Quantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation

Transcription

1 arxiv:math/ v1 [math.qa] 1 Mar 2007 Quantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation Koji HASEGAWA Mathematical Institute, Tohoku University Sendai JAPAN Dedicated to Professor Akihiro Tsuchiya on the occasion of his retirement Abstract Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as symmetries or the Bäcklund transformations in Painlevé equations. We thereby propose a quantization of discrete Painlevé VI equation as a discrete Hamiltonian flow commuting with the action of W(D (1) 1 4 ). 1 Introduction Let A = [a ij ] l i,j=0 be a generalized Cartan matrix of affine type and W(A) be the corresponding Weyl group. We denote the generators by s i (i = 0,,l). Let F cl := C(a 0,,a l,f 0,,f l ) be the field of rational functions generated by commuting variables a 0,,a l,f 0,,f l. Let u ij be integers that satisfy (i) u ij = 0 if i = j or a ij = 0, (ii) u ij : u ji = a ij : a ji otherwise. Theorem 1 (KNY, case A (1) l ) For i,j = 0,,l put s i (a j ) := a j a a ij i, s i (f j ) := f j ( ai +f i 1+a i f i Then these formulas define a group homomorphism W(A (1) l ) Aut(F cl ). address : kojihas@math.tohoku.ac.jp 1 AMS subject classifications(2000): 37K60, 39A70, 81R50. Key words: Weyl groups, discrete Painlevé equations, quantum integrable systems. ) uij. (1)

2 This is the typical formula of the affine Weyl group symmetry or the Bäcklund transforationfor difference Painlevé equation inits symmetric form: the case A (1) 2 gives the symmetry of the difference Painlevé IV equation[kny]. Moreover, this action is a Poisson map with respect to the bracket {f i,f j } = u ij f i f j, {a i,a j } = {a i,f j } = 0. (2) A naive expect is that there exist a quantization of this representation realized as adjoint actions of some suitable operators (quantum Hamiltonian action). One of the aim of this note is to answer this problem. In the type A case, we introduce the letters F 0,,F l subject to the quantized relation of (2), F i F i+1 = q 1 F i+1 F i, F i F j F j F i = 0 (j i±1) as well as central letters a 0,,a l. Let F be the skew field defined by these relations. We will construct the affine Weyl group action on F in the form s i (φ) = S i φs 1 i for any φ F. The Hamiltonian S i is actually given by some infinite product which is rather familiar in q- analysis, despite that it involves non-commutative letters (Section 2, Theorem 2). It is also shown that the construction works for other affine Weyl groups as well (Section 3, Theorems 3 and 4). Rescent studies of Painlevé systems enabled us to understand their discrete symmetries (Bäcklund transformations) and the (discrete) time evolution tranformation from the one, namely the affine Weyl group actions of the above type [NY1][S]. Based on this knowledge together with our quantization of the affine Weyl group action, we can quantize discrete (multiplicative) Painlevé type equations. In principle, if we choose some lattice direction in the affine Weyl group as the generator of a discrete time evolution, then this discrete dynamics commutes with the simple reflections corresponding to the roots that are perpendicular to the evolution direction in the lattice. We apply this idea to quantize the q- difference Painlevé III equation studied by Kajiwara and Kimura [KK] and also to quantize Jimbo-Sakai s q- difference Painlevé VI system[js]: See(12), (13) in Section 2 and Theorems 5, 6 for results. 2 Quantizing the Weyl group action : Type A case Let Q = Zα 0 + +Zα l be the root lattice of type A (1) l with simple roots α 0,,α l and C[Q] = C[e α 0,,e α l ] be its group algebra. The Weyl group action s i (α j ) = α j a ij α i gives rise to the action on C[Q], and we can identify the previously used letter a j as e α j : s i (a j ) = a a ij i a j. Let K be the quotient field of the group algebra C[Q], namely K = C(a 0,,a l ).

