Drinfeld-Sokolov hierarchies of type A and fourth order Painlevé systems

Transcription

1 Drinfeld-Sokolov hierarchies of type A and fourth order Painlevé systems arxiv: v1 [math-ph] 22 Apr 2009 Kenta Fuji and Takao Suzuki Department of Mathematics Kobe University Rokko Kobe Japan suzukit@math.kobe-u.ac.jp Abstract We study the Drinfeld-Sokolov hierarchies of type A (1) n associated with the regular conjugacy classes of W(A n ). A class of fourth order Painlevé systems is derived from them by similarity reductions. 1 Introduction Three types of fourth order Painlevé type ordinary differential equations have been studied [FS NY1 S]. They are extensions of the Painlevé equations P II...P VI and expressed as Hamiltonian systems H X(1) n : (1) dq i dt = HX n p i with the Coupled Hamiltonians (1) n dp i dt = HX (i = 12) q i H A(1) 4 = H IV (q 1 p 1 ;α 2 α 1 )+H IV (q 2 p 2 ;α 4 α 1 +α 3 )+2q 1 p 1 p 2 th A(1) 5 = H V (q 1 p 1 ;α 2 α 1 α 1 +α 3 ) +H V (q 2 p 2 ;α 4 α 1 +α 3 α 1 +α 3 )+2q 1 p 1 (q 2 1)p 2 t(t 1)H D(1) 6 = H VI (q 1 p 1 ;α 0 α 3 +α 5 α 3 +α 6 α 2 (α 1 +α 2 )) +H VI (q 2 p 2 ;α 0 +α 3 α 5 α 6 α 4 (α 1 +2α 2 +α 3 +α 4 )) +2(q 1 t)p 1 q 2 {(q 2 1)p 2 +α 4 } 1

2 Lie algebra Partition Painlevé system A (1) 1 (2) P II (11) P IV A (1) 2 (3) P IV (21) P V (111) P VI A (1) 3 (4) P V A (1) 4 (5) H A(1) 4 A (1) 5 (6) H A(1) 5 Table 1: Relation between A (1) n -hierarchies and Painlevé systems where H IV (qp;ab) = qp(p q t) aq bp H V (qp;abc) = q(q 1)p(p+t)+atq +bp cqp H VI (qp;abcd) = q(q 1)(q t)p 2 {(a 1)q(q 1) +bq(q t)+c(q 1)(q t)}p+dq. But complete classification of fourth order Painlevé systems is not achieved so that the existence of unknown ones is expected. In this article we derive a class of fourth order Painlevé systems from the Drinfeld-Sokolov hierarchies of type A (1) n by similarity reductions. The Drinfeld-Sokolov hierarchies are extensions of the KdV (or mkdv) hierarchy for the affine Lie algebras [DS]. For type A (1) n they imply several Painlevé systems by similarity reductions [AS KIK KK1 KK2 NY1]; see Table 1. Such fact clarifies the origines of several properties of the Painlevé systems Lax pairs affine Weyl group symmetries and particular solutions in terms of the Schur polynomials. The Drinfeld-Sokolov hierarchies are characterized by the Heisenberg subalgebras that is maximal nilpotent subalgebras of the affine Lie algebras. And the isomorphism classes of the Heisenberg subalgebras are in one-to-one correspondence with the conjugacy classes of the finite Weyl group [KP]. In this article we choose the regular conjugacy classes of W(A n ) and consider their associated hierarchies called type I hierarchies [GHM]. In the notation of [DF] the regular conjugacy classes of W(A n ) correspond to the partitions (p...p) and (p...p1). For the derivation of fourth order Painlevé sys- 2

