An Integral Basis for the Universal Enveloping Algebra of the Onsager Algebra
aa r X i v : . [ m a t h . R A ] J un AN INTEGRAL BASIS FOR THE UNIVERSAL ENVELOPING ALGEBRAOF THE ONSAGER ALGEBRA
ANGELO BIANCHI AND SAMUEL CHAMBERLIN
Abstract.
We construct an integral form for the universal enveloping algebra of the
Onsageralgebra and an explicit integral basis for this integral form. We also formulate straighteningidentities among some products of basis elements.
MSC2020: Primary 17B65; Secondary 17B05
Contents
Introduction 11. Preliminaries 21.1. The sl -algebra, its loop algebra, and a Chevalley involution 21.2. Universal enveloping algebras 31.3. The Onsager Algebra and one of its historical realizations 31.4. The Onsager Algebra as an equivariant map algebra. 32. A better realization of the Onsager Algebra O U ( O ). 42.2. Identities for D Γ , ± u,v ( j, l ). 52.3. Straightening Identities 63. Proof of the main result 73.1. Proof of Theorem 2.2 73.2. Proof of Proposition 2.5 (8) and (9). 93.3. Proof of Proposition 2.5 (7) 11References 22 Introduction
The focus of this paper is the Onsager algebra, which took on the name of the Nobel Prize win-ning Lars Onsager (1903-1976). This algebra appeared for the first time in Onsager’s solution ofthe two-dimensional Ising model in a zero magnetic field, [19]. In this work, two non-commutingmatrices appeared related to the transfer matrix associated to this model. By analyzing thestructure of the algebra generated by these matrices, Onsager derived a complex Lie algebra.There are a few different realizations of the Onsager algebra. We refer the reader to [8] for asurvey with details. The realization used here is as an equivariant map algebra. Equivariantmap algebras have the form g A := ( g ⊗ C A ) Γ , where g is a finite-dimensional complex simpleLie algebra, A is a commutative, associative, unital C -algebra, and Γ is a group acting on g and A (and hence diagonally on g A ).We highlight the work of Roan [20] who first investigated the relationship between the On-sager algebra and the sl -loop algebra, folowed by Benkart and Terwilliger [2] and Hartwig andTerwilliger [12]. Since then, a few authors have written papers on this topic, see for instance[9, 10].Suitable integral forms and bases for the universal enveloping algebras of the complex simplefinite-dimensional Lie algebras were formulated by Kostant in 1966 (cf. [15]), after Chevalleyinvestigated integral forms for the classical Lie algebras in 1955. This led to the constructionof the classical Chevalley groups. In 1978, Garland formulated useful integral forms and basesfor the universal enveloping algebras of the (untwisted) loop algebras, [11]. Mitzman extended Garland’s integral bases to the universal enveloping algebras of the twisted loop algebras in1983, [16]. The second author formulated suitable integral forms and bases for the universalenveloping algebras of the (untwisted) map algebras and (untwisted) map superalgebras, [1, 6].Once one has access to suitable integral forms and bases for a particular Lie algebra one canstudy its representation theory in positive characteristic via its hyperalgebra. This was done forthe (untwisted) hyper-loop algebras by Jakeli´c and Moura using Garland’s integral bases, [14].The first author and Moura extended this to the twisted hyper-loop algebras using Mitzman’sintegral bases, [5]. The authors extended these results to the (untwisted) hyper-multiloop,hyper-multicurrent, and hyper-map algebras this year using the integral bases formulated bythe first author, [3, 4].When we view the Onsager algebra as an equivariant map algebra, the defining automorphism(the standard Chevalley involution) is not an isomorphism of the Dynkin diagram. No integralforms for any equivariant map algebras have been formulated in such a case in the literature.Section 1 is dedicated to the algebraic preliminaries, including a review of the Onsager algebra.In Section 2, we construct an integral form and an integral basis for the Onsager algebra, themain result of the paper is stated as Theorem 2.2. Section 3 contains all relevant proofs for themain result of the paper, which includes the necessary straightening identities.
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Preliminaries
Throughout this work we denote by C , Z , Z + and N the sets of complex numbers, integers,nonnegative integers and positive integers, respectively.If A is an C -algebra, an integral form A Z of A is a Z -algebra such that A Z ⊗ Z C = A . An integral basis for A is a Z -basis for A Z .1.1. The sl -algebra, its loop algebra, and a Chevalley involution. Let sl be the setof complex traceless order two matrices with Lie C -algebra structure with bracket given by thecommutator. Recall the standard basis of sl , (cid:26) x + = (cid:18) (cid:19) , h = (cid:18) − (cid:19) , x − = (cid:18) (cid:19)(cid:27) , satisfying [ x + , x − ] = h , [ h, x + ] = 2 x + , and [ h, x − ] = − x − .In what follows an unadorned tensor means a tensor product over the field C . Let C [ t, t − ]be the set of complex Laurent polynomials in one variable. Then the (untwisted) loop algebraof sl is the Lie algebra f sl := sl ⊗ C [ t, t − ] with Lie bracket given by bilinearly extending thebracket [ z ⊗ f, z ′ ⊗ g ] = [ z, z ′ ] ⊗ f g, where z, z ′ ∈ sl and f, g ∈ C [ t, t − ].The Chevalley involution on sl is defined by the map M M := (cid:18) (cid:19) M (cid:18) (cid:19) for all M ∈ sl . Therefore, under this involution we have x + x − , x − x + , h
7→ − h. Notice that the Chevalley involution is an involution defined as an automorphism of order two.The Chevalley involution on sl induces an involution on f sl , also called the Chevalley invo-lution on f sl , and denoted by ω where ω ( g ⊗ f ( t )) := g ⊗ f ( t − ) , for f ( t ) ∈ C [ t, t − ] , g ∈ sl . N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 3
Universal enveloping algebras.
For a Lie algebra a , we denote by U ( a ) the correspond-ing universal enveloping algebra of the Lie algebra a . Given u ∈ U ( a ) and k ∈ Z define thedivided powers and binomials of u , respectively, as follows: u ( k ) = (cid:0) uk (cid:1) = 0, if k <
0, and u ( k ) := u k k ! and (cid:18) uk (cid:19) := u ( u − . . . ( u − k + 1) k ! , if k ≥ T ( a ) := C , and for all j ≥
1, define T j ( a ) := a ⊗ j , T ( a ) := L ∞ j =0 T j ( a ), and T j ( a ) := L jk =0 T k ( a ). Then, set U j ( a ) to be the image of T j ( a ) under the canonical surjection T ( a ) → U ( a ), and for any u ∈ U ( a ) define the degree of u bydeg u := min j { u ∈ U j ( a ) } . The Onsager Algebra and one of its historical realizations.
We begin by with theoriginal definition of the Onsager algebra and its relation to f sl . We use the notation in [8]instead of the original notation in [19].The Onsager algebra, denoted O , is the (nonassociative) C -algebra with basis { A m , G l | m ∈ Z , l ∈ N + } and antisymmetric product given by:[ A l , A m ] = 2 G l − m , for l > m, [ G l , A m ] = A m + l − A m − l , [ G l , G m ] = 0 . By considering the C -linear map γ : O → f sl defined by A m x + ⊗ t m + x − ⊗ t − m G l (cid:16) h ⊗ ( t l − t − l ) (cid:17) (1)for m ∈ Z and l ∈ N , one can see that the Onsager algebra is isomorphic to the Lie subalgebraof f sl fixed by the Chevalley involution ω introduced in the Subsection 1.1 (see details in [20]).1.4. The Onsager Algebra as an equivariant map algebra.