3 Remark Let 0,, l be the dual letters such that then we have [ j,α k ] := j α k α h j = a jk, e π 1α i i α j e π 1α i i = s i (α j ). That is, the Weyl group action on K can be realized as adjoint actions. This remark applies to the latter cases as well. For type A (1) l case (l > 1), we introduce the cannonically quantized letters F 0,,F l corresponding to (2): F i F i+1modl+1 = q 1 F i+1modl+1 F i, F i F j = F j F i (i j ±1). (3) (Here and in what follows we regard the subscripts as elements in Z/(l+1)Z.) Let K F 0,,F l be the K- algebra generated by the above letters (3). It can be shown in a standard way that this algebra is an Ore domain (cf. [B]). Let F := K(F 0,,F l ) be the quotient skew field of K F 0,,F l. The above relations (3) actually quantize the Poisson bracket (2). In fact, letting q 1 and think of the Poisson structure 1 {φ,ψ} := lim q 1 q 1 [φ,ψ] on the commutative algebra F mod(q 1). Then we have {F i,f i+1 } 1 q 1 (F if i+1 F i+1 F i ) = F i F i+1 according to the defining relation (3). Note that a i K is central in F. Let us introduce the following multiplication operator Ψ(z,F i ) = Ψ q (z,f i ) := (qf i,q) (F 1 i,q) (zqf i,q) (zf 1 i,q) (4) where z and q are central letters and (x,q) := m=0 (1+xq m ). The right hand side of (4) should be understood in the q- adic completion F((q)) of F. We put ρ i := e 1 2 π 1α i i, S i := Ψ(z,F i )ρ i. Note that ρ i commutes with the variables F j, 0 j l.we are interested in the adjoint action of S i, Ad(S i ) : φ F((q)) S i φsi 1 F((q)). The statement of the following theorem essentially goes back to [FV]. Theorem 2 We have Ad(S i ) 2 = id, (5) S i S j = S j S i (j i±1), (6) S i S i+1 S i = S i+1 S i S i+1 (7) where the index should read modulo l+1. Hence s i Ad(S i ) defines a group homomorphism W(A (1) l ) Aut(F((q))).

4 Let us calculate Ad(S i )F j = S i F j Si 1 first. We have Ad(S i )(F i 1 ) = 1+a if i a i +F i F i 1 Ad(S i )(F i+1 ) = F i+1 a i +F i 1+a i F i (8) Ad(S i )(F j ) = F j (i j ±1) In fact these are the defining recurrence relation for the multiplication operator (4), that is, one can recover the formula of Ψ(a i,f i ) from these modulo pseudo constants. Now we can check AdS 2 i = id(5) from these formula. Equivalently, since S 2 i = Ψ(a i,f i )Ψ(a 1 i,f i )ρ 2 i, (5) follows from the fact that Ψ(a i,f i )Ψ(a 1 i,f i ) is a pseudo constant: Ψ(a i,f i )Ψ(a 1 i,f i ) = Ψ(a i,qf i )Ψ(a 1 i,qf i ). (9) As for (7), we can do the similar computation to check it as the adjoint action on F((q)), namely, compare the result when adjointly applied to generaters F i. However, (7) is satisfied as an identity of elements in F((q)). Actually (7) follows from the dilogarithmic identity : suppose F and G satisfies FG = qgf, then we have From this we can show ([FV], [Ki]) (G,q) (F,q) = (F,q) (GF,q) (G,q). Ψ(x,F i )Ψ(xy,F i+1 )Ψ(y,F i ) = Ψ(y,F i+1 )Ψ(xy,F i )Ψ(x,F i+1 ) (10) where x,y are central letters, which is equivalent to (7). Introduce the diagram automorphism by ω : a i a i+1, F i F i+1 modl+1, then ω and s i := Ad(S i ) generate the extended affine Weyl group As is well known, we have the commuting elements T 1 := s 1 s 2 s l ω 1 T 2 := s 2 s l ω 1 s 1. T l := s l ω 1 s 1 s l 1. W(A (1) l ) acting on F((q)). (11) They are mutually conjugate. If we take T 1 as a discrete time evolution operator, then the group s 0 s 1 s 0,s 2,s 3,,s l W(A (1) l 1 ) commutes with the T 1 action. This gives the