3 Lie algebra Partition Painlevé system A (1) 3 (22) P VI (31) H A(1) 4 A (1) 4 (41) H A(1) 5 (2 2 1) system (1.1) with (1.2) A (1) 5 (3 3) system (1.1) with (1.2) Table 2: List of Painlevé systems obtained in this article tems we investigate the partitions (22) (31) (41) (221) and (33); see Table 2. One of impotant results in this article is the derivation of a new Painlevé system. It is expressed as a Hamiltonian system dq i dt = H c p i dp i dt = H c q i (i = 12) (1.1) with a Coupled Hamiltonian t(t 1)H c = H VI (q 1 p 1 ;α 2 α 0 +α 4 α 3 +α 5 ηηα 1 ) +H VI (q 2 p 2 ;α 0 +α 2 α 4 α 1 +α 3 ηηα 5 ) +(q 1 t)(q 2 1){(q 1 p 1 +α 1 )p 2 +p 1 (p 2 q 2 +α 5 )}. (1.2) This system admits affine Weyl group symmetry of type A (1) 5 ; see Appendix B. On the other hand the system H D(1) 6 admits one of type D (1) 6. The relation between those two coupled Painlevé VI systems is not clarified. Remark 1.1. For the partition (1...1) of n + 2 we have the Garnier system in n-variables [KK2]. Also for each partition (51) and (222) a system of sixth order is derived; we do not give the explicit formula here. Thus we conjecture that any more fourth order Painlevé system do not arise from the type I hierarchy. This article is organized as follows. In Section 2 we recall the affine Lie algebra of type A (1) n and realize it in a flamework of a central extension of the loop algebra sl n+1 [zz 1 ]. In Section 3 the Heisenberg subalgebra of ŝl n+1 corresponding to the partition n is introduced. In Section 4 we formulate the Drinfeld-Sokolov hierarchies and their similarity reductions. In Section 5 and 6 the Painlevé systems are derived from the Drinfeld-Sokolov hierarchies. In 3

4 Appendix A we give explicit descriptions of Lax pairs by means of a bases of ŝl n+1. In Appendix B we discuss a group of symmetries for the system (1.1) with (1.2). 2 Affine Lie algebra In this section we recall the affine Lie algebra of type A (1) n and realize it in a flamework of a central extension of the loop algebra sl n+1 [zz 1 ]. In the notation of [Kac] the affine Lie algebra g = g(a (1) n ) is generated by the Chevalley generators e i f i αi (i = 0...n) and the scaling element d with the fundamental relations (ade i ) 1 a ij (e j ) = 0 (adf i ) 1 a ij (f j ) = 0 (i j) [α i α j ] = 0 [α i e j] = a ij e j [α i f j] = a ij f j [e i f j ] = δ ij α i [dα i ] = 0 [de i] = δ i0 e 0 [df i ] = δ i0 f 0 for ij = 0...n. The generalized Cartan matrix A = [a ij ] n ij=0 defined by for g is a ii = 2 (i = 0...n) a ii+1 = a n0 = a i+1i = a 0n = 1 (i = 0...n 1) a ij = 0 We denote the Cartan subalgebra of g by (otherwise). h = Cα 0 Cα 1 Cα n Cd = h Cd. The normalized invariant form ( ) : g g C is determined by the conditions (α i α j) = a ij (e i f j ) = δ ij (α i e j ) = (α i f j ) = 0 (d d) = 0 (d α j ) = δ 0j (d e j ) = (d f j ) = 0 for ij = 0...n. Letn + andn bethesubalgebrasofggeneratedbye i andf i (i = 0...n) respectively. Then the Borel subalgebra b + of g is defined by b + = h n +. Note that we have the triangular decomposition g = n h n + = n b +. The corresponding infinite demensional groups are defined by N ± = exp(n ± ) H = exp(h ) B + = HN + 4

5 where n ± are completions of n ± respectively. Let s = (s 0...s n ) be a vector of non-negative integers. We consider a gradation g = k Z g k(s) of type s by setting degh = 0 dege i = s i degf i = s i (i = 0...n). With an element ϑ(s) h such that (ϑ(s) α i ) = s i (i = 0...n) this gradation is defined by We denote by g k (s) = { x g [ϑ(s)x] = kx } (k Z). g <k (s) = l<k g l (s) g k (s) = l k g l (s). Note that a gradation s p = (1...1) called the principal gradation implies g <0 (s p ) = n g 0 (s p ) = b +. The affine Lie algebra g can be identified with ŝl n+1 = sl n+1 [zz 1 ] Cz d dz CK wherek isacanonicalcentralelement. Inaflameworkofŝl n+1 thechevalley generators and the scaling element are given by e i = E ii+1 f i = E i+1i α i = E ii E i+1i+1 (i = 1...n) e 0 = ze n+11 f 0 = z 1 E 1n+1 α 0 = E n+1n+1 E 11 +K d = z d dz where E ij = (δ ir δ js ) n+1 rs=1 are matrix units. The Lie bracket is defined by where XY sl n+1. [z k Xz l Y] = z k+l (XY YX)+kδ k+l0 tr(xy)k 5