Following the language ofequivariant map algebras as in [18], let Γ := h σ i ∼ = Z be a group of order two. We act on sl by the involution: σ · ( x ± ) = − x ∓ σ · ( h ) = − h, and on C [ t, t − ] by σ · ( t ) = t − . This induces the diagonal action of Γ on f sl given by σ · ( g ⊗ f ) = σ · ( g ) ⊗ σ · ( f ) , for all g ∈ sl and all f ∈ C [ t, t − ]. Therefore, the Onsager algebra O is also isomorphic tothe Lie subalgebra of f sl which is fixed by this action of Γ, according to [17, Remark 3.11 andLemma 3.3]. 2. A better realization of the Onsager Algebra O The realization of the Onsager algebra in (1) is not suitable to construct an integral formfor O due to the intrinsic difficulty to deal with brackets between elements in (1) and, hence,to produce useful identities and Z -subalgebras in U ( O ). The following realization of O allowsstraightening identities and the construction of suitable subalgebras, which then gives rise to a“triangular type decomposition” and an integral form for U ( O ). We start with the followingbasis for sl and then build the appropriate basis for O from this one. ANGELO BIANCHI AND SAMUEL CHAMBERLIN
We define the following elements in sl : h Γ := − i (cid:0) x + − x − (cid:1) , x Γ+ := 12 (cid:0) x + + x − − ih (cid:1) , and x Γ − := 12 (cid:0) x + + x − + ih (cid:1) . Then, span (cid:8) h Γ (cid:9) ∼ = C , h ∗ Γ ∼ = C , span { x Γ − , h Γ , x Γ+ } ∼ = sl , and σ (cid:0) x Γ ± (cid:1) = − x Γ ± .Given j, k, l ∈ Z + , define h Γ k := h Γ ⊗ (cid:16) t k + t − k (cid:17) , x Γ , + j := x Γ+ ⊗ (cid:0) t j − t − j (cid:1) , and x Γ , − l := x Γ − ⊗ (cid:16) t l − t − l (cid:17) , and note that h Γ − k = h Γ k , x Γ , ±− j = − x Γ , ± j , and x Γ , ± = 0.Therefore, by defining u j := (cid:0) x + − x − (cid:1) ⊗ (cid:0) t j + t − j (cid:1) , v k := (cid:0) x + + x − (cid:1) ⊗ (cid:16) t k − t − k (cid:17) , and w l := h ⊗ (cid:16) t l − t − l (cid:17) , for all j, k, l ∈ Z , then, using a similar argument to that in [8, Proposition 3.2.2], we see thatthe set { u j , v k , w l | j ∈ Z + , k, l ∈ N } is a C -basis for O and, hence, the relations u j = ih Γ j , v k = x Γ , + k + x Γ , − k , and w l = − i (cid:16) x Γ , − l − x Γ , + l (cid:17) demonstrate that the set B := n x Γ , − l , h Γ k , x Γ , + j | k ∈ Z + , j, l ∈ N o spans O . Furthermore, since B is clearly linearly independent it is a basis for O .We can define an ordering, ≤ , on B such that x Γ , − l ≤ h Γ k ≤ x Γ , + j whenever k ∈ Z + and j, l ∈ N , x Γ , ± l ≤ x Γ , ± j whenever l, j ∈ N with l ≤ j , and h Γ k ≤ h Γ m whenever k, m ∈ Z + with k ≤ m . Notethat h x Γ , + j , x Γ , − l i = h Γ j + l − h Γ j − l = h Γ j + l − h Γ | j − l | ; h h Γ k , x Γ , + j i = 2 (cid:16) x Γ , + j + k + x Γ , + j − k (cid:17) = 2 (cid:16) x Γ , + j + k − x Γ , + k − j (cid:17) ; h h Γ k , x Γ , − l i = − (cid:16) x Γ , − l + k − x Γ , − k − l (cid:17) = − (cid:16) x Γ , − l + k + x Γ , − l − k (cid:17) . An integral form and integral basis for U ( O ) . Define U Z ( O ) to be the Z -subalgebraof U ( O ) generated by (cid:26)(cid:16) x Γ , ± k (cid:17) ( s ) (cid:12)(cid:12)(cid:12)(cid:12) k ∈ N , s ∈ Z + (cid:27) ,U + Z ( O ) and U − Z ( O ) to be the Z -subalgebras of U Z ( O ) generated respectively by (cid:26)(cid:16) x Γ , + k (cid:17) ( s ) (cid:12)(cid:12)(cid:12)(cid:12) k ∈ N , s ∈ Z + (cid:27) and (cid:26)(cid:16) x Γ , − k (cid:17) ( s ) (cid:12)(cid:12)(cid:12)(cid:12) k ∈ N , s ∈ Z + (cid:27) , and U Z ( O ) = U (cid:16)(cid:0) g ⊗ C [ t, t − ] (cid:1) Γ (cid:17) T U Z ( O ).We need the following definitions in order to give our integral basis and to state our straight-ening identities.Given j, l ∈ N , define Λ Γ j,l, = − (cid:16) h Γ j + l − h Γ j − l (cid:17) and D Γ , ± u, ( j, l ) ∈ U ( O ), for u ∈ Z + , recursivelyas follows: D Γ , +0 , ( j, l ) = (cid:16) x Γ , + j (cid:17) ; D Γ , − , ( j, l ) = (cid:16) x Γ , − l (cid:17) ; D Γ , ± u, ( j, l ) = ± h D Γ , ± u − , ( j, l ) , Λ Γ j,l, i . Furthermore, for j, k, l ∈ N , set p Γ k ( j, l ) := h x Γ , + j , D Γ , − k − , ( j, l ) i and for j, l ∈ N , and k ∈ Z ,define Λ Γ j,l,k by equating coefficients in the following formal series:Λ Γ j,l ( u ) := ∞ X r =0 Λ Γ j,l,r u r = exp − ∞ X s =1 p Γ s ( j, l ) s u s ! . N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 5
In particular, Λ Γ j,l,k = 0 for k <
0, and Λ Γ j,l, = 1. Proposition 2.1.
Given j, k, l ∈ N ,Λ Γ j,l,k = − k k X i =1 p Γ i ( j, l )Λ Γ j,l,k − i . Proof.
We use an argument similar to that in [7, Lemma 3.2]. Differentiating both sides oflog Λ Γ j,l ( u ) = − ∞ X s =1 p Γ s ( j, l ) s u s with respect to u and then multiplying by u gives u (cid:16) Λ Γ j,l (cid:17) ′ ( u )Λ Γ j,l ( u ) = − ∞ X s =1 p Γ s ( j, l ) u s . Multiplying both sides of this equation by Λ Γ j,l ( u ) gives ∞ X k =1 k Λ Γ j,l,k u k = − ∞ X s =1 p Γ s ( j, l ) u s Λ Γ j,l ( u ) . Expanding Λ Γ j,l ( u ) and the product on the right side and equating coefficients gives the result. (cid:3) Given a PBW monomial with respect to the order on B , we construct an ordered monomialin the elements of the set M := (cid:26)(cid:16) x Γ , + j (cid:17) ( r ) , Λ Γ j,l,k , (cid:16) x Γ , − l (cid:17) ( s ) (cid:12)(cid:12)(cid:12)(cid:12) j, l ∈ N , k, r, s ∈ Z + (cid:27) via the correspondence (cid:16) x Γ , ± j (cid:17) k ↔ (cid:16) x Γ , ± j (cid:17) ( k ) and (cid:0) p Γ1 ( l, m ) (cid:1) r ↔ Λ Γ l,m,r . The main goal of this paper is to prove the following theorem, whose proof is in Section 3(see [1, 6, 11, 13, 16] for analogs in different settings).
Theorem 2.2.
The subalgebra U Z ( O ) is a free Z -module and the set of ordered monomialsconstructed from M is a Z basis of U Z ( O ).This theorem implies the following: C ⊗ Z U Z ( O ) ∼ = U ( O ) , C ⊗ Z U ± Z ( O ) ∼ = U (cid:16) span n x Γ , ± j | j ∈ N o(cid:17) , C ⊗ Z U Z ( O ) ∼ = U (cid:0) span (cid:8) h Γ k | k ∈ Z + (cid:9)(cid:1) . In particular, U Z ( O ) is an integral form of U ( O ).2.2. Identities for D Γ , ± u,v ( j, l ) . Define D Γ , ± u,v ( j, l ) ∈ U ( O ) recursively as follows: D Γ , ± u,v ( j, l ) = 0 if v < D Γ , ± u, ( j, l ) = δ u, ; D Γ , ± u,v ( j, l ) = 1 v u X i =0 D Γ , ± i, ( j, l ) D Γ , ± u − i,v − ( j, l ) . The following proposition gives the D Γ , ± u,v ( j, l ) as the coefficients of a power series in theindeterminate w . The proof of the first equation is done by induction on v and we omit thedetails. The second equation follows by the Multinomial Theorem. ANGELO BIANCHI AND SAMUEL CHAMBERLIN
Proposition 2.3.