5 quantization of the q- difference version of type A discrete system with Painlevé type symmetry which is extensively studied by Noumi and Yamada [NY1][NY2]. (1) Example Let l = 2. Note that a 0 a 1 a 2 =: p is invariant under W(A 2 ), and the same holds for F 0 F 1 F 2 =: c since c commutes with everything. The action of T 1 = s 1 s 2 ω 1 is given by T 1 (a 0 ) = p 1 a 0, T 1 (a 1 ) = pa 1, T 1 (a 2 ) = a 2 (12) and F 0 T 1 (F 0 ) = c 1+a 1F1 1, T1 1 (F 1 )F 1 = c 1+a 1 0 F 0 1+a 1 F 1 1+a 1 0 F0 1. (13) This T 1 action commutes with s 0 s 1 s 0,s 2 W(A (1) 1 ) and gives a quantization of the qp III system studied in [KK] (where p should be regarded as q). In this l = 2 case, the Hamiltonian for the diagram automorphism ω can be found as follows, so that the above T 1 flow is actually a discrete Hamiltonian flow. We put and where θ(x) := (X,q) (qx 1,q) Ω := ( θ(f 1 0 F 1 )θ(qf 1 ) 2 θ(f 1 2 F 1 0 ) ) 1 p 1 ρ1 ρ 2, [ 1,α 0] = 1,[ 1,α 1] = 1,[ 1,α 2] = 0. (Note : If we realize the A (1) 2 root system in R 3 Rδ by α 0 = e 3 e 1 +δ,α 1 = e 1 e 2,α 2 = e 2 e 3, where e 1,e 2,e 3 are the standard orthogonal basis of R 3 and δ the canonical null root so that p = e δ, then 1 stands for the derivation corresponding to e 1. ) Then we can easily check Ωa i Ω 1 = a i+1mod3, ΩF i Ω 1 = F i+1mod3 and therefore We have T 1 = Ad(S 1 S 2 Ω 1 ). (14) S 1 S 2 Ω 1 = Ψ(a 1,F 1 )ρ 1 Ψ(a 2,F 2 )ρ 2 (p 1 ρ1 ρ 2 ) 1 θ(f0 1 F 1 )θ(qf 1 ) 2 θ(f2 1 F0 1 ) = Ψ(a 1,F 1 )Ψ(a 1 a 2,F 2 )θ(f0 1 F 1 )θ(qf 1 ) 2 θ(f2 1 F0 1 ) p 1. 3 General case If the Dynkin diagram for the genralized Cartan matrix is simply laced, the construction in the last section applies to obtain the corresponding Weyl group action. As for the nonsimply laced case, the construction can be reduced to the rank two cases : B 2 type and G 2 type (cf. [NY3]).

6 As before let K := C(a 1,a 2 ) be the quotient field of the group algebra C[Q], where Q stands for the rank two root lattice in problem and we identify the letter a j with e α j C[Q]. We introduce j (j = 1,2) such that [ j,α k ] = a jk and put ρ j := e 1 2 π 1α j j. Then the Weyl group action s j on K is given by the adjoint action of ρ j : B 2 case. [ ] [ a11 a Let 12 = a 21 a 22 G 2 case. [ ] [ a11 a Let 12 = a 21 a 22 s j (α k ) := α k a jk α j = ρ j α k ρ 1 j, s j (a k ) = a a jk j a k = ρ j a k ρ 1 j ] ] and define the skew field F = K(F 1,F 2 ), where F 2 F 1 = q 2 F 1 F 2. and define the skew field F = K(F 1,F 2 ), where F 2 F 1 = q 3 F 1 F 2. Using these, we have Hamiltonian Weyl group action on F in both cases : Theorem 3 For type B 2 case, put Then we have S 1 := Ψ q (a 1,F 1 )ρ 1, S 2 := Ψ q 2(a 2 2,F 2)ρ 2. (15) Ad(S j ) 2 = id, S 1 S 2 S 1 S 2 = S 2 S 1 S 2 S 1. (16) Theorem 4 For type G 2 case, put S 1 := Ψ q (a 1,F 1 )ρ 1, S 2 := Ψ q 3(a 3 2,F 2)ρ 2. (17) We have Ad(S j ) 2 = id, S 1 S 2 S 1 S 2 S 1 S 2 = S 2 S 1 S 2 S 1 S 2 S 1. (18) For example, in the B 2 case we have S 1 F 2 S 1 1 = Ψ q (a 1,F 1 )F 2 Ψ q (a 1,F 1 ) 1 = F 2 Ψ q (a 1,q 2 F 1 )Ψ q (a 1,F 1 ) 1 = F 2 (F 1 q 1,q) (F 1 1 q 2,q) (a 1 F 1 q 1,q) (a 1 F 1 = F 2 (1+F 1 q 1 )(1+F 1 ) (1+a 1 F 1 q 1 )(1+a 1 F 1 ) = F 2 (a 1 +F 1 )(a 1 +F 1 q 1 ) (1+a 1 F 1 )(1+a 1 F 1 q 1 ). 1 1,q) (a 1 F 1 q,q) (a 1 F 1 q 2,q) (F 1 q,q) (F1 1,q) (1+a 1 F1 1 )(1+a 1 F1 1 q) (1+F1 1 )(1+F1 1 q)

