 1 j (2) r + 1 ′ j ′ r +1−j (−1) qS ′ (−1) qS [S ] [S][S ] =0= [S]r+1−j [S ′ ][S]j j j q ′ qS ′ +1 rX j=0 j (3.13) j=0 S Propositions 3.6 and 3.9 immediately yield Theorem 3.11. Let E be an exceptional brick and let SE be the minimal multiplicatively closed j [E] for all j ∈ Z. Then SE is an Ore set in HAdd (E) subset of HA containing 1 + (qE − 1)qE A and in HAdd A (E) phism of HAdd and the assignment [M ] 7→ ExpA ([E])[M ] ExpA ([E])−1 extends to an automor- A (E) [SE−1 ] and HAdd A (E) [SE−1 ]. In particular, if E is a projective or an injective indecomposable, this assignment defines an automorphism of HA [SE−1 ]. Therefore, Theorem 1.8 is proved. QUANTUM CHEVALLEY GROUPS 19 4. Preprojective and preinjective factorization 4.1. Let A be a hereditary finitary category with the finite length property and with a projective generator and an injective co-generator. We may assume, up to equivalence, that A is the category mod Λ of finitely generated left modules over a hereditary Artin k-algebra Λ. We now gather the properties of preprojective and preinjective objects in the category A that will be needed later. Let τ + : A → A be the Auslander-Reiten translation and τ − be its left adjoint functor. By [1, Corollary IV.4.7], for any objects X, Y ∈ Ob A we have dimk Ext1A (X, Y ) = dimk HomA (Y, τ + X). (4.1) An indecomposable M ∈ Ob A is called preprojective (respectively, preinjective) if for some nonnegative integer n, (τ + )n M is a non-zero projective (respectively, (τ − )n M is a non-zero injective) f ) (respectively, If(A )) be the set of isomorphism classes of indecomposable preobject. Let P(A projective (respectively, preinjective) objects. We say that M ∈ Ob A is preprojective (respectively, preinjective) if all its indecomposable summands are preprojective (respectively, preinjective), or, equivalently, if (τ + )n M = 0 (respectively, (τ − )n M = 0) for some n ≥ 0. For an indecomposable preprojective P (respectively, preinjective I) we denote by δ+ (P ) (respectively, δ− (I)) the non-negative integer, easily seen to be unique, such that (τ + )δ+ (P ) P is projective (respectively, (τ − )δ− (I) I is injective). If an indecomposable X is not preprojective (respectively, preinjective) we set δ+ (X) = ∞ (respectively, δ− (X) = ∞). Clearly we can regard δ± as functions δ± : Ind A → Z≥0 ∪ {∞}. We will need the following properties of preprojectives and preinjectives (the details can be found in [1, §VIII.1–2]) Proposition 4.1. (a) For any n ≥ 0, the sets {α ∈ Ind A : δ+ (α) ≤ n} and {α ∈ Ind A : f ) and If(A ) are countable. δ− (α) ≤ n} are finite. In particular, P(A f ) and [I] ∈ If(A ). Then Let X, P , I be indecomposable, with [P ] ∈ P(A (b) HomA (X, P ) 6= 0 =⇒ δ+ (X) ≤ δ+ (P ). (c) Ext1A (P, X) 6= 0 =⇒ δ+ (X) ≤ δ+ (P ) − 1. (d) HomA (I, X) 6= 0 =⇒ δ− (X) ≤ δ− (I). (e) Ext1A (X, I) 6= 0 =⇒ δ− (X) ≤ δ− (I) − 1. (f) P and I are exceptional bricks. (g) If |X| = |P | (respectively, |X| = |I|) then X ∼ = P (respectively, X ∼ = I). Lemma 4.2. Suppose that E, E ′ are non-isomorphic preprojective or preinjective indecomposables in A . Then HomA (E, E ′ ) 6= 0 implies that HomA (E ′ , E) = 0 and Ext1A (E ′ , E) 6= 0 implies Ext1A (E, E ′ ) = 0. Proof. We prove the statement for preprojectives only, since the duality functor implies the dual statement for preinjectives. If HomA (E, E ′ ) 6= 0 then δ+ (E) ≤ δ+ (E ′ ) by Proposition 4.1 and so the strict inequality immediately yields HomA (E ′ , E) = 0. Assume that δ+ (E) = δ+ (E ′ ) = r. Thus, E = (τ − )r P and E ′ = (τ − )r P ′ where P, P ′ are projective and indecomposable. Since E ∼ 6 E′, = P ∼ 6 P ′ by [1, Proposition VIII.1.3]. Since the functor τ − is fully faithful on the subcategory = of modules without injective summands ([1, Lemma VIII.1.1]), it is thus enough to prove that HomA (P, P ′ ) 6= 0 implies that HomA (P ′ , P ) = 0. By [1, Lemma II.1.12], HomA (P, P ′ ) 6= 0 implies 20 A. BERENSTEIN AND J. GREENSTEIN that P is isomorphic to a submodule of P ′ and similarly HomA (P ′ , P ) 6= 0 implies that P ′ is isomorphic to a submodule of P . Since all involved modules are finite dimensional over k, we conclude that P ∼  = P ′. 