6 3 Heisenberg subalgebra For type A (1) n the isomorphism classes of the Heisenberg subalgebras are in one-to-one correspondence with the partitions of n + 1. In this section we introduce the Heisenberg subalgebra of ŝl n+1 corresponding to the partition n following the manner in [KL]. Let n = (n 1 n 2...n r n r+1...n s ) be a partition of n + 1 with n 1 n 2... n r > n r+1 =... = n s = 1. Consider a partition of matrix corresponding to n B 11 B 12 B 1s B 21 B 22 B 2s B s1 B s2 B ss where each block B ij is an n i n j -matrix. With this blockform we define matricies Λ i ŝl n+1 (i = 1...r) by O O Λ i =. B ii. B ii = O O z diagonal matricies H j ŝl n+1 (i = j...s 1) by and a diagonal matrix η n ŝl n+1 by H j = n j+1 z 1 (Λ j) n j n j z 1 (Λ j+1) n j+1 B ii = 1 2n i diag(n i 1n i 3... n i +1) (i = 1...r). Denoting the matrix η n by diag(η 1 η 2...η n+1 ) we consider a permutation ( ) η 1 η 2... η n+1 σ = η 1 η 2... η n+1 such that η 1 η 2... η n+1. This permutation can be lifted to the transformation σ acting on the matricies Λ i and H j. We set Λ i = σ(λ i) (i = 1...r) H j = σ(h j) (j = 1...s 1). 6

7 Then the Heisenberg subalgebra of ŝl n+1 corresponding to the partition n is defined by s n = r i=1 k Z\n i Z CΛ k i s 1 j=1 k Z\{0} Cz k H j CK. Let N n be the least common multiple of n 1...n s. Also let ( 1 N n if N n + 1 ) 2Z for (ij) N n = n i n j. 2N n otherwise We consider a operator corresponding to n ( ϑ n = N n z d ) dz +adη n where η n = σ(η n ). Then the operator ϑ n implies a gradation s = (s 0...s n ) as follows: ϑ n (e i ) = s i e i (i = 0...n). Note that the Heisenberg subalgebra s n admits the gradation s defined by ϑ n. 4 Drinfeld-Sokolov hierarchy In this section we formulate the Drinfeld-Sokolov hierarchy associated with the Heisenberg subalgebra s n. Its similarity reduction is also formulated. Let Λ i and H j be the generators for s n given in Section 3. Introducing timevariablest ik (i = 1...r;k N) we consider ann B + -valuedfunction G = G(t 11 t 12...) defined by ( r ) G = exp t ik Λ k i G(0). i=1 k=1 Here we assume the n-reduced condition t il = 0 (i = 1...r;l n i N). Then we have a system of partial differential equations ik (G) = Λ k ig (i = 1...r;k N) (4.1) 7

8 where ik = / t ik Via the trianglar decomposition G = W 1 Z W N Z B + the system (4.1) implies a Sato equation ik (W) = B ik W WΛ k i (i = 1...r;k N) (4.2) where B ik stands for the b + -component of WΛ k i W 1. The compatibility condition of (4.2) gives the Drinfeld-Sokolov hierarchy [ ik B ik jl B jl ] = 0 (ij = 1...r;kl N). (4.3) Under the system (4.2) we consider an equation (ϑ n adρ)(w) = r d i kt ik ik (W) (4.4) i=1 k=1 where d i = degλ i (i = 1...r) and ρ = s 1 j=1 ρ jh j. Note that each ρ j is independent of time vatiables t ik. The compatibility condition of (4.2) and (4.4) gives where [ϑ n M ik B ik ] = 0 (i = 1...r;k N) (4.5) M = ρ+ r i=1 d i kt ik B ik. k=1 We call the systems (4.3) and (4.5) a similarity reduction of the Drinfeld- Sokolov hierarchy. Remark 4.1. The similarity reduction can be regarded as the compatibility condition of a Lax form ik (Ψ) = B ik Ψ (i = 1...r;k N) ϑ n (Ψ) = MΨ. Here an N B + -valued function Ψ is given by ( r ) Ψ = W exp t ik Λ k i. i=1 k=1 5 Derivation of Coupled P VI In this section we derive the Painlevé system (1.1) with (1.2) from the Drinfeld-Sokolov hierarchies for s (33) and s (221) by similarity reductions. 8