For all j, l, u, v ∈ N D Γ , ± u,v ( j, l ) = X k ,...,k u ∈ Z + k + ...k u = vk +2 k + ··· + uk u = u (cid:16) D Γ , ± , ( j, l ) (cid:17) ( k ) . . . (cid:16) D Γ , ± u, ( j, l ) (cid:17) ( k u ) = X m ≥ D Γ , ± m, ( j, l ) w m +1 ( v ) u + v (cid:3) It can be easily shown by induction that D Γ , +0 ,v ( j, l ) = (cid:16) x Γ , + j (cid:17) ( v ) and D Γ , − ,v ( j, l ) = (cid:16) x Γ , − l (cid:17) ( v ) .A straightforward calculation shows that h(cid:16) x Γ , + k (cid:17) , Λ Γ j,l, i = 2 (cid:16) x Γ , + k + j + l (cid:17) + 2 (cid:16) x Γ , + k − j − l (cid:17) − (cid:16) x Γ , + k + j − l (cid:17) − (cid:16) x Γ , + k − j + l (cid:17) . (2)The following proposition gives the D Γ , ± u, ( j, l ) as Z -linear combinations of the (cid:16) x Γ , ± n (cid:17) . Proposition 2.4.
For all u ∈ Z + and j, l ∈ N D Γ , + u, ( j, l ) = ⌊ u − ⌋ X k =0 u +1 X i =0 ( − k + i (cid:18) uk (cid:19)(cid:18) u + 1 i (cid:19) (cid:16) x Γ , +( u +1 − i ) j +( u − k ) l (cid:17) + (( u + 1) mod 2) u X i =0 ( − u + i (cid:18) u u (cid:19)(cid:18) u + 1 i (cid:19) (cid:16) x Γ , +( u +1 − i ) j (cid:17) ; (3) D Γ , − u, ( j, l ) = ⌊ u − ⌋ X k =0 u +1 X i =0 ( − k + i (cid:18) uk (cid:19)(cid:18) u + 1 i (cid:19) (cid:16) x Γ , − ( u +1 − i ) l +( u − k ) j (cid:17) + (( u + 1) mod 2) u X i =0 ( − u + i (cid:18) u u (cid:19)(cid:18) u + 1 i (cid:19) (cid:16) x Γ , − ( u +1 − i ) l (cid:17) . (4)The proof of (3) has been omitted to shorten the text due to its straightforwardness. (4)follows from (3) by applying the automorphism τ : O → O given by τ (cid:16) x Γ , ± j (cid:17) = x Γ , ∓ j and τ (cid:0) h Γ k (cid:1) = − h Γ k and switching j and l .2.3. Straightening Identities.
In this section we state all necessary straightening identities.Given u ∈ Z + and j, k, m ∈ N define D Γ , ± u ( j, k, m ) = u X n =0 u X v =0 ( − n + v (cid:18) un (cid:19)(cid:18) uv (cid:19) (cid:16) x Γ , ± j +( u − n ) k +( u − v ) m (cid:17) . Proposition 2.5.
For all j, l, k, m ∈ N and n, r, s ∈ Z + Λ Γ j,l,r Λ Γ k,m,n = Λ Γ k,m,n Λ Γ j,l,r (5) (cid:16) x Γ , ± j (cid:17) ( r ) (cid:16) x Γ , ± j (cid:17) ( s ) = (cid:18) r + ss (cid:19) (cid:16) x Γ , ± j (cid:17) ( r + s ) (6) (cid:16) x Γ , + j (cid:17) ( r ) (cid:16) x Γ , − l (cid:17) ( s ) = X m,n,q ∈ Z + m + n + q ≤ min { r,s } ( − m + n + q D Γ , − m,s − m − n − q ( j, l )Λ Γ j,l,n D Γ , + q,r − m − n − q ( j, l ) (7) (cid:16) x Γ , + j (cid:17) ( r ) Λ Γ k,m,n = n X i =0 Λ Γ k,m,n − i Y v ...v u ∈ Z + P v u = r P uv u = i (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) (8) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 7 Λ Γ k,m,n (cid:16) x Γ , − l (cid:17) ( s ) = n X i =0 Y v ...v u ∈ Z + P v u = s P uv u = i (cid:0) ( u + 1) D Γ , − u ( l, k, m ) (cid:1) ( v u ) Λ Γ k,m,n − i (9)(5) and (6) are clear. Before proving (7)–(9) we will state and prove a necessary corollary andprove Theorem 2.2. Corollary 2.6.
For all j, l ∈ N and u, v ∈ Z + ( i ) u D Γ , ± u,v ( j, l ) ∈ U Z ( O );( ii ) u Λ Γ j,l,u ∈ U Z ( O ) . Proof.
We will prove both statements simultaneously by induction on u according to the scheme( ii ) u ⇒ ( i ) u ⇒ ( ii ) u +1 where ( i ) u is the statement that ( i ) m holds for all m ≤ u and similarly for ( ii ) u . ( i ) , ( ii ) and ( ii ) are all clearly true. Assume that ( ii ) u and ( i ) u − hold for some u ∈ Z + . Then, byProposition 2.5(7) we have (cid:16) x Γ , + j (cid:17) ( u + v ) (cid:16) x Γ , − l (cid:17) ( u ) = X m,n,q ∈ Z + m + n + q ≤ u ( − m + n + q D Γ , − m,u − m − n − q ( j, l )Λ Γ j,l,n D Γ , + q,u + v − m − n − q ( j, l )= ( − u D Γ , + u,v ( j, l ) + X m,n,q ∈ Z + m + n + q ≤ um,q
Proof of Theorem 2.2.
In this section we will prove Theorem 2.2. The proof will proceedby induction on the degree of monomials in U Z ( O ) and the following lemmas and proposition. Lemma 3.1.
For all r, s, n ∈ Z + and j, k, l, m ∈ N (1) (cid:20)(cid:16) x Γ , + j (cid:17) ( r ) , (cid:16) x Γ , − l (cid:17) ( s ) (cid:21) is in the Z -span of B and has degree less that r + s .(2) (cid:20)(cid:16) x Γ , + j (cid:17) ( r ) , Λ Γ k,m,n (cid:21) is in the Z -span of B and has degree less that r + n .(3) (cid:20) Λ Γ k,m,n , (cid:16) x Γ , − l (cid:17) ( s ) (cid:21) is in the Z -span of B and has degree less that s + n . Proof.