0 4.2. Let E be an indecomposable preprojective object in A . Let Ind< E A (respectively, IndE A , + + + + Ind> E A ) be the subset of Ind A consisting of α with δ (α) < δ (E) (respectively, δ (α) = δ (E), 0 < < 0 δ+ (α) > δ+ (E)). Set AE> = AddA Ind> E A , AE = AddA IndE A and AE = AddA IndE A and let A ≥0 = AddA (Ind0E A ∪ Ind> E A ). For an indecomposable preinjective object, we define similar + subcategories with δ replaced by δ− . It turns out that (AE< , AE0 , AE> ) is a factorizing triple. Moreover, we have the following 0 Proposition 4.3. (a) Ind< E A and IndE A are finite sets; ≥ (b) AE> , AE0 , AE< and AE are closed under extensions; (c) The multiplication map HA < ⊗ HA 0 ⊗ HA > → HA is an isomorphism of vector spaces and E E E ExpA = ExpA < ExpA 0 ExpA > ; E E E (d) For all [M ], [N ] ∈ Ind0E A , [M ][N ] = q r [N ][M ], q = |k|, r ∈ Z, and HA 0 is generated by the E [M ] ∈ Ind0E A ; (e) SE defined in Proposition 3.11 is an Ore set in HA and ExpA ([E]) acts on HA [SE−1 ] by conjugation. Proof. Part (a) is immediate from Proposition 4.1(a). > Let [M ] ∈ Ind< E A , [N ] ∈ Ind0 A , [K] ∈ IndE A . Then HomA (K, N ) = HomA (K, M ) = HomA (N, M ) = 0 by Proposition 4.1(b), and part (b) follows from Lemma 2.1. Furthermore, A = AE< By Proposition 4.1(c) AE< ⊂ (AE≥0 ) , W AE0 W AE> . AE0 ⊂ (AE> ) . Then (c) follows from Proposition 2.5. To prove (d), note that by Proposition 4.1(c), Ext1A (L, L′ ) = 0 = Ext1A (L′ , L) for all L, L′ with [L], [L′ ] ∈ Ind0E A and by Lemma 4.2 HomA (L, L′ ) 6= 0 implies that HomA (L′ , L) = 0. Thus, by Corollary 2.4 [L][L′ ] = | HomA (L′ , L)| ′ [L ][L]. | HomA (L, L′ )| Since HomA (L, L′ ) is a finite dimensional k-vector space, this fraction is a power of q. In particular, every [N ] ∈ Ind A0E , can be written, up to a power of q, as a product of the [L] ∈ Ind0E A . To prove (e), note that by part (c), every basis element [M ] of HA can be written as a product of a basis element of HAdd (E) and of a basis element of HAdd (E) . It remains to apply Proposition 3.11. A A  f ) such that [M ] ≺ [N ] implies Lemma 4.4. There exists a total order ≺, called normal, on P(A that Ext1A (M, N ) = 0 = HomA (N, M ). Similarly, there exists a total order ≺ on If(A ) such that [M ] ≺ [N ] implies that Ext1A (N, M ) = 0 = HomA (M, N ). f ) : δ+ (α) = n} Proof. Fix n ≥ 0. We first proceed to define an order ≺n on the set {α ∈ P(A f ) by setting α ≺ β if δ+ (α) < δ+ (β) or δ+ (α) = δ+ (β) = n then extend it to a total order on P(A QUANTUM CHEVALLEY GROUPS 21 and α ≺n β. To define ≺n , it is enough to define ≺0 , since τ − is a fully faithful functor on the full subcategory category of A whose objects have no injective summands. Thus, if P , P ′ are projective indecomposables, set P ≺0 P ′ if P is a subobject of P ′ . Since A is acyclic, this can be completed to a total order on the set of projective indecomposables with the desired property.  4.3. Suppose that A is indecomposable in the sense that it cannot be written as a direct sum of abelian blocks. Then Ind A is a finite set if and only if there exists an indecomposable module which is both preinjective and preprojective ([1, Proposition VIII.1.14]). Otherwise, there exists an indecomposable module, called regular, which is neither preprojective, nor preinjective. Let R(Λ) be the set of isomorphism classes of such modules. More generally, a module is called regular if all its indecomposable summands are regular. f ), A+ = AddA If(A ) and A0 = AddA R(Λ). Thus, A− (respectively, Let A− = AddA P(A A+ , A0 ) is the full subcategory of preprojective (respectively, preinjective, regular) modules. We