9 5.1 For the partition (3 3) At first we define the Heisenberg subalgebra s (33) of g(a (1) 5 ). Let where Λ 1 = e 12 +e 34 +e 50 Λ 2 = e 01 +e 23 +e 45 H 1 = α 1 +α 3 +α 5 Then we have s (33) = e i1 i 2...i n 1 i n = ade i1 ade i2...ade in 1 (e in ). k Z\3Z CΛ k 1 k Z\3Z The grade operator for s (33) is given by CΛ k 2 k Z\{0} ( ϑ (33) = 3 z d ) dz +adη (33) Cz k H 1 CK. where η (33) = 1 3 (α 1 +2α 2 +2α 3 +2α 4 +α 5 ). It follows that s (33) admits the gradation of type s = (101010) namely Note that ϑ (33) (e i ) = e i (i = 024) ϑ (33) (e j ) = 0 (j = 135). g 0 (101010) = Cf 1 Cf 3 Cf 5 b +. We now assume t 21 = 1 and t 1k = t 2k = 0 (k 2). Then the similarity reduction (4.3) and (4.5) for s (33) is expressed as [ ϑ(33) M 11 B 11 ] = 0. (5.1) Here the b + -valued functions M and B 11 are defined by M = ϑ (33) (W)W 1 +W(ρ 1 H 1 +t 11 Λ 1 +Λ 2 )W 1 B 11 = 11 (W)W 1 +WΛ 1 W 1 (5.2) where W is an N -valued function; its explicit formula is given below. In the following we derive the Painlevé system from the system (5.1) with (5.2). We denote by W = exp(ω 0 )exp(ω 1 )exp(ω < 1 ) 9

10 where ω 0 = w 1 f 1 w 3 f 3 w 5 f 5 ω 1 = w 0 f 0 w 2 f 2 w 4 f 4 w 01 f 01 w 12 f 12 w 23 f 23 w 34 f 34 w 45 f 45 w 50 f 50 w 123 f 123 w 345 f 345 w 501 f 501 and ω < 1 g < 1 (101010). Then the b + -valued function M is described as M = κ 0 α 0 +κ 1α 1 +κ 2α 2 +κ 3α 3 +κ 4α 4 +κ 5α 5 (t 11w 5 w 1 )e 0 +ϕ 1 e 1 (t 11 w 1 w 3 )e 2 +ϕ 3 e 3 (t 11 w 3 w 5 )e 4 +ϕ 5 e 5 +t 11 Λ 1 +Λ 2 with dependent variables and parameters ϕ 1 = t 11 w 2 w 0 ϕ 3 = t 11 w 4 w 2 ϕ 5 = t 11 w 0 w 4 κ 0 = t 11 w 50 w 01 κ 1 = t 11 (w 1 w 2 w 12 ) (w 0 w 1 +w 01 )+ρ 1 κ 2 = t 11 w 12 w 23 κ 3 = t 11 (w 3 w 4 w 34 ) (w 2 w 3 +w 23 )+ρ 1 κ 4 = t 11 w 34 w 45 κ 5 = t 11 (w 0 w 5 w 50 ) (w 4 w 5 +w 45 )+ρ 1. Note that We also remark that 11 (κ i ) = 0 (i = 0...5). w 1 ϕ 1 +w 3 ϕ 3 +w 5 ϕ 5 +κ 0 κ 1 +κ 2 κ 3 +κ 4 κ 5 +3ρ 1 = 0. The b + -valued function B 11 is described as where B 11 = u 0 K +(u 1 +w 1 x 1 )α 1 +u 2α 2 +(u 3 +w 3 x 3 )α 3 +u 4α 4 +w 5 x 5 α 5 w 5 e 0 +x 1 e 1 w 1 e 2 +x 3 e 3 w 3 e 4 +x 5 e 5 +Λ 1 u 1 = 2w 1ϕ 1 +w 3 ϕ 3 +w 5 ϕ 5 2κ 0 +2κ 1 +κ 2 κ 3 +κ 4 κ 5 3t 11 u 2 = w 1ϕ 1 +κ 0 κ 1 +ρ 1 t 11 u 3 = w 1ϕ 1 w 3 ϕ 3 +2w 5 ϕ 5 κ 0 +κ 1 κ 2 +κ 3 +2κ 4 2κ 5 3t 11 u 4 = w 5ϕ 5 +κ 4 κ 5 +ρ 1 t 11 x 1 = t2 11ϕ 1 +t 11 ϕ 5 +ϕ 3 t x 3 = t2 11ϕ 3 +t 11 ϕ 1 +ϕ 5 t x 5 = t2 11ϕ 5 +t 11 ϕ 3 +ϕ 1. t