For (1), by Proposition 2.5(7) we have (cid:20)(cid:16) x Γ , + j (cid:17) ( r ) , (cid:16) x Γ , − l (cid:17) ( s ) (cid:21) = X m,n,q ∈ Z + By Proposition 2.4 and the definition of D Γ , ± u,v ( j, l ) we see that the terms of the summation arein the Z -span of B and have degree r + s − m − n − q < r + s . For (2) and (3) we have, byProposition 2.5(8) and (9), (cid:20)(cid:16) x Γ , + j (cid:17) ( r ) , Λ Γ k,m,n (cid:21) = n X i =1 Λ Γ k,m,n − i Y v ...v u ∈ Z + P v u = r P uv u = i (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) , (cid:20) Λ Γ k,m,n , (cid:16) x Γ , − l (cid:17) ( s ) (cid:21) = n X i =1 Y v ...v u ∈ Z + P v u = s P uv u = i (cid:0) ( u + 1) D Γ , − u ( l, k, m ) (cid:1) ( v u ) Λ Γ k,m,n − i . In either case the right-hand side is in the Z -span of B and has degree n − i + r < n + r or n − i + s < n + s respectively by the definition of D Γ , ± u ( j, k, m ). (cid:3) The following proposition gives the p Γ u ( j, l ) as Z -linear combinations of the h Γ n . Proposition 3.2. Given u ∈ N p Γ u ( j, l ) = ⌊ u − ⌋ X k =0 u X i =0 ( − k + i (cid:18) uk (cid:19)(cid:18) ui (cid:19) (cid:16) h Γ( u − i ) j +( u − k ) l (cid:17) + (( u + 1) mod 2) u X i =0 ( − u + i (cid:18) u − u − (cid:19)(cid:18) ui (cid:19) (cid:16) h Γ( u − i ) j (cid:17) (cid:3) The proof is omitted due to straightforwardness. Lemma 3.3. For all j, l ∈ N and k, m ∈ N Λ Γ j,l,k Λ Γ j,l,m = (cid:18) k + mk (cid:19) Λ Γ j,l,k + m + u where u is in the Z -span of the set n Λ Γ i ,n ,q . . . Λ Γ i s ,n s ,q s (cid:12)(cid:12)(cid:12) i , . . . , i s , n . . . , n s ∈ N , q , . . . , q s ∈ Z + o and has degree less than k + m . Proof. Λ Γ j,l,k corresponds to Λ k − from [11] under the map τ : U (cid:0) ( g ⊗ C [ t, t − ]) Γ (cid:1) → C [ X , X , . . . ]given by τ ( p Γ i ( j, l )) = − X i . Therefore, by Lemma 9.2 in [11], it suffices to show that p Γ i ( j, l ) isan integer linear combination of Λ Γ i,n,q . To that end we have, for all n ∈ Z + p Γ2 n +1 ( j, l ) = n X i =0 n X k =0 ( − k + i +1 (cid:18) n + 1 i (cid:19)(cid:18) n + 1 k (cid:19) Λ Γ(2 n +1 − i ) j, (2 n +1 − k ) l, . (10) p Γ2 n ( j, l ) = n − X i =0 2 n X k =0 ( − k + i +1 (cid:18) n − i (cid:19)(cid:18) nk (cid:19) Λ Γ(2 n − − i ) j +(2 n − k ) l,j, . (11)The proofs of (10) and (11) are straightforward and have been omitted to shorten the text. (cid:3) We can now prove Theorem 2.2. Proof. Let B be the set of ordered monomials constructed from M . The PBW Theorem impliesthat B is a C -linearly independent set. Thus it is a Z -linearly independent set.The proof that the Z -span of B is U Z ( O ) will proceed by induction on the degree of monomialsin U Z ( O ). Since Λ Γ j,l, = − (cid:16) h Γ j + l − h Γ j,l (cid:17) , any degree one monomial is in the Z -span of B . Nowtake any monomial, m , in U Z ( O ). If m ∈ B , then we are done. If not then either the factors of m are not in the correct order or m has “repeated” factors with the following forms (cid:16) x Γ , ± j (cid:17) ( r ) and (cid:16) x Γ , ± j (cid:17) ( s ) or Λ Γ j,l,r and Λ Γ j,l,s , j, l, r, s ∈ N . (12) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 9 If the factors of m are not in the correct order, then we can rearrange the factors of m usingthe straightening identities in Proposition 2.5. Once this is done Lemma 3.1 guarantees thateach rearrangement will only produce Z -linear combinations of monomials in the correct orderwith lower degree. These lower degree monomials are then in the Z -span of B by the inductionhypothesis.If (possibly after rearranging factors as above) m contains the products of the pairs of factorsin (12) we apply Proposition 2.5(6) or Lemma 3.3 respectively to consolidate these pairs offactors into single factors with integral coefficients.In the end we see that m ∈ Z -span B . Thus the Z -span of B is U Z ( O ) and hence B is anintegral basis for U Z ( O ). (cid:3) All that remains is to prove Proposition 2.5(7)-(9).3.2. Proof of Proposition 2.5 (8) and (9). We first need the following propositions. Proposition 3.4. For all k, u ∈ Z + p Γ k + u +1 ( j, l ) = h D Γ , + u, ( j, l ) , D Γ , − k, ( j, l ) i Proof. To prove the claim we proceed by induction on u . If u = 0 the claim is true by definition.Now assume the claim for some u ∈ Z + . Then we have h D Γ , + u +1 , ( j, l ) , D Γ , − k, ( j, l ) i = 12 hh D Γ , + u, ( j, l ) , Λ Γ j,l, i , D Γ , − k, ( j, l ) i = − h D Γ , − k, ( j, l ) , h D Γ , + u, ( j, l ) , Λ Γ j,l, ii = 12 h D Γ , + u, ( j, l ) , h Λ Γ j,l, , D Γ , − k, ( j, l ) ii + 12 h Λ Γ j,l, , h D Γ , − k, ( j, l ) , D Γ , + u, ( j, l ) ii by the Jacobi Identitiy= h D Γ , + u, ( j, l ) , D Γ , − k +1 , ( j, l ) i − (cid:2) Λ Γ j,l, , p Γ k + u +1 ( j, l ) (cid:3) by the induction hypothesis= p Γ k + u +1 ( j, l ) by the induction hypothesis again (cid:3) We now give the proof of Proposition 2.5 (8). The proof of (9) is similar. Proof. We proceed by induction on r . The case r = 0 is clear. In the case r = 1 we need toshow (cid:16) x Γ , + j (cid:17) Λ Γ k,m,n = n X i =0 i X r =0 i X s =0 ( − r + s ( i + 1) (cid:18) ir (cid:19)(cid:18) is (cid:19) Λ Γ k,m,n − i (cid:16) x Γ , + j +( i − r ) k +( i − s ) m (cid:17) . (13)We will prove this by induction on n . The case n = 1 is a straightforward calculation. Assumefor some n ∈ N that (13) holds for all 1 ≤ q ≤ n . Then we have (cid:16) x Γ , + j (cid:17) Λ Γ k,m,n = − n n X i =1 (cid:16) x Γ , + j (cid:17) p Γ i ( k, m )Λ Γ k,m,n − i = − n n X i =1 p Γ i ( k, m ) (cid:16) x Γ , + j (cid:17) Λ Γ k,m,n − i + 2 n n X i =1 i X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − r ) k +( i − s ) m (cid:17) Λ Γ k,m,n − i by Proposition 3 . − n n X i =1 n − i X u =0 u X r =0 u X s =0 ( − r + s ( u + 1) (cid:18) ur (cid:19)(cid:18) us (cid:19) p Γ i ( k, m )Λ Γ k,m,n − i − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 2 n n X i =1 i X r =0 i X s =0 n − i X u =0 u X v =0 u X w =0 ( u + 1)( − r + s ( − v + w (cid:18) ir (cid:19)(cid:18) is (cid:19)(cid:18) uv (cid:19)(cid:18) uw (cid:19) Λ Γ k,m,n − i − u × (cid:16) x Γ , + j +( u + i − v + r )) k +( u + i − w + s )) m (cid:17) by the induction hypothesis= − n n − X u =0 u X r =0 u X s =0 ( − r + s ( u + 1) (cid:18) ur (cid:19)(cid:18) us (cid:19) n − u X i =1 p Γ i ( k, m )Λ Γ k,m,n − i − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 2 n n X i =1 i X r =0 i X s =0 n X u = i u − i X v =0 u − i X w =0 ( u − i + 1)( − r + s ( − v + w (cid:18) ir (cid:19)(cid:18) is (cid:19)(cid:18) u − iv (cid:19)(cid:18) u − iw (cid:19) Λ Γ k,m,n − u × (cid:16) x Γ , + j +( u − v + r )) k +( u − w + s )) m (cid:17) = 1 n n − X u =0 u X r =0 u X s =0 ( − r + s ( u + 1)( n − u ) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 2 n n X i =1 i X r =0 i X s =0 n X u = i u + r − i X v = r u + s − i X w = s ( u − i + 1)( − r + s ( − v − r + w − s (cid:18) ir (cid:19)(cid:18) is (cid:19)(cid:18) u − iv − r (cid:19)(cid:18) u − iw − s (cid:19) × Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − v ) k +( u − w ) m (cid:17) = 1 n n − X u =0 u X r =0 u X s =0 ( − r + s ( u + 1)( n − u ) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 2 n n X u =1 u X v =0 u X w =0 u X i =1 v X r =0 w X s =0 ( − v + w ( u − i + 1) (cid:18) ir (cid:19)(cid:18) is (cid:19)(cid:18) u − iv − r (cid:19)(cid:18) u − iw − s (cid:19) Λ Γ k,m,n − u × (cid:16) x Γ , + j +( u − v ) k +( u − w ) m (cid:17) = 1 n n − X u =0 u X r =0 u X s =0 ( − r + s ( u + 1)( n − u ) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 2 n n X u =1 u X v =0 u X w =0 ( − v + w u X i =1 ( u − i + 1) (cid:18) uv (cid:19)(cid:18) uw (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − v ) k +( u − w ) m (cid:17) by the Chu-Vandermonde identity= Λ Γ k,m,n (cid:16) x Γ , + j (cid:17) + 1 n n X u =1 u X r =0 u X s =0 ( − r + s ( u + 1)( n − u ) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 2 n n X u =1 u X v =0 u X w =0 ( − v + w (cid:18) u ( u + 1) − u ( u + 1)2 (cid:19) (cid:18) uv (cid:19)(cid:18) uw (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − v ) k +( u − w ) m (cid:17) = Λ Γ k,m,n (cid:16) x Γ , + j (cid:17) + 1 n n X u =1 u X r =0 u X s =0 ( − r + s ( u + 1)( n − u ) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) + 1 n n X u =1 u X v =0 u X w =0 ( − v + w u ( u + 1) (cid:18) uv (cid:19)(cid:18) uw (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − v ) k +( u − w ) m (cid:17) = Λ Γ k,m,n (cid:16) x Γ , + j (cid:17) + n X u =1 u X r =0 u X s =0 ( − r + s ( u + 1) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 11 = n X u =0 u X r =0 u X s =0 ( − r + s ( u + 1) (cid:18) ur (cid:19)(cid:18) us (cid:19) Λ Γ k,m,n − u (cid:16) x Γ , + j +( u − r ) k +( u − s ) m (cid:17) This concludes the proof of the r = 1 case. Assume the proposition for some r ∈ N . Then( r + 1) (cid:16) x Γ , + j (cid:17) ( r +1) Λ Γ k,m,n = (cid:16) x Γ , + j (cid:17) (cid:16) x Γ , + j (cid:17) ( r ) Λ Γ k,m,n = n X i =0 (cid:16) x Γ , + j (cid:17) Λ Γ k,m,n − i Y v ...v u ∈ Z + P v u = r P uv u = i (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) by the induction hypothesis= n X i =0 n − i X s =0 Λ Γ k,m,n − i − s ( s + 1) D Γ , + s ( j, k, m ) Y v ...v u ∈ Z + P v u = r P uv u = i (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) = n X i =0 n − i X s =0 Λ Γ k,m,n − i − s Y v ...v u ∈ Z + u = s P v u = r − v s P uv u = i − sv s (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) × (cid:0) ( s + 1) D Γ , + s ( j, k, m ) (cid:1) ( v s ) ( s + 1) D Γ , + s ( j, k, m )= n X i =0 n − i X s =0 Λ Γ k,m,n − i − s Y v ...v u ∈ Z + u = s P v u = r − v s P uv u = i − sv s (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) × ( v s + 1) (cid:0) ( s + 1) D Γ , + s ( j, k, m ) (cid:1) ( v s +1) = n X i =0 n X s = i Λ Γ k,m,n − s Y v ...v u ∈ Z + P v u = r +1 P uv u = s ( v s − i ) (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) = n X s =0 Λ Γ k,m,n − s Y v ...v u ∈ Z + P v u = r +1 P uv u = s s X i =0 ( v s − i ) (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) = n X s =0 Λ Γ k,m,n − s Y v ...v u ∈ Z + P v u = r +1 P uv u = s s X i =0 ( v s ) (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) = ( r + 1) n X s =0 Λ Γ k,m,n − s Y v ...v u ∈ Z + P v u = r +1 P uv u = s (cid:0) ( u + 1) D Γ , + u ( j, k, m ) (cid:1) ( v u ) (cid:3) Proof of Proposition 2.5 (7). We will proceed following the method of proof of Lemma5.4 in [6], which was in turn modeled on the proof of Lemma 5.1 in [7]. We begin with somenecessary propositions. Proposition 3.5. For all i, j, k, l, m ∈ N (1) h(cid:16) x Γ , + j (cid:17) , p Γ i ( k, m ) i = − D Γ , + i ( j, k, m )(2) h p Γ i ( k, m ) , (cid:16) x Γ , − l (cid:17)i = − D Γ , − i ( l, k, m ) Proof. We will only prove (1) in detail, because the proof of (2) is similar. By Proposition 3.2we have h(cid:16) x Γ , + j (cid:17) , p Γ i ( k, m ) i = ⌊ i − ⌋ X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) h(cid:16) x Γ , + j (cid:17) , (cid:16) h Γ( i − s ) k +( i − r ) m (cid:17)i + (( i + 1) mod 2) i X s =0 ( − i + s (cid:18) i − i − (cid:19)(cid:18) is (cid:19) h(cid:16) x Γ , + j (cid:17) , (cid:16) h Γ( i − s ) k (cid:17)i = − ⌊ i − ⌋ X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16)(cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) + (cid:16) x Γ , + j − ( i − s ) k − ( i − r ) m (cid:17)(cid:17) − i + 1) mod 2) i X s =0 ( − i + s (cid:18) i − i − (cid:19)(cid:18) is (cid:19) (cid:16)(cid:16) x Γ , + j +( i − s ) k (cid:17) + (cid:16) x Γ , + j − ( i − s ) k (cid:17)(cid:17) = − ⌊ i − ⌋ X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) − ⌊ i − ⌋ X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j − ( i − s ) k − ( i − r ) m (cid:17) − i + 1) mod 2) i X s =0 ( − i + s (cid:18) i − i − (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k (cid:17) − i + 1) mod 2) i X s =0 ( − i + s (cid:18) i − i − (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j − ( i − s ) k (cid:17) = − ⌊ i − ⌋ X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) − i X r = ⌊ i +22 ⌋ i X s =0 ( − i − r + i − s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) − i + 1) mod 2) i X s =0 ( − i + s (cid:18) i − i − (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k (cid:17) − i + 1) mod 2) i X s =0 ( − i − s (cid:18) i − i − (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k (cid:17) = − ⌊ i − ⌋ X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) − i X r = ⌊ i +22 ⌋ i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 13 − i + 1) mod 2) i X s =0 ( − i + s (cid:18) i i (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k (cid:17) = − i X r =0 i X s =0 ( − r + s (cid:18) ir (cid:19)(cid:18) is (cid:19) (cid:16) x Γ , + j +( i − s ) k +( i − r ) m (cid:17) (cid:3) Proposition 3.6. For all i, j, k, m ∈ N and u ∈ Z + h p Γ m ( j, l ) , D Γ , + u, ( j, l ) i = 2 D Γ , + m + u, ( j, l ) Proof. We will proceed by induction on u . The case u = 0 is Proposition 