7 The property Ad(S j ) 2 = id can be checked by continuing this computation, or we can conclude it immediately from the pseudoconstant property (9). In principle we can calculate the expressions for Ad(S 1 S 2 S 1 S 2 )F j and Ad(S 2 S 1 S 2 S 1 )F j (j = 1,2) also to check (16) at the adjoint level, though it is a quite lengthy way. In fact (16) holds as an identity in F((q)). Let us introduce a square root of F 2, namely let F 2 be the letter satisfying Then we have F 2 2 = F2, F 1 F 2 = q 1 F 2 F 1. Ψ q 2(a 2 2,F 2) = Ψ q (a 2, F 2 )Ψ q (a 2, F 2 ), so that (16) can be reduced to the type A identity (10). For short, we write a 1 =: a,a 2 =: b, Ψ q (a,f 1 ) =: Ψ a 1, Ψ q 2(b 2,F 2 ) =: Ψ b 2, Ψ q (b,± F 2 ) =: Ψ b ±. Then S 1 S 2 S 1 S 2 = S 2 S 1 S 2 S 1 is equivalent to Ψ a 1 Ψab 2 Ψab2 1 Ψ b 2 = Ψb 2 Ψab2 1 Ψ ab 2 Ψa 1, or Ψ a 1 Ψab + Ψab Ψab2 1 Ψ b + Ψb = Ψb + Ψb Ψab2 1 Ψ ab + Ψab Ψa 1. This can be verified as follows, which uses (10): Ψ x ± Ψxy 1 Ψ y ± = Ψ y 1Ψ xy ± Ψ x 1 as well as Ψx + Ψy = Ψ y Ψ x + at the underlined places. LHS = Ψ a 1Ψ ab +Ψ ab Ψ ab2 1 Ψ b Ψ b + = Ψ a 1Ψ ab +Ψ b 1Ψ ab2 Ψ ab 1 Ψ b + = Ψ b +Ψ ab 1 Ψ a +Ψ ab2 Ψ ab 1 Ψ b + = Ψ b +Ψ ab Ψ a +Ψ ab 1 Ψ b + = Ψ b +Ψ ab Ψ b 1Ψ ab +Ψ a 1 = Ψ b +Ψ b Ψ ab2 1 Ψ ab Ψ ab +Ψ a 1 = RHS. Proof for the G 2 case can be quite similary done as in the B 2 case : we use the cubic root ζ 1 of unity and 3 F 2 of F 2 that satisfies F 1 3 F 2 = q 1 3 F 2 F 1. We have Ψ q 3(a 3 2,F 2) = Ψ q ( a 2, 3 F 2 ) Ψ q ( a 2,ζ 3 F 2 ) Ψ q ( a 2,ζ 1 3 F 2 ). As before, write a 1 =: a,a 2 =: b, Ψ q (a,f 1 ) =: Ψ a 1,Ψ q 3(a 3,F 2 ) =: Ψ b 2 and also for short. Then (18) is equivalent to Ψ q (b, 3 F 2 ) =: Ψ b 0, Ψ q (b,ζ ±1 3 F 2 ) =: Ψ b ± Ψ a 1 Ψab 2 Ψa2 b 3 2 Ψ ab3 1 Ψ b 2 = Ψb 2 Ψab3 2 Ψ a2 b 3 1 Ψ ab 2 Ψa 1. (19)