11 Hence the system (5.1) with (5.2) can be expressed as a system of ordinary differential equations in terms of the variabes ϕ 1 ϕ 5 w 1 w 3 w 5 ; we do not give its explicit formula. Let q 1 = w 1 p t 2 1 = t2 11 w 3ϕ 1 q 2 = w 5 p 2 = t 11w 3 ϕ 5 t = 1. 11w 3 3 t 11 w 3 3 t 3 11 We also set α 0 = 1 3 (1 2κ 0 +κ 1 +κ 5 ) α 1 = 1 3 (κ 0 2κ 1 +κ 2 ) α 2 = 1 3 (1+κ 1 2κ 2 +κ 3 ) α 3 = 1 3 (κ 2 2κ 3 +κ 4 ) α 4 = 1 3 (1+κ 3 2κ 4 +κ 5 ) α 5 = 1 3 (κ 0 +κ 4 2κ 5 ) and η = ρ (α 1 +α 3 +α 5 ). Then we have Theorem 5.1. The system (5.1) with (5.2) gives the Painlevé system (1.1) with (1.2). Furthermore w 3 satisfies the completely integrable Pfaffian equation t(t 1) d dt logw 3 = (q 1 1)(q 1 t)p 1 (q 2 1)(q 2 t)p 2 α 1 q 1 α 5 q (α 1 +α 2 α 3 α 4 +2η)t 1 3 (α 1 +α 2 +2α 3 α 4 4η). 5.2 For the partition (221) The Heisenberg subalgebra s (221) of g(a (1) 4 ) is defined by s (221) = CΛ k 1 CΛ k 2 Cz k H 1 k Z\2Z k Z\2Z k Z\{0} k Z\{0} Cz k H 2 CK with Λ 1 = e 40 +e 123 Λ 2 = e 01 +e 234 H 1 = α 1 +α 2 α 3 H 2 = α 2 +α 3 +α 4. 11