F f ) F R(A ); thus, objects of A≥0 also set A≥0 = AddA (If(A ) R(A )) and A≤0 = AddA (P(A (respectively, A≤0 ) are modules which have no preprojective (respectively, preinjective) summands. Since G G f ) R(Λ) If(A ) Ind A = P(A and HomA (R, P ) = HomA (I, P ) = HomA (I, R) = 0 for all indecomposable P , R, I such that f ), [R] ∈ R(Λ) and [I] ∈ If(A ), it follows from Lemma 2.1 that A± , A≥0 , A≤0 and A0 [P ] ∈ P(A are closed under extensions. ∼ HA ⊗ HA ⊗ HA as an algebra, Proposition 4.5. Suppose that Ind A is infinite. Then HA = − 0 + and ExpA = ExpA− ExpA≥0 = ExpA− ExpA0 ExpA+ = ExpA≤0 ExpA+ . Moreover, ExpA± = → Y ExpA ([M ]), [M ]∈Ind A± where the product is taken in the normal order, that is, if [M ] ≺ [M ′ ] then ExpA ([M ]) occurs to the left of ExpA ([M ′ ]). W W Proof. Clearly, A = A− A0 A+ . Since by Proposition 4.1 Ext1A (M− , M0 ) = Ext1A (M− , M+ ) = Ext1A (M0 , M+ ) = 0 and HomA (M0 , M− ) = HomA (M+ , M− ) = HomA (M+ , M0 ) = 0 the first assertion follows from Proposition 2.5. To prove the second, note that [P ] ≺ [Q] implies that HomA (Q⊕b , P ⊕a ) = 0 = Ext1A (P ⊕a , Q⊕b ). f ) is countable, number its elements according to the normal order as Since P(A f ) = {[P1 ], [P2 ], . . . }, P(A [P1 ] ≺ [P2 ] ≺ · · · Then we have by Corollary 2.4 [P1⊕a1 ⊕ P2⊕a2 ⊕ · · · ] = [P1⊕a1 ][P2⊕a1 ] · · · and we obtain the desired factorization for ExpA− .  22 A. BERENSTEIN AND J. GREENSTEIN Thus, we have ExpA− ExpA0 ExpA+ = ExpA = ExpA ([S1 ]) · · · ExpA ([Sr ]), (4.2) where the product in the right hand side is written in the source order. 4.4. Fix an ordering of the isomorphism classes of simples, say the one corresponding to the source order. Since A has the finite length property, S = {|Si |}1≤i≤r is a basis of K0 (A ) and X |M | = [M : Si ]|Si |. i It is well-known (cf. [1, Lemma VIII.2.1]) that P = {|Pi |}1≤i≤r (respectively, I = {|Ii |}1≤i≤r ) where Pi (respectively, Ii ) is the projective cover (respectively, the injective envelope) of a simple object Si , are bases of K0 (A ). The automorphism c of K0 (A ) defined by c(|Pi |) = −|Ii | is called the Coxeter transformation. Consider the non-symmetric Euler form (· , ·) : K0 (A ) × K0 (A ) → Z defined by (|M |, |N |) = dimk HomA (M, N ) − dimk Ext1A (M, N ). Thus, h|M |, |N |i = |k|(|M |,|N |). Let di = (|Si | , |Si |). Since for any M ∈ Ob A dimEndA Si HomA (Pi , M ) = [M : Si ] = dimEndA Si HomA (M, Ii ) we have (|Pi |, |Sj |) = di δi,j = X [Pi : Sk ](|Sk |, |Sj |) k which implies that the basis change matrix from P to S is equal to (C T )−1 D, where C = ((|Si |, |Sj |))1≤i,j≤r is the matrix of the Euler form with respect to S and D = diag(d1 , . . . , dr ). Similarly, the basis change matrix from I to S equals C −1 D. Thus, we obtain Lemma 4.6. The matrix of the Coxeter transformation with respect to S equals −C −1 C T . Since (|Si |, |Sj |) = 0 if i > j and di = dimk EndA Si 6= 0, we conclude that the form (·, ·) is non-degenerate. We will need the following properties of the Coxeter transformation. Proposition 4.7 ([1, Proposition VIII.2.2 and Corollary VIII.2.3]). Let M be an indecomposable object. (a) If M is not projective then c(|M |) = |τ + M | ∈ K0+ (A ). Otherwise, c(|M |) ∈ −K0+ (A ). (b) If M is not injective then c−1 (|M |) = |τ − M | ∈ K0+ (A ). Otherwise, c−1 (|M |) ∈ −K0+ (A ). (c) For any preprojective (respectively, preinjective) object X, there exists r > 0 such that cr (|X|) (respectively, c−r (|X|)) is in −K0+ (A ). δ Proof of Lemma 1.14 and Theorem 1.15. Since h|Pi |, αj i = qi ij = hαj , |Ii |i and the form h·, ·i is nondegenerate, it follows that γ−i = |Pi | and γi = |Ii |, 1 ≤ i ≤ r. Therefore, the automorphism c defined before Lemma 1.14 coincides with the Coxeter transformation. Then γi,k = ck (|Ii |) = |(τ + )k Ii | and γ−i,−k = c−k (|Pi |) = |(τ − )k Pi | for 1 ≤ i ≤ r and k ≥ 0. By Proposition 4.7, for