8 This time (10) means Ψ x k Ψxy 1 Ψ y k = Ψy 1Ψ xy k Ψx 1 for k = 0,± and Ψx 0,Ψy +,Ψ z for any central x,y,z. We have LHS of (19) = Ψ a 1Ψ ab 0 Ψ ab +Ψ ab Ψ a2 b 3 0 }{{} Ψab2 + Ψ ab2 Ψ ab3 1 Ψ b 0 }{{} Ψb +Ψ b = Ψ a 1 Ψab 0 Ψab + Ψab2 1 Ψ a2 b 3 Ψab 1 Ψab2 + Ψb 1 Ψab3 0 Ψ ab2 1 Ψ b + Ψb = Ψ a 1 Ψab 0 Ψab + Ψab2 1 Ψ a2 b 3 Ψb + Ψab2 = Ψ a 1 Ψab 0 Ψb 1 Ψab2 + Ψab b 3 1 Ψa2 1 Ψ ab + }{{} Ψab3 0 Ψ ab2 1 Ψ b + }{{} Ψb Ψab2 1 Ψ ab3 0 Ψ b 1 }{{} Ψab2 + Ψab 1 Ψb = Ψ b 0Ψ ab 1 Ψ a 0Ψ ab2 + Ψ ab 1 Ψ a2 b 3 Ψ b 0Ψ ab3 0 Ψ ab2 + Ψ ab 1 Ψ b = Ψ b 0Ψ ab + Ψ b 1Ψ ab 0 Ψ a 1Ψ a2 b 3 Ψ ab3 1 }{{} Ψab2 0 Ψ ab2 + Ψ ab 1 Ψ b = Ψ b 0 Ψb + Ψab2 1 Ψ ab + Ψab b 3 0 Ψab3 Ψa2 1 Ψ a }{{} Ψab2 0 Ψ ab2 + Ψab 1 Ψb }{{} = Ψ b 0 Ψb + Ψab2 1 Ψ ab + Ψab3 Ψab b 3 0 Ψa2 0 Ψ ab2 + Ψa }{{} Ψab 1 Ψb }{{} = Ψ b 0 Ψb + Ψab2 1 Ψ ab + Ψab3 Ψab2 1 Ψ a2 b 3 0 Ψ ab 1 Ψab2 + Ψb 1 Ψab Ψa 1 = Ψ b 0 Ψb + Ψab2 1 Ψ ab + Ψab3 Ψab2 1 Ψ a2 b 3 0 Ψ b + Ψab2 1 Ψ ab + Ψab Ψa 1 = Ψ b 0 Ψb + Ψab2 1 Ψ ab3 Ψb 1 Ψab2 + Ψab b 3 1 Ψa2 0 Ψ ab2 1 }{{} Ψab + Ψab Ψa 1 = Ψ b 0Ψ b +Ψ b Ψ ab3 Ψ ab2 + Ψ ab2 0 Ψ a2 b 3 1 Ψ ab 0 Ψ ab = Ψ b 2 Ψab3 2 Ψ a2 b 3 1 Ψ ab 2 Ψa 1. = RHS of (19) +Ψ ab Ψ a 1 are commuting 4 Quantizing the discrete Painlevé equation There is a discretization of the Painlevé VI equation proposed by Jimbo and Sakai [JS] and later it was reformulated under the affine Weyl group symmetry of type D (1) 5 : [S], [TM]. Using the ideas in the previous sections, here we propose its quantum (non-commutative) version in a quite straightforward way. Let us introduce the Dynkin diagram of type D (1) 5 and its numbering: 0 5 \ / 2 3 / \ 1 4 We denote the corresponding generalized Cartan matrix by [a ij ] 5 i,j=0 and the simple roots by {α i } 5 0. Let q be a formal central letter (or a complex parameter, q < 1). We introduce the field of rationals K = C(a 0,,a 5 ), where a i = e α i as in the previous sections. The Weyl group W = W(D (1) 5 ) acts on K by s i (a j ) = a a ij i a j.