12 The subalgebra s (221) admits the gradation of type s = (20110) with the grade operator ( ϑ (221) = 4 z d ) dz +adη (221) η (221) = 1 4 (α 1 +2α 2 +2α 3 +α 4 ). Note that g 0 (20110) = Cf 1 Cf 4 b +. We now assume t 12 = 1 and t 1k = t 2k = 0 (k 3). Then the similarity reduction (4.5) for s (221) is expressed as [ ϑ(221) M 11 B 11 ] = 0 (5.3) with M = ϑ (221) (W)W 1 +W(ρ 1 H 1 +ρ 2 H 2 +2t 11 Λ 1 +2Λ 2 )W 1 B 11 = 11 (W)W 1 +WΛ 1 W 1. (5.4) Let where W = exp(ω 0 )exp(ω 1 )exp(ω 2 )exp(ω < 2 ) ω 0 = w 1 f 1 w 4 f 4 ω 1 = w 2 f 2 w 3 f 3 w 12 f 12 w 34 f 34 ω 2 = w 0 f 0 w 01 f 01 w 23 f 23 w 40 f 40 w 123 f 123 w 234 f 234 w 401 f 401 w 1234 f 1234 and ω < 2 g < 2 (20110). Then the system (5.4) gives explicit formulas of MB 11 as follows: where M = κ 0 α 0 +κ 1α 1 +κ 2α 2 +κ 3α 3 +κ 4α 4 +2(w 1 t 11 w 4 )e 0 +ϕ 1 e 1 +(ϕ 2 w 1 ϕ 12 )e 2 +(ϕ 3 +w 4 ϕ 34 )e 3 +ϕ 4 e 4 +ϕ 12 e 12 +2(t 11 w 1 w 4 )e 23 ϕ 34 e 34 +2t 11 Λ 1 +2Λ 2 B 11 = u 0 K +(u 2 +w 1 x 1 )α 1 +u 2α 2 +u 3α 3 +w 4x 4 α 4 w 4e 0 +x 1 e 1 w 1 x 12 e 2 + ϕ 3 2t 11 e 3 +x 4 e 4 +x 12 e 12 w 1 e 23 +Λ 1 ϕ 1 = 2w 0 +t 11 w 2 w 3 2t 11 w 23 ϕ 2 = 2w 34 ϕ 3 = 2t 11 w 12 ϕ 4 = 2t 11 w 0 +w 2 w 3 +2w 23 ϕ 12 = 2t 11 w 3 ϕ 34 = 2w 2 12

13 and u 2 = w 1ϕ 1 +κ 0 κ 1 +ρ 1 2t 11 u 3 = w 4ϕ 4 +κ 3 κ 4 +ρ 1 2t 11 x 1 = (t 11ϕ 1 +ϕ 4 )ϕ 3 +(w 1 ϕ 1 +w 4 ϕ 4 +κ 0 κ 1 +κ 3 κ 4 +2ρ 1 )ϕ 34 2(t )ϕ 3 x 4 = (ϕ 1 +t 11 ϕ 4 )ϕ 3 +t 11 (w 1 ϕ 1 +w 4 ϕ 4 +κ 0 κ 1 +κ 3 κ 4 +2ρ 1 )ϕ 34 2(t )ϕ 3 x 12 = w 1ϕ 1 +w 4 ϕ 4 +κ 0 κ 1 +κ 3 κ 4 +2ρ 1 ϕ 3. Note that κ 0...κ 4 are constants. We also remark that ϕ 2 ϕ 34 +2(w 1 ϕ 1 +w 4 ϕ 4 +κ 0 κ 1 +κ 2 κ 4 +2ρ 2 ) = 0 ϕ 3 ϕ 12 2t 11 (w 1 ϕ 1 +w 4 ϕ 4 +κ 0 κ 1 +κ 3 κ 4 +2ρ 1 ) = 0. Hence the system (5.3) can be expressed as a system of ordinary differential equations in terms of the variables ϕ 1 ϕ 3 ϕ 4 ϕ 34 w 1 w 4. Let We also set Then we have q 1 = t2 11 ϕ 34w 4 p 1 = ϕ 3ϕ 4 ϕ 3 4t 2 11 ϕ 34 q 2 = t 11ϕ 34 w 1 ϕ 3 p 2 = ϕ 3ϕ 1 4t 11 ϕ 34 t = t α 0 = 1 4 (2 2κ 0 +κ 1 +κ 4 ) α 1 = 1 4 (κ 0 +κ 3 2κ 4 ) α 2 = 1 4 (1+κ 2 2κ 3 +κ 4 ) α 3 = 1 4 ( κ 2 +κ 3 +2ρ 1 2ρ 2 ) α 4 = 1 4 (1+κ 1 κ 2 2ρ 1 +2ρ 2 ) α 5 = 1 4 (κ 0 2κ 1 +κ 2 ) η = 1 4 (2κ 0 2κ 1 +2κ 3 2κ 4 +3ρ 1 ρ 2 ). Theorem 5.2. The system (5.3) with (5.4) gives the