each γ ∈ Γ± QUANTUM CHEVALLEY GROUPS 23 f )= there exists a unique, up to an isomorphism, indecomposable Eγ with |Eγ | = γ. Thus, P(A {[Eγ ] : γ ∈ Γ− } and If(A ) = {[Eγ ] : γ ∈ Γ+ }. The remaining assertions of the Lemma follow the fact that the category A has finitely many isomorphism classes of indecomposables if and only if f ) ∩ If(A ) 6= ∅ and this happens if and only if P(A f ) = If(A ) (cf. [1, Propositions VIII.1.13– P(A 14]). f ) Moreover, the order on Γ± defined in Theorem 1.15 coincides with the normal order on P(A and If(A ). It only remains to apply Proposition 4.5.  We have dimk HomA (Pi , (τ − )k Pj ) = di [(τ − )k (Pj ) : Si ] = (|Pi |, c−k (|Pj |)) and dimk Ext1A ((τ − )k Pj , Pi ) = dimk HomA (Pi , (τ − )k−1 Pj ) = di [(τ − )k−1 Pj : Si ] = (|Pi |, c−k+1 (|Pj |)). Therefore, we have dimk HomA (Eγ−i,−k , Eγ−j,−r ) = di [(τ − )r−k (Pj ) : Si ] = (γ−i , ck−r (γ−j )), r≥k (4.3) r > k. (4.4) and, by (4.1) dimk Ext1A (Eγ−j,−r , Eγ−i,−k ) = di [(τ − )r−k−1 (Pj ) : Si ](γ−i , ck+1−r (γ−j )), This, together with (3.11), (3.12), can be used to write explicit commutator relations between indecomposable preprojective objects. Namely, (ad∗(γ−i ,ck−r (γ−j )) q (1,q di ,q 2di ,...,q (γ−i ,c k+1−r (γ −j )) ) [Eγ−i,−k ])([Eγ−j,−r ]) = 0, and (ad 4.5. q (γ−i ,c k+1−r (γ k−r (γ −j )) ) −j )) (1,q dj ,q 2dj ,...,q (γ−i ,c [Eγ−j,−r ])([Eγ−i,−k ]) = 0. We finish this section with a proof of Corollary 1.16. Lemma 4.8. Suppose that A is not of finite type. Then K0+ (A− ) ∩ K0+ (A≥0 ) = 0 = K0+ (A≤0 ) ∩ K0+ (A+ ). Proof. Take γ ∈ K0+ (A− ) ∩ K0+ (A≥0 ). Since γ ∈ K0+ (A− ), by Proposition 4.7(c) there exists r > 0 such that cr (γ) ∈ −K0+ (A ). On the other hand, again by Proposition 4.7, since γ ∈ K0+ (A≥0 ), we have cs (γ) ∈ K0+ (A ) for all s ≥ 0 hence cr (γ) ∈ K0+ (A ) ∩ (−K0+ (A )) = 0. Since c is an automorphism, γ = 0. The second statement is proved similarly.  Now we have all ingredients to prove Corollary 1.16. Recall that we denote by CA the composition algebra of A and by EA the subalgebra of HA generated by the homogeneous components of ExpA . Note that in this case CA = EA by (4.2). Proposition 4.9. (a) HA± ⊂ CA ; (b) ExpA0 |γ ∈ CA for all γ ∈ K0 (A )0 . 24 A. BERENSTEIN AND J. GREENSTEIN Proof. If A has finitely many indecomposables, A = A± and so EA± = EA = CA . Otherwise, (A− , A≥0 ) and (A≤0 , A+ ) are factorizing pairs for A by Proposition 4.5. Then by Lemmata 4.8 and 2.9, EA± ⊂ EA = CA . Proposition 4.1(f,g) and Corollary 2.4(b) imply that A+ and A− satisfy the assumptions of Lemma 2.10, hence HA± = EA± and thus are subalgebras of CA . This proves (a). To prove (b), it remains to observe that by (4.2) ExpA0 = ExpA− −1 ExpA ExpA+ −1 .  5. Examples 5.1. We will now discuss the case of a hereditary acyclic category with only two non-isomorphic (a1 ,a0 ) simples in more detail. Thus, EA is the valued graph 1 −−−−→ 0, a0 a1 > 0. Then S1 = I1 , S0 = P0 , |P1 | = α10 = α1 + a0 α0 and |I0 | = α01 = a1 α1 + α0 . We have f ) = {[P2n+i ]}0≤n≤δ− (P ),i∈I , P(A i If(A ) = {[I2n+i ]}0≤n≤δ+ (Ii ),i∈I where P2n+i = (τ − )n (Pi ) and I2n+i = (τ + )n Ii . In this case, it is possible to describe the images of preinjectives and preprojectives in K0 (A ) very explicitly via a rather simple recursion. Consider the Auslander-Reiten quiver of the preprojective component ([1, Propositions VIII.1.15–16]) [P1 ] o❴ ❴ ❴ ❴ ❴ [P3 ] o❴ ❴ ❴ ❴ ❴ ·B · · ❀❀ ❀❀ A A ✆ ✄ (a1 ,a0 ) ✄ ❀❀(a0 ,a1 ) ✆✆✆ ❀❀(a0 ,a1 ) ✄✄✄ ✄ ❀ ❀ ✄ ✄ ✆ ❀ ✆✆ ✄✄ ✄✄ (a1 ,a0 ) ❀ ❴ o ❴ ❴ ❴ ❴ ❴ o ❴ ❴ ❴ ❴ [P0 ] [P2 ] [P4 ] o❴ ❴ ❴ ❴ · · · where the dashed arrows denote the Auslander-Reiten translation. Thus, for all k ≥ 0, we have short exact sequences ⊕a1 0 → P2k → P2k+1 → P2k+2 → 0, ⊕a0 0 → P2k+1 → P2k+2 → P2k+3 → 0, whence |Pr+1 | + |Pr−1 | = ar |Pr |, r ≥ 1, where ar = ar (mod 2) . Similar considerations yield a recursion |Ir+1 | + |Ir−1 | = ar |Ir |, r ≥ 