9 One checks that a 0 a 1 a 2 2 a2 3 a 4a 5 =: p is invariant under the action of W. Moreover, this action can be extended by the diagram automorphisms σ 01,σ 45,τ : σ 01 : a 0 a 1 1,a j a 1 j (j 0,1), τ : a j a 1 5 j We denote the extended Weyl group by W := W,σ 01,σ 45,τ. Let F = K(F,G) be the skew field, where σ 45 : a 4 a 1 5,a j a 1 j (j 4,5), (j = 0,,5). (20) FG = qgf. (21) The action of W on K can be extended to F = K(F,G) as follows. Theorem 5 We can extend the automorphisms s j (j = 0,,5) of K as algebra automorphisms of F by putting and s 2 (F) := F a 0a 1 1 G+a 2 2 a 0 a 1 1 a 2 2G+1, s j(f) := F (j 2) s 3 (G) := a2 3a 4 a 1 5 F +1 a 4 a 1 G, s 5 F +a 2 j (G) := G (j 3). 3 They give rise to a homomorphism W Aut skew field (F). Moreover, the action of the diagram automorphisms σ 01,σ 45,τ on K can be extended as involutive antiautomorphisms on F by σ 01 : F q 1 F 1, G G σ 45 : F F, G q 1 G 1 τ : F G, G F so that we have a homomorphism W Aut C lin (F). The proof is straightforward. Because of the noncommutativity, it seems inevitable to define σ 01,σ 45,τ actions on F as antiautomorphisms. For example, if we want to extend the action of τ on K to F by τ : F G (cf. [TM]), this cannot be compatible with the relation FG = qgf if we insist τ to be an automorphism: τ(f)τ(g) qτ(g)τ(f). Note that σ := σ 01 σ 45 is an automorphism of F satisfying σs j = s σ(j) τ, where We are interested in the action of (σ(0),σ(1),σ(2),σ(3),σ(4),σ(5))= (1,0,2,3,5,4). T 3 = s 2 s 1 s 0 s 2 σ 01 s 3 s 4 s 5 s 3 σ 45 = s 2 s 1 s 0 s 2 s 3 s 4 s 5 s 3 σ W, since in the commutative case this recovers the discrete Painlevé VI system [S]. Put t := a 2 3a 4 a 5, then t is invariant under s 0,s 1,s 2 s 3 s 2 = s 3 s 2 s 3,s 4,s 5 = W T 3.

10 Theorem 6 We have (T 3 (a 0 ),T 3 (a 1 ),T 3 (a 2 ),T 3 (a 3 ),T 3 (a 4 ),T 3 (a 5 )) = (a 0,a 1,pa 2,p 1 a 3,a 4,a 5 ), T 3 (t) = p 1 t, T 3 (F) = q 1 p 2 t 2 G+tp 1 a 2 1 G+t 1 pa 2 0 T 1 3 (G) = q 1 t 2 G 1 F +ta 2 4 F +t 1 a 2 5 G+tp 1 a 2 1 G+t 1 pa 2 0 F +ta 2 4 F +t 1 a 2 5 F 1 (22). (23) This T 3 flow allows the symmetry of W(D (1) 4 ), namely T 3 commutes with the subgroup s 0,s 1,s 2 s 3 s 2 = s 3 s 2 s 3,s 4,s 5,σ 01,τ W(D (1) 4 ),σ 01,τ of W. Put Z := tf,y := tg,t := ( ) 2 t p p and let us use the notation T3 (X) =: X. We have ZY = qyz, T = p 2 T and the formula (22), (23) can be rewritten as follows: ZZ = p q Y +Ta 2 1 Y +a 2 0 Y +Ta 2 1 Y +a 2 0, ȲY = 1 pq Z +Ta 2 4 Z +a 2 5 Z +Ta 2 4 Z +a 2 5. (24) This system should be regarded as a quantization of the discrete Painlevé VI equation. Likewise in the previous sections, we have the Hamiltonians for the W(D (1) 5 )- action written in terms of infinite product Ψ. Put S 2 = Ψ(a 2 2,a 0a 1 1 G)ρ 2, S 3 = Ψ(a 2 3,a 5a 1 4 F)ρ 3, S j = ρ j (j 2,3), where [ j,a k ] = a jk a k and ρ j = e π 2 1αj j. Then we have s j = Ad(S j ) for j = 0,,5. We can also find a Hamiltonian for the diagram automorphism σ = σ 01 σ 45. Let us introduce the letters j by the relation [ j,α k] = δ jk and put Σ := θ(qfg)θ(g 1 F)θ(qF) 4 1(α0 +α 1 )( ) e π 1α 2 2 e π 1α 3 3 π e e π 2 2 1(α4 +α 5 )( ). Then we have σ = Ad(Σ), that is, ΣFΣ 1 = F 1, ΣGΣ 1 = G 1 and Σa j Σ 1 = a 1 σ(j) hold. Thus we have Theorem 7 The quantum discrete Painlevé VI equation is a discrete Hamiltonian flow, T 3 = Ad(S 2 S 1 S 0 S 2 S 3 S 4 S 5 S 3 Σ). Explicitly, we have S 2 S 1 S 0 S 2 S 3 S 4 S 5 S 3 Σ = Ψ(a 2 2,a 0 a 1 1 G)ρ 2 ρ 1 ρ 0 Ψ(a 2 2,a 0 a 1 1 G)ρ 2 Ψ(a 2 3,a 1 4 a 5 F)ρ 3 ρ 4 ρ 5 Ψ(a 2 3,a 1 4 a 5 F)ρ 3 Σ = Ψ(a 2 2,a 0a 1 1 G)Ψ((a 0a 1 a 2 ) 2,a 1 0 a 1G)Ψ(p 2 (a 3 a 4 a 5 ) 2,a 4 a 1 5 F)Ψ(p2 a 2 2 a 2 3,a 4a 1 5 F) θ(qfg)θ(g 1 F)θ(qF) 4 p,