3.5. Assume the claimfor some u ∈ N . Then we have h p Γ m ( j, l ) , D Γ , + u +1 , ( j, l ) i = 12 h p Γ m ( j, l ) , h D Γ , + u, ( j, l ) , Λ Γ j,l, ii = − (cid:16)h D Γ , + u, ( j, l ) , (cid:2) Λ Γ j,l, , p Γ m ( j, l ) (cid:3)i + h Λ Γ j,l, , h p Γ m ( j, l ) , D Γ , + u, ( j, l ) ii(cid:17) by the Jacobi Identity= − h Λ Γ j,l, , h p Γ m ( j, l ) , D Γ , + u, ( j, l ) ii = − h Λ Γ j,l, , D Γ , + m + u, ( j, l ) i by the induction hypothesis= h D Γ , + m + u, ( j, l ) , Λ Γ j,l, i = 2 D Γ , + m + u +1 , ( j, l ) (cid:3) Proposition 3.7. For all u, n ∈ Z + and j, l ∈ N D Γ , + u, ( j, l )Λ Γ j,l,n = n X i =0 ( i + 1)Λ Γ j,l,n − i D Γ , + i + u, ( j, l ) Proof. We will proceed by induction on u . The case u = 0 is Proposition 2.5(8). Assume theclaim for some u ∈ Z + . Then we have h D Γ , + u +1 , ( j, l ) , Λ Γ j,l,n i = 12 hh D Γ , + u, ( j, l ) , Λ Γ j,l, i , Λ Γ j,l,n i = 12 h D Γ , + u, ( j, l ) , (cid:2) Λ Γ j,l, , Λ Γ j,l,n (cid:3)i + 12 h Λ Γ j,l, , h Λ Γ j,l,n , D Γ , + u, ( j, l ) ii by the Jacobi Identity= − n X i =1 ( i + 1) h Λ Γ j,l, , Λ Γ j,l,n − i D Γ , + i + u, ( j, l ) i by the induction hypothesis= − n X i =1 ( i + 1)Λ Γ j,l, Λ Γ j,l,n − i D Γ , + i + u, ( j, l ) + 12 n X i =1 ( i + 1)Λ Γ j,l,n − i D Γ , + i + u, ( j, l )Λ Γ j,l, = 12 n X i =1 ( i + 1)Λ Γ j,l,n − i h D Γ , + i + u, ( j, l ) , Λ Γ j,l, i = n X i =1 ( i + 1)Λ Γ j,l,n − i D Γ , + i + u +1 , ( j, l ) (cid:3) Proposition 3.8. For all i, k ∈ Z + and j, l ∈ N we haveΛ Γ j,l,i D Γ , + k, ( j, l ) = D Γ , + k, ( j, l )Λ Γ j,l,i − D Γ , + k +1 , ( j, l )Λ Γ j,l,i − + D Γ , + k +2 , ( j, l )Λ Γ j,l,i − 24 ANGELO BIANCHI AND SAMUEL CHAMBERLIN Proof. We will proceed by induction on i . The cases i = 0 , i ∈ N . Then we have( i + 1)Λ Γ j,l,i +1 D Γ , + k, ( j, l ) = − i +1 X m =1 p Γ m ( j, l )Λ Γ j,l,i +1 − m D Γ , + k, ( j, l )= − i +1 X m =1 p Γ m ( j, l ) (cid:16) D Γ , + k, ( j, l )Λ Γ j,l,i +1 − m − D Γ , + k +1 , ( j, l )Λ Γ j,l,i − m + D Γ , + k +2 , ( j, l )Λ Γ j,l,i − − m (cid:17) by the induction hypothesis= − i +1 X m =1 p Γ m ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,i +1 − m + 2 i +1 X m =1 p Γ m ( j, l ) D Γ , + k +1 , ( j, l )Λ Γ j,l,i − m − i +1 X m =1 p Γ m ( j, l ) D Γ , + k +2 , ( j, l )Λ Γ j,l,i − − m = − D Γ , + k, ( j, l ) i +1 X m =1 p Γ m ( j, l )Λ Γ j,l,i +1 − m − i +1 X m =1 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m + 2 D Γ , + k +1 , ( j, l ) i +1 X m =1 p Γ m ( j, l )Λ Γ j,l,i − m + 4 i +1 X m =1 D Γ , + k + m +1 , ( j, l )Λ Γ j,l,i − m − D Γ , + k +2 , ( j, l ) i +1 X m =1 p Γ m ( j, l )Λ Γ j,l,i − − m − i +1 X m =1 D Γ , + k + m +2 , ( j, l )Λ Γ j,l,i − − m by Proposition 3 . 6= ( i + 1) D Γ , + k, ( j, l )Λ Γ j,l,i +1 − D Γ , + k +1 , ( j, l )Λ Γ j,l,i − D Γ , + k +2 , ( j, l )Λ Γ j,l,i − − i +1 X m =3 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m − iD Γ , + k +1 , ( j, l )Λ Γ j,l,i + 4 i +2 X m =2 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m + ( i − D Γ , + k +2 , ( j, l )Λ Γ j,l,i − − i +3 X m =3 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m = ( i + 1) D Γ , + k, ( j, l )Λ Γ j,l,i +1 − i +1 X m =3 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m − i + 1) D Γ , + k +1 , ( j, l )Λ Γ j,l,i + 4 D Γ , + k +2 , ( j, l )Λ Γ j,l,i − + 4 i +1 X m =3 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m + ( i − D Γ , + k +2 , ( j, l )Λ Γ j,l,i − − i +1 X m =3 D Γ , + k + m, ( j, l )Λ Γ j,l,i +1 − m = ( i + 1) D Γ , + k, ( j, l )Λ Γ j,l,i +1 − i + 1) D Γ , + k +1 , ( j, l )Λ Γ j,l,i + ( i + 1) D Γ , + k +2 , ( j, l )Λ Γ j,l,i − (cid:3) Proposition 3.9. For all j, l ∈ N and u, v ∈ Z + (1) uD Γ , ± u,v ( j, l ) = u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v − ( j, l )(2) ( u + v ) D Γ , ± u,v ( j, l ) = u X i =0 ( i + 1) D Γ , ± i, ( j, l ) D Γ , ± u − i,v − ( j, l ) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 15 Proof. We will proceed by induction on v . The cases v = 0 , v ∈ Z + . Then, for (1), we have uD Γ , ± u,v +1 ( j, l ) = uv + 1 u X i =0 D Γ , ± i, ( j, l ) D Γ , ± u − i,v ( j, l )= 1 v + 1 u X i =0 D Γ , ± i, ( j, l )( u − i ) D Γ , ± u − i,v ( j, l ) + 1 v + 1 u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v ( j, l )= 1 v + 1 u X i =0 D Γ , ± i, ( j, l ) u − i X k =0 kD Γ , ± k, ( j, l ) D Γ , ± u − i − k,v − ( j, l ) + 1 v + 1 u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v ( j, l )by the induction hypothesis= 1 v + 1 u X k =0 kD Γ , ± k, ( j, l ) u − k X i =0 D Γ , ± i, ( j, l ) D Γ , ± u − i − k,v − ( j, l ) + 1 v + 1 u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v ( j, l )= vv + 1 u X k =0 kD Γ , ± k, ( j, l ) D Γ , ± u − k,v ( j, l ) + 1 v + 1 u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v ( j, l )= u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v ( j, l ) . For (2), we have( u + v ) D Γ , ± u,v ( j, l ) = uD Γ , ± u,v ( j, l ) + vD Γ , ± u,v ( j, l )= u X i =0 iD Γ , ± i, ( j, l ) D Γ , ± u − i,v − ( j, l ) + u X i =0 D Γ , ± i, ( j, l ) D Γ , ± u − i,v − ( j, l ) by (1)= u X i =0 ( i + 1) D Γ , ± i, ( j, l ) D Γ , ± u − i,v − ( j, l ) (cid:3) Lemma 3.10. For all n, v ∈ Z + and j, l ∈ N n X i =0 Λ Γ j,l,i D Γ , + n − i,v ( j, l ) (cid:16) x Γ , − l (cid:17) = − ( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v − ( j, l )+ n X m =0 n − m X k =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − m − k D Γ , + k,v ( j, l ) Proof. We will proceed by induction on v . If v = 0 we haveΛ Γ j,l,n (cid:16) x Γ , − l (cid:17) = n X m =0 ( m + 1) D Γ , − m ( l, j, l )Λ Γ j,l,n − m . Assume the result for some v ∈ Z + . Then we have( v + 1) n X i =0 Λ Γ j,l,i D Γ , + n − i,v +1 ( j, l ) (cid:16) x Γ , − l (cid:17) Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) = n X i =0 n − i X k =0 Λ Γ j,l,i D Γ , + k, ( j, l ) D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) = n X k =0 n − k X i =0 Λ Γ j,l,i D Γ , + k, ( j, l ) D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) = n X k =0 n − k X i =0 (cid:16) D Γ , + k, ( j, l )Λ Γ j,l,i − D Γ , + k +1 , ( j, l )Λ Γ j,l,i − + D Γ , + k +2 , ( j, l )Λ Γ j,l,i − (cid:17) D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) by Proposition 3 . n X k =0 D Γ , + k, ( j, l ) n − k X i =0 Λ Γ j,l,i D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) − n X k =0 D Γ , + k +1 , ( j, l ) n − k X i =0 Λ Γ j,l,i − D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) + n X k =0 D Γ , + k +2 , ( j, l ) n − k X i =0 Λ Γ j,l,i − D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) = n X k =0 D Γ , + k, ( j, l ) n − k X i =0 Λ Γ j,l,i D Γ , + n − i − k,v ( j, l ) (cid:16) x Γ , − l (cid:17) − n X k =0 D Γ , + k +1 , ( j, l ) n − k − X i =0 Λ Γ j,l,i D Γ , + n − i − k − ,v ( j, l ) (cid:16) x Γ , − l (cid:17) + n X k =0 D Γ , + k +2 , ( j, l ) n − k − X i =0 Λ Γ j,l,i D Γ , + n − i − k − ,v ( j, l ) (cid:16) x Γ , − l (cid:17) = n X k =0 D Γ , + k, ( j, l ) − ( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k +1 − u D Γ , + u,v − ( j, l )+ n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l ) ! − n X k =0 D Γ , + k +1 , ( j, l ) − ( n − k ) n − k X u =0 Λ Γ j,l,n − k − u D Γ , + u,v − ( j, l )+ n − k − X m =0 n − k − − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − − m − r D Γ , + r,v ( j, l ) ! + n X k =0 D Γ , + k +2 , ( j, l ) − ( n − k − n − k − X u =0 Λ Γ j,l,n − k − − u D Γ , + u,v − ( j, l )+ n − k − X m =0 n − k − − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − − m − r D Γ , + r,v ( j, l ) ! by the induction hypothesis= − n X k =0 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k +1 − u D Γ , + u,v − ( j, l )+ n X k =0 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l )+ 2 n X k =0 D Γ , + k +1 , ( j, l )( n − k ) n − k X u =0 Λ Γ j,l,n − k − u D Γ , + u,v − ( j, l ) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 17 − n X k =0 D Γ , + k +1 , ( j, l ) n − k − X m =0 n − k − − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − − m − r D Γ , + r,v ( j, l ) − n X k =0 D Γ , + k +2 , ( j, l )( n − k − n − k − X u =0 Λ Γ j,l,n − k − − u D Γ , + u,v − ( j, l )+ n X k =0 D Γ , + k +2 , ( j, l ) n − k − X m =0 n − k − − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − − m − r D Γ , + r,v ( j, l )= − n X k =0 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k +1 − u D Γ , + u,v − ( j, l )+ n X k =0 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l )+ 2 n +1 X k =1 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k − u +1 D Γ , + u,v − ( j, l ) − n +1 X k =1 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l ) − n +2 X k =2 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k +1 − u D Γ , + u,v − ( j, l )+ n +2 X k =2 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l )= − D Γ , +0 , ( j, l )( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v − ( j, l ) − D Γ , +1 , ( j, l ) n n X u =0 Λ Γ j,l,n − u D Γ , + u,v − ( j, l ) − n X k =2 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k +1 − u D Γ , + u,v − ( j, l )+ D Γ , +0 , ( j, l ) n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l )+ D Γ , +1 , ( j, l ) n − X m =0 n − − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − − m − r D Γ , + r,v ( j, l )+ n X k =2 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l )+ 2 D Γ , +1 , ( j, l ) n n X u =0 Λ Γ j,l,n − u D Γ , + u,v − ( j, l )+ 2 n X k =2 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k − u +1 D Γ , + u,v − ( j, l ) − D Γ , +1 , ( j, l ) n − X m =0 n − − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − − m − r D Γ , + r,v ( j, l ) − n X k =2 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l ) − n X k =2 D Γ , + k, ( j, l )( n − k + 1) n − k +1 X u =0 Λ Γ j,l,n − k +1 − u D Γ , + u,v − ( j, l )+ n X k =2 D Γ , + k, ( j, l ) n − k X m =0 n − k − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − k − m − r D Γ , + r,v ( j, l )= − ( n + 1) D Γ , +0 , ( j, l ) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v − ( j, l ) + nD Γ , +1 , ( j, l ) n X u =0 Λ Γ j,l,n − u D Γ , + u,v − ( j, l )+ n X m =0 n − m X r =0 ( m + 1) D Γ , +0 , ( j, l ) D Γ , − m, ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n − X m =0 n − − m X r =0 ( m + 1) D Γ , +1 , ( j, l ) D Γ , − m, ( j, l )Λ Γ j,l,n − − m − r D Γ , + r,v ( j, l )= − ( n + 1) n +1 X u =0 n +1 − u X i =0 ( i + 1)Λ Γ j,l,n +1 − u − i D Γ , + i, ( j, l ) D Γ , + u,v − ( j, l )+ n n X u =0 n − u X i =0 ( i + 1)Λ Γ j,l,n − u − i D Γ , + i +1 , ( j, l ) D Γ , + u,v − ( j, l )+ n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l )+ n X m =0 n − m X r =0 ( m + 1) p Γ m +1 ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l ) − n − X m =0 n − − m X r =0 ( m + 1) p Γ m +2 ( j, l )Λ Γ j,l,n − − m − r D Γ , + r,v ( j, l ) by Propositions 3 . . − ( n + 1) n +1 X u =0 n +1 − u X i =0 ( i + 1)Λ Γ j,l,n +1 − u − i D Γ , + i, ( j, l ) D Γ , + u,v − ( j, l )+ n n +1 X u =0 n − u +1 X i =0 i Λ Γ j,l,n − u − i +1 D Γ , + i, ( j, l ) D Γ , + u,v − ( j, l )+ n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l )+ n X m =0 n − m X r =0 ( m + 1) p Γ m +1 ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l ) − n X m =0 n − m X r =0 mp Γ m +1 ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l )= − n +1 X u =0 n +1 − u X i =0 ( n + i + 1)Λ Γ j,l,n +1 − u − i D Γ , + i, ( j, l ) D Γ , + u,v − ( j, l ) N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 19 + n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) + n X m =0 n − m X r =0 p Γ m +1 ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l )= − n +1 X i =0 n +1 − i X u =0 ( n + i + 1)Λ Γ j,l,n +1 − u − i D Γ , + i, ( j, l ) D Γ , + u,v − ( j, l )+ n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) + n X r =0 n − r X m =0 p Γ m +1 ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l )= − n +1 X i =0 n +1 X u = i ( n + i + 1)Λ Γ j,l,n +1 − u D Γ , + i, ( j, l ) D Γ , + u − i,v − ( j, l )+ n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) + n X r =0 n − r +1 X m =1 p Γ m ( j, l )Λ Γ j,l,n − m − r +1 D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l )= − nv n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l ) − n +1 X u =0 ( u + v )Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l )+ n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n +1 X r =0 ( n − r + 1)Λ Γ j,l,n − r +1 D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l ) + n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n − X m =0 n − m − X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l ) + n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) − n X m =0 n − m X r =1 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r − ,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l ) + n X m =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m D Γ , +0 ,v ( j, l )+ n X m =0 1 X k =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − D Γ , +1 − k,v ( j, l )+ n X m =0 n − m X r =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , +0 