1. Combining these, we obtain Lemma 5.1. Let β0 = α0 , β−1 = −α1 and define βn , n ∈ Z \ {−1, 0} by βn+1 + βn−1 = an βn . Then β2r  |P |, 2r = −|I−2r+2 |, 0 ≤ r < δ− (P0 ) −δ+ (I0 ) ≤ r < 0 , β2r+1 =  |P 2r+1 |, −|I−2r−1 |, 0 ≤ r < δ− (P1 ) −δ+ (I1 ) ≤ r < 0. f ) with {βr : r ≥ 0} ⊂ K0 (A ) and If(A ) with {−βr : r < 0} ⊂ K0 (A ). Thus, we can identify P(A It is not hard to write an explicit formula for βr , r ∈ Z. Let Un be the Chebyshev polynomial of the second kind Un (t) = 2tUn−1 − Un−2 , n ≥ 1, U−1 = 0, U0 = 1 and set λn (t) = Un−1 (t/2 − 1), µn (t) = Un (t/2 − 1) + Un−1 (t/2 − 1). Then βr = ar+1 λ⌊(r+1)/2⌋ (a0 a1 )αr+1 + µ⌊r/2⌋ (a0 a1 )αr , r ∈ Z, QUANTUM CHEVALLEY GROUPS 25 where αk = αk (mod 2) . If r = s (mod 2), r ≤ s we have (in this case End Pr ∼ = End Ps ) dimEnd Pr HomA (Pr , Ps ) = µ(s−r)/2 (a0 a1 ) dimEnd Pr Ext1A (Ps , Pr ) = µ(s−r)/2−1 (a0 a1 ). while for r < s with r = s + 1 (mod 2) dimEnd Pr HomA (Pr , Ps ) = ar λ(s+1−r)/2 (a0 a1 ), dimEnd Ps HomA (Pr , Ps ) = as λ(s+1−r)/2 (a0 a1 ) dimEnd Pr Ext1A (Ps , Pr ) = ar λ(s−r−1)/2 (a0 a1 ), dimEnd Ps HomA (Pr , Ps ) = as λ(s−r−1)/2 (a0 a1 ). where we used (4.3), (4.4). This, together with (3.11) and (3.12), allows us to write the commutation relations among all preprojective objects. Namely, if r ≤ s and r = s (mod 2)     x+1 x+1 j j X X j x+1−j j (2)−jy x + 1 j (2)+jy x + 1 (−1) qr [Ps ]x+1−j [Pr ][Ps ]j (−1) qr [Pr ] [Ps ][Pr ] =0= j j qr qr j=0 j=0 (5.1) where x = µ 1 (s−r)−1 (a0 a1 ) and y = µ(s−r)/2 (a0 a1 ). If r < s and s = r + 1 (mod 2), 2   (2j )+jar w ar z + 1 (−1) qr [Pr ]j [Ps ][Pr ]ar z+1−j = 0 j qr aX r z+1 j=0 j =   (2j )−jas w as z + 1 (−1) qs [Ps ]as z+1−j [Pr ][Ps ]j , (5.2) j qs aX s z+1 j=0 j where z = λ(s−r−1)/2 (a0 a1 ) and w = λ(s−r+1)/2 (a0 a1 ). f ), If(A ) are finite (and hence Ind A is finite) if and only if a0 a1 < 4. The Both sets P(A f ) (respectively, If(A )) is given by βr ≺ βs (respectively, β−s ≺ β−r ) if r < s. normal order on P(A If a0 a1 ≥ 4, the identity (4.2) can be written as follows → ←  Y  Y Expqr (Eβr ) ExpA0 Expqr (E−β−r ) = [Expq0 (Eα0 ), Expq1 (Eα1 )] r>0 r>1 a−1 bab−1 . where [a, b] = If a0 a1 < 4, we have A+ = A− = A and ExpA = → Y Expqr (Eβr ) = r≥0 ← Y Expqr (E−β−r ) r>0 (both products are finite) which yields → Y Expqr (Eβr ) = [Expq0 (Eα0 ), Expq1 (Eα1 )]. r>0 More explicitly, we have [expq0 (Eα0 ), expq1 (Eα1 )] =    expq1 (Eα10 ),    a0 = a1 = 1 expq1 (Eα10 ) expq0 (Eα01 ), a0 a1 = 2 expq1 (Eα10 ) expq0 (Eα01 +α0 ) expq1 (Eα1 +α10 ) expq0 (Eα01 ), a0 a1 = 3. 26 A. BERENSTEIN AND J. GREENSTEIN It should be noted that these Chevalley-type relations hold in higher ranks. 5.2. Consider the category A of k-representations of the quiver 1 → 2 → · · · → n. If αi = |Si |, then isomorphism classes of indecomposable objects are uniquely determined by their P images in K0 (A ) which are given by αi,j = jk=i αk . Denote the corresponding indecomposable by Ei,j ; in particular, Ei,i = Si . Then dimk HomA (Ej,k , Si ) = δi,j , dimk HomA (Si , Ej,k ) = δi,k and dimk Ext1A (Si , Ej,k ) = δi,j−1 , dimk Ext1A (Ej,k , Si ) = δi,k+1 . Fix 1 ≤ i < n and let A+ (respectively, A− ) be the full subcategory of A whose objects satisfy [M : Sj ] = 0 if j ≤ i (respectively, j > i). It is immediate from Corollary 2.8 that (A+ , A− ) is a factorizing pair. We claim that it in fact a pentagonal pair. Indeed, clearly A− ⊂ A+ . Moreover, W if M is an indecomposable object that is not in A− A+ , then M ∼ = Ej,k with either j ≤ i or k > i. W W In particular, every such object is in A0 = A− ∩ A+ and so A = A− A0 A+ . It is now easy to see, using Proposition 2.5, that (A− , A0 , A+ ) is a factorizing triple and hence (A+ , A− ) is a pentagonal pair. Also, let E = Eαij . Then AE0 = AddA {[αk,j ] : k < j}, AE< = AddA {[αrs ] : s > j} and finally AE> = AddA {[αrs ] : s < j}. 