11 where = 1 2 ( 3 2 ) so that p tp = tp 1. Note that we can realize the D (1) 5 root lattice in Rδ R 5 = Rδ Re 1 Re 5 by α 0 = δ e 1 e 2, α 1 = e 1 e 2, α 2 = e 2 e 3, α 3 = e 3 e 4, α 4 = e 4 e 5, α 5 = e 4 +e 5, where e j are regarded as the orthonormal basis. Then we have t = e e 3, = / e 3. The root subsystem pependicular to e 3 is generated by α 0,α 1,α 2 + α 3 = e 2 e 4,α 4,α 5 and isomorphic to D (1) 4. As for the classical(commutative) case, the discrete Painlevé system allows rather simple, so-called seed solutions, from which one can construct rich explicit solutions via Bäcklund transformations. In our quantized system, it seems however that such seed solutions are difficult to find because of the noncommutativity. For such issues as well as the consideration of the continuous limit, we hope to discuss elsewhere. Acknowledgement. Part of this work is based on a talk at Newton institute, Cambridge (EuroConference Application of the Macdonald polynomials, April 2001) and thanks are due to the organizers, including Professor Noumi, of the conference. He also express a sincere gratitude for Gen Kuroki, Tetsuya Kikuchi and Hajime Nagoya for valuable discussions and informations. This work has been supported by the grants-in-aid for scientific research, Japan Society for the Promotion of Science, no , no References [B] J. E. Björk, Rings of differential operators, North-Holland publishing company (1979), p25 [FV] L. Faddeev and A. Volkov, Abelian current algebra and the Virasoro algebra on the lattice. Phys. Lett. B 3150, (1993) [JS] M. Jimbo and H. Sakai, A q-analog of the sixth Painlevé equation. Lett. Math. Phys. 38, (1996) [KK] K. Kajiwara, K. Kimura, On a q- difference Painlevé III equation : I. Derivation, Symmetry and Riccati Type Solutions. J. Nonlin. Math. Phys (2003) [KNY] K. Kajiwara, M. Noumi and Y. Yamada, A study on the fourth q-painlevé equation, J. Phys. A : Math. Gen. 34 (2001) [Ki] A. N. Kirillov, Dilogarithm identities. Lecture note, University of Tokyo [NY1] M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), [NY2] M. Noumi and Y. Yamada, Higher order Painlevé equations oftype A (1) l. Funkcial. Ekvac. 41 (1998), no. 3,

12 [NY3] M. Noumi and Y. Yamada, Birational Weyl group action arising from a nilpotent Poisson algebra, in Physics and combinatorics 1999, Proceedings of the Nagoya 1999 Workshop (A. N. Kirillov, A. Tsuchiya and H. Umemura eds.) , World Scientific [S] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220(2001), [TM] T.Tsuda and T. Masuda, q-painlevé VI equation arising from q-uc hierarchy, Comm. Math. Phys. 262(2006),