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v ( j, l ) + n X m =0 n − m X r =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r − ,v ( j, l )+ n X m =0 n − m X r =2 r X k =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l ) − n X m =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − D Γ , +0 ,v ( j, l ) − n X m =0 n − m X r =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , +1 , ( j, l )Λ Γ j,l,n − m − r D Γ , + r − ,v ( j, l ) − n X m =0 n − m X r =2 r X k =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l )+ n X m =0 n − m X r =2 r X k =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l )+ n X m =0 n − m X r =0 r X k =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l ) − n X m =0 n − m X r =1 r X k =1 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l )+ n X m =0 n − m X r =2 r X k =2 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l )+ n X m =0 n − m X r =0 r X k =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r D Γ , + r − k,v ( j, l ) − n X m =0 n − m X r =0 r X k =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k +1 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r − k,v ( j, l )+ n X m =0 n − m X r =0 r X k =0 ( m + 1) D Γ , − m, ( j, l ) D Γ , + k +2 , ( j, l )Λ Γ j,l,n − m − r − D Γ , + r − k,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l )+ n X m =0 n − m X r =0 r X k =0 ( m + 1) D Γ , − m, ( j, l ) (cid:16) D Γ , + k, ( j, l )Λ Γ j,l,n − m − r − D Γ , + k +1 , ( j, l )Λ Γ j,l,n − m − r − + D Γ , + k +2 , ( j, l )Λ Γ j,l,n − m − r − (cid:17) D Γ , + r − k,v ( j, l )= − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l ) + n X m =0 n − m X r =0 r X k =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − m − r D Γ , + k, ( j, l ) D Γ , + r − k,v ( j, l )by Proposition 3 . N INTEGRAL BASIS FOR THE ONSAGER ALGEBRA 21 = − ( v + 1)( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,v ( j, l ) + ( v + 1) n X m =0 n − m X r =0 ( m + 1) D Γ , − m, ( j, l )Λ Γ j,l,n − m − r D Γ , + r,v +1 ( j, l ) (cid:3) Now, we prove Proposition 2.5 (7): Proof. We proceed by induction on s . The case s = 0 holds, because D Γ , +0 ,r ( j, l ) = (cid:16) x Γ , + j (cid:17) ( r ) .Assume the formula is true for some s ∈ Z + . Then we have( s + 1) (cid:16) x Γ , + j (cid:17) ( r ) (cid:16) x Γ , − l (cid:17) ( s +1) = (cid:16) x Γ , + j (cid:17) ( r ) (cid:16) x Γ , − l (cid:17) ( s ) (cid:16) x Γ , − l (cid:17) = X m,n,q ∈ Z + m + n + q ≤ min { r,s } ( − m + n + q D Γ , − m,s − m − n − q ( j, l )Λ Γ j,l,n D Γ , + q,r − m − n − q ( j, l ) (cid:16) x Γ , − l (cid:17) by the induction hypothesis= X m,i,q ∈ Z + m + i + q ≤ min { r,s } ( − m + i + q D Γ , − m,s − m − i − q ( j, l )Λ Γ j,l,i D Γ , + q,r − m − i − q ( j, l ) (cid:16) x Γ , − l (cid:17) = X m,i,n ∈ Z + m + n ≤ min { r,s } ( − m + n D Γ , − m,s − m − n ( j, l )Λ Γ j,l,i D Γ , + n − i,r − m − n ( j, l ) (cid:16) x Γ , − l (cid:17) = X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n D Γ , − m,s − m − n ( j, l ) n X i =0 Λ Γ j,l,i D Γ , + n − i,r − m − n ( j, l ) (cid:16) x Γ , − l (cid:17) = X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n D Γ , − m,s − m − n ( j, l ) (cid:18) − ( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,r − m − n − ( j, l )+ n X i =0 n − i X k =0 ( i + 1) D Γ , − i, ( j, l )Λ Γ j,l,n − i − k D Γ , + k,r − m − n ( j, l ) (cid:19) by Lemma 3 . X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n +1 D Γ , − m,s − m − n ( j, l )( n + 1) n +1 X u =0 Λ Γ j,l,n +1 − u D Γ , + u,r − m − n − ( j, l )+ X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n n X i =0 ( i + 1) D Γ , − i, ( j, l ) D Γ , − m,s − m − n ( j, l ) n − i X k =0 Λ Γ j,l,n − i − k D Γ , + k,r − m − n ( j, l )= X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n +1 D Γ , − m,s − m − n ( j, l )( n + 1) n +1 X u =0 Λ Γ j,l,u D Γ , + n +1 − u,r − m − n − ( j, l )+ X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n n X i =0 ( i + 1) D Γ , − i, ( j, l ) D Γ , − m,s − m − n ( j, l ) n − i X k =0 Λ Γ j,l,k D Γ , + n − i − k,r − m − n ( j, l )= X m,n,u ∈ Z + m + n ≤ min { r,s } ( − m + n +1 ( n + 1) D Γ , − m,s − m − n ( j, l )Λ Γ j,l,u D Γ , + n +1 − u,r − m − n − ( j, l ) + X m,n ∈ Z + m + n ≤ min { r,s } ( − m + n n X k =0 n − k X i =0 ( i + 1) D Γ , − i, ( j, l ) D Γ , − m,s − m − n ( j, l )Λ Γ j,l,k D Γ , + n − i − k,r − m − n ( j, l )= X m,u,q ∈ Z + m + q + u ≤ min { r,s } ( − m + q + u +1 ( q + u + 1) D Γ , − m,s − m − q − u ( j, l )Λ Γ j,l,u D Γ , + q +1 ,r − m − q − u − ( j, l )+ X m,q,k ∈ Z + m + q + k ≤ min { r,s } ( − m + q + k q X i =0 ( i + 1) D Γ , − i, ( j, l ) D Γ , − m,s − m − q − k ( j, l )Λ Γ j,l,k D Γ , + q − i,r − m − q − k ( j, l )= X m,u,q ∈ Z + ≤ m + q + u ≤ min { r,s } +1 ( − m + q + u ( q + u ) D Γ , − m,s +1 − m − q − u ( j, l )Λ Γ j,l,u D Γ , + q,r − m − q − u ( j, l )+ min { r,s } X i =0 X m,q,k ∈ Z + m + q + k ≤ min { r,s } ( − m + q + k ( i + 1) D Γ , − i, ( j, l ) D Γ , − m,s − m − q − k ( j, l )Λ Γ j,l,k D Γ , + q − i,r − m − q − k ( j, l )= X m,u,q ∈ Z + ≤ m + q + u ≤ min { r,s +1 } ( − m + q + u ( q + u ) D Γ , − m,s +1 − m − q − u ( j, l )Λ Γ j,l,u D Γ , + q,r − m − q − u ( j, l )+ min { r,s } X i =0 X m,q,k ∈ Z + m + q + k ≤ min { r,s } ( − m + q + k ( i + 1) D Γ , − i, ( j, l ) D Γ , − m − i,s − m − q − k ( j, l )Λ Γ j,l,k D Γ , + q,r − m − q − k ( j, l )= X m,u,q ∈ Z + ≤ m + q + u ≤ min { r,s +1 } ( − m + q + u ( q + u ) D Γ , − m,s +1 − m − q − u ( j, l )Λ Γ j,l,u D Γ , + q,r − m − q − u ( j, l )+ X m,q,k ∈ Z + m + q + k ≤ min { r,s } ( − m + q + k m X i =0 ( i + 1) D Γ , − i, ( j, l ) D Γ , − m − i,s − m − q − k ( j, l )Λ Γ j,l,k D Γ , + q,r − m − q − k ( j, l )= X m,u,q ∈ Z + ≤ m + q + u ≤ min { r,s +1 } ( − m + q + u ( q + u ) D Γ , − m,s +1 − m − q − u ( j, l )Λ Γ j,l,u D Γ , + q,r − m − q − u ( j, l )+ X m,q,k ∈ Z + m + q + k ≤ min { r,s } ( − m + q + k ( s + 1 − q − k ) D Γ , − m,s +1 − m − q − k ( j, l )Λ Γ j,l,k D Γ , + q,r − m − q − k ( j, l )by Proposition 3 . ii )= X m,n,q ∈ Z + m + n + q ≤ min { r,s +1 } ( − m + n + q ( q + n ) D Γ , − m,s +1 − m − n − q ( j, l )Λ Γ j,l,n D Γ , + q,r − m − n − q ( j, l )+ X m,n,q ∈ Z + m + n + q ≤ min { r,s +1 } ( − m + n + q ( s + 1 − q − n ) D Γ , − m,s +1 − m − n − q ( j, l )Λ Γ j,l,n D Γ , + q,r − m − n − q ( j, l )= ( s + 1) X m,n,q ∈ Z + m + n + q ≤ min { r,s +1 } ( − m + n + q D Γ , − m,s +1 − m − n − q ( j, l )Λ Γ j,l,n D Γ , + q,r − m − n − q ( j, l ) (cid:3) References [1] I. 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