5.3. We now discuss a non-hereditary example. Consider the following quiver 1 a12 3 2 ❃❃ ❃❃ a24 ❃ ❃❃ a13 ❃❃ ❃ a34 4 with relation a34 a13 = a24 a12 . Let A be the category of finite dimensional representations of that quiver over k with |k| = q. Denote αi = |Si |, 1 ≤ i ≤ 4. The isomorphism classes of indecomposables in A are uniquely determined by their images in K0 (A ). The Auslander-Reiten quiver of A is (cf. [1, §VII.2]) α1 +α2 +α3 +α4 ✁@ ✁✁ ✁ ✁ α2 +α4 o❴ ❴ ❴ ❴ ❴ ✁❴✁✁❴ ❴8 α3 ●● B✆ ✁ ♣♣♣ ●● ✆ ✁♣✁♣♣♣♣ ● ✆ ✁ ● # ✁♣ ✆✆ α4 o❴ ❴ ❴ ❴ α2 +α3 +α4 o❴ ❴ ❴ ❴ ✾✾ ; ◆◆◆ ✇✇ ✾✾ ◆◆◆ ✾ ✇✇✇✇ ◆◆ o❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴' α3 +α4 α2 ❂❂ ❂❂ ❂❂ ❂ o❴ ◆◆◆❴ ❴❂❂❴ ❴ ❴ ❴ ❴ α; 1 +α✾2 ◆◆◆ ❂❂ ✾✾ ✇✇ ◆◆◆❂❂ ✾✾ ✇✇ '  ✇✇ ❴ ❴ ❴ α1 +α2 +α3 o❴ ❴ ❴ ❴ α 1 ●● ♣8 ✆B ●● ♣♣♣ ✆✆ ● ♣ ✆ ♣ ● ♣ # ✆ ♣ o❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ α1 +α3 where dashed arrows denote the Auslander-Reiten translation. Then we have expq (E4 ) expq (E24 ) expq (E34 ) expq (E234 ) expq (E2 ) expq (E3 ) expq (E1234 ) expq (E123 )× expq (E12 ) expq (E13 ) expq (E1 ) = expq (E1 ) expq (E2 ) expq (E3 ) expq (E4 ) QUANTUM CHEVALLEY GROUPS 27 where Ei1 i2 ... denotes [αi1 + αi2 + · · · ]. In particular, if we set A+ = AddA S1 and let A− be the largest full subcategory of A whose objects satisfy [M : S1 ] = 0 then (A+ , A− ) is a pentagonal pair by Proposition 2.5 (note that A− is equivalent to the category of k-representations of the quiver 2 → 4 ← 3). Consider now the same quiver a12 1 2 ❃❃ ❃❃ a24 ❃ ❃❃ a13 ❃❃ ❃ 3 a34 4 but this time with the relation a24 a12 = 0. The Auslander-Reiten quiver in this case is o❴ ❴ ❴ ❴ ❴ ❴ ❴ α1 +α2 α2 +α4 o❴ ❴ ❴ ❴ ❴ ❴ ❴ ♠6 α3 ◗◗◗◗ ❀❀ ❑❑❑ s9 ◗◗◗ ♠♠♠ ✄A ♠ ❑ ❀ ✄ ♠ sss ( % ♠ ✄ o❴ ❴ ❴ ❴ o ❴ ❴ ❴ o ❴ ❴ ❴ ❴ ❴ α1 α2 +α3 +α4 α4 α1 +α2 +α3 ❀❀ 9 ◗◗◗ ❏❏❏ ✄A ♠♠6 ◗◗( ♠ ❀ tttt ✄ ❏ ♠ ♠ % ✄ α3 +α4 o❴ ❴ ❴ ❴ α1 +2α2 +α3 +α4 o❴ ❴ ❴ ❴ α1 +α3 ❏❏ ◗ 9 6 ◗◗◗ ♠ ❏❏ tt ◗ ♠♠♠ % tt ♠♠♠❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ◗❴◗( ❴ o P ♠♠6 I4 ❏❏❏❏ t9 1 ◗◗◗◗◗ ♠♠♠ ◗◗( tt ❏❏ ♠ ♠ t ♠ to❴ ❴ ❴ ❴ ❴ ❴ o❴ ❴ ❴ ❴ ❴ ❴ % α2 α1 +α3 +α4 α2 where the two copies of S2 in the lowest row are to be identified and P1 (respectively, I4 ) is the projective cover (respectively, the injective envelope) of S1 (respectively, S4 ), are non-isomorphic and |P1 | = |I4 | = α1 + α2 + α3 + α4 . In this case, no partition of the set of isomorphism classes of simples gives a pentagonal pair, since the indecomposable X with |X| = α1 + α3 + α4 satisfies Ext1A (X, S2 ) 6= 0 and Ext1A (S2 , X) 6= 0, while it involves all other simples as its composition factors. 5.4. Suppose that we have have an autoequivalence on A which induces a permutation σ of the set of isomorphism classes of simples in A and a natural action of σ on K0 (A ). Thus, on the level of EA , σ must induce an automorphism of valued graphs. Then we can consider the full subcategory A σ of A whose objects M satisfy |M | ∈ K0 (A )σ . This category is clearly closed under extensions. L The simple objects in A σ are of the form Si = i∈i Si where i ∈ {1, . . . , r}/σ. There is a natural source order on the set of σ-orbits and we get ExpA σ = ExpA σ (Si1 ) · · · ExpA σ (Sir ). 5.5. Let R be a principal ideal domain such that R/m is a finite field for every maximal ideal m. L Consider the category A of R-modules of finite length. It is well-known that A = m∈Spec R A (m) where A (m) is the full subcategory of A whose objects are finite length R-modules M with Ann M = mj for some j ≥ 0. Each of the categories A (m) is hereditary, and HomA (M, N ) = Ext1A (M, N ) = 0 if M ∈ Ob A (m), N ∈ Ob A (m′ ) with m 6= m′ . Thus, Y ExpA = ExpA (m) . m∈Spec R Fix m ∈ Spec R. Then for each r > 0, there exists a unique indecomposable Ir := Ir (m) = R/mr of length r, and for every partition λ = (λ1 ≥ λ2 ≥ · · · ) there is a unique object Iλ = Iλ (m) = Iλ1 ⊕ Iλ2 ⊕ · · · . 28 A. BERENSTEIN AND J. GREENSTEIN The Hall algebra of A (m) is in fact very well understood (cf., for example, [6, 10]). It is commutative, is isomorphic to the Hall algebra of the category of nilpotent finite length modules over k[x] where k ∼ = R/m and is the classical Hall-Steiniz algebra. It is freely generated by [I(1r ) ] = [I1⊕r ]. Then X r (−1)r q (2) [I(1r ) ], ExpA (m) −1 = r≥0 where q = |R/m|. There is a well-known homomorphism Φ : HA (m) → Sym, where Sym is the r ring of symmetric polynomials in infinitely many variables x1 , x2 , . . . , given by Φ([I(1r ) ]) 7→ q −(2) er P d be the where er = 1≤i1 <i2 <···<ir xi1 · · · xir is the rth elementary symmetric function. Let Sym b A (m) → Sym. d Then natural completion of Sym and extend Φ to an isomorphism H X Y Φ(ExpA (m) −1 ) = (−1)r er = (1 − xi ). r≥0 This implies that Φ(ExpA (m) ) = i≥1 Y i≥1 X 1 = hr , 1 − xi r≥0 where hr is the rth complete symmetric function. Let r X  Y σ xλ1 1 · · · xλr r [mi ]q !−1 Pλ (x1 , . . . , xr ; q) = i=0 σ∈Sr Y 1≤i<j≤r xi − qxj  xi − xj where λ = (λ1 ≥ · · · ≥ λr ≥ 0) and mi = #{1 ≤ j ≤ r : λj = i}. Since Φ([Iλ ]) = q −n(λ) Pλ (x; q −1 ), P where n(λ) = i (i − 1)λi , we obtain X X q n(λ) Pλ (x; q) (−1)r er = 1. r≥0 λ This identity specializes to a well-known identity in the ring of symmetric polynomials in finitely many variables (in which the second sum becomes finite). In particular (cf. [6]), X q n(λ) Pλ (x; q) = hr . λ⊢r It is not hard to see that | Aut(Iλ )| = q P i,j (min(i,j)−δij )ai aj Y | GL(ai , k)| i where λ = (1a1 2a2 · · · ). Let aλ (q) = | Aut(Iλ )|. Then the homomorphism given by [Iλ ] 7→ aλ (q)−1 t|λ| and yields X r≥0 or tr X λ⊢r r Y s X s=0 aλ (q)−1  X s (−1)s q (2) (1 − q j )−1 j=1 s≥0  X λ⊢r−s  ts =1 | GL(s, q)| aλ (q)−1 = 0, r > 0. R b A (m) → Q[[t]] is :H QUANTUM CHEVALLEY GROUPS 29 5.6. Retain the notation of 5.1 and assume that A is of tame representation type. In that case, A0 is abelian and is a direct sum of countably many abelian subcategories Aρ , known as stable tubes, since for any ρ there exists rρ ≥ 0 such that for any indecomposable M ∈ Ob Aρ , τ rρ M ∼ = M . The minimal rρ with that property is called the rank of Aρ , and it is well-known that rρ = 1 for all but finitely many ρ. Thus, O HAρ HA0 ∼ = ρ as an algebra and moreover ExpA0 = Y ExpAρ . ρ There exists an indecomposable Sρ ∈ Ob Aρ of minimal length which is a simple object in Aρ . In particular, it is a brick. Then all non-isomorphic simple objects in Aρ are of the form (τ − )r Sρ , 0 ≤ r < rρ . Since τ is an exact autoequivalence Aρ → Aρ , it induces an automorphism of HAρ of order rρ defined by [M ] 7→ [τ M ]. Thus, we have P a P X ( i) ExpAρ −1 = (−1) i ai qρ i 2 [Sρ (a)] r ρ a=(a1 ,...,arρ )∈Z≥0 Lr ρ where qρ = | EndA Sρ | and Sρ (a) = i=1 (τ i Sρ )⊕ai . In particular, if rρ = 1, Aρ is equivalent to the category of nilpotent representations of k[x] as described in 5.5. Appendix A. Coproduct b A by: Let A be a finitary category and define a linear map ∆ : HA → HA ⊗H X |HomA (A, B)| A,B F · [A] ⊗ [B] , ∆([C]) = | Ext1A (A, B)| C (A.1) [A],[B]∈Iso A where FCA,B is the dual Hall number given by FCA,B = | AutA (A)|| AutA (B)| C | Ext1A (A, B)C | FA,B = | AutA (C)| | HomA (A, B)| where we used (2.1). Lemma A.1. We have ∆(ExpA ) = ExpA ⊗ ExpA . Proof. Since for all A, B ∈ Ob A X |HomA (A, B)| [A],[B] = F | Ext1A (A, B)| [C] [C]∈Iso A X [C]∈Iso A | Ext1A (A, B)C | =1 |Ext1A (A, B)| we obtain ∆(ExpA ) = X [C]∈Iso A = X [A],[B]∈Iso A = X [A],[B]∈Iso A X |HomA (A, B)| [A],[B] [A] ⊗ [B] F | Ext1A (A, B)| [C] [A],[B],[C]∈Iso A   X |HomA (A, B)| [A],[B]  [A] ⊗ [B]  F | Ext1A (A, B)| [C] ∆([C]) = [C]∈Iso A [A] ⊗ [B] = ExpA ⊗ ExpA .  30 A. BERENSTEIN AND J. GREENSTEIN In general ∆ is not coassociative. However, it is easy to see that ∆ becomes coassociative if A is hereditary. Moreover, in that case it is a homomorphism of braided algebras. In order to make this statement precise, we need to introduce some terminology from the theory of braided categories. Let C be an F-linear braided tensor category with the braiding operator ΨU,V : U ⊗ V → V ⊗ U for all objects U, V of C . If A, B are associative algebras in C , then A ⊗ B acquires a natural structure of an associative algebra in C via mA⊗B := (mA ⊗ mB )(1A ⊗ ΨB,A ⊗ 1B ), where mA , mB are the respective multiplication morphisms. Our main example of a category C is as follows. Let Γ be an additive monoid and let χ : Γ × Γ → L × V (γ) such k be a bicharacter. Let Cχ be the tensor category of Γ-graded vector spaces V = γ∈Γ that each component V (γ) is finite-dimensional. The following Lemma is immediate. Lemma A.2. The category Cχ is a braided tensor category with the braiding ΨU,V : U ⊗ V → V ⊗ U for each objects U and V given by ΨU,V (uγ ⊗ vδ ) = χ(γ, δ) · vδ ⊗ uγ for any uγ ∈ U (γ), vδ ∈ V (δ). In particular, if A, B are associative algebras in Cχ , the multiplication in A ⊗ B is given by (a ⊗ b)(a′ ⊗ b′ ) = χ(β, α′ )(aa′ ⊗ bb′ ), where a′ ∈ A(α′ ), b ∈ B(β). Now let A be a hereditary finitary category. Let Γ = K0 (A ) be the Grothendieck group of A and let χ = χA : Γ × Γ → Q be the bicharacter given by: χA (|M |, |N |) = | HomA (N, M )| , |Ext1A (N, M )| M, N ∈ Ob A . (the bicharacter χA is well-defined because it is essentially the Euler form on K0 (A )). Thus, the coproduct ∆ can be written as X ∆([C]) = χ(|B|, |A|)FCA,B [A] ⊗ [B]. [A],[B] The following is a reformulation of Green’s theorem ([3]) for Hall algebras. Theorem A.3. Let A be a finitary hereditary category. Then HA is a bialgebra in CχA with the coproduct ∆ given by (A.1). Proof. We have ∆([C])∆([C ′ ])   X  X ′ ′ = χ(|B|, |A|)FCA,B · [A] ⊗ [B] χ(|B ′ |, |A′ |)FCA′ ,B · [A′ ] ⊗ [B ′ ] [A′ ],[B ′ ] [A],[B] = X ′ ′ χ(|B|, |A|)χ(|B ′ |, |A′ |)χ(|B|, |A′ |)FCA,B FCA′ ,B · [A][A′ ] ⊗ [B][B ′ ] [A],[B],[A′],[B ′ ] = X [A],[B],[A′] [B ′ ],[A′′ ],[B ′′ ] ′ ′ ′′ ′′ A B ′′ ′′ χ(|B|, |A|)χ(|B ′ |, |A′ |)χ(|B|, |A′ |)FCA,B FCA′ ,B FA,A ′ FB,B ′ · [A ] ⊗ [B ] QUANTUM CHEVALLEY GROUPS On the other hand, X C ′′ ′′ ∆([C][C ′ ]) = FC,C ′ ∆([C ]) = [C ′′ ] X 31 ′′ ′′ ′′ A ,B C · [A′′ ] ⊗ [B ′′ ] χ(|B ′′ |, |A′′ |)FC,C ′ F ′′ C [C ′′ ],[A′′ ],[B ′′ ] We need to compare the coefficients of [A′′ ] ⊗ [B ′′ ] in these expressions. Since |A′′ | = |A| + |A′ | and |B ′′ | = |B| + |B ′ |, χ(|B|, |A|)χ(|B ′ |, |A′ |)χ(|B|, |A′ |)χ(|B ′′ |, |A′′ |)−1 = χ(|B ′ |, |A|)−1 By [3, Theorem 2] X ′′ ′′ ′ ′ A ,B A B FCA,B = χ(|B ′ |, |A|)−1 · FA,A ′ FB,B ′ F ′ C X ′′ ′′ A ,B C FC,C ′ F ′′ C ′′ (A.2) [C ′′ ]∈Iso A [A],[B],[A′],[B ′ ]∈Iso A for any objects A′′ , B ′′ , C, C ′ in A , and it is now immediate that ∆([C])∆([C ′ ]) = ∆([C][C ′ ]) that is, ∆ is a homomorphism of algebras.  References [1] M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. [2] X. Chen and J. Xiao, Exceptional sequences in Hall algebras and quantum groups, Compositio Math. 117 (1999), no. 2, 161–187. [3] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), no. 2, 361–377. [4] J. Fei, Counting Using Hall Algebras I. Quivers, Preprint, arXiv:1111.6452 (2011). [5] B. Keller, On cluster theory and quantum dilogarithm identities, Preprint, arXiv:1102.4148 (2011). [6] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press Oxford University Press, New York, 1979. [7] M. Reineke, Counting rational points of quiver moduli, Int. Math. Res. Not. 2006 (2006), 1–19. [8] C. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), no. 2, 526–546. [9] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. [10] O. Schiffmann, Lectures on Hall algebras, Preprint, arXiv:math.RT/0611617 (2006). [11] A. Sevostyanov, Regular nilpotent elements and quantum groups, Comm. Math. Phys. 204 (1999), no. 1, 1–16. Department of Mathematics, University of Oregon, Eugene, OR 97403, USA E-mail address: [email protected] Department of Mathematics, University of California, Riverside, CA 92521. E-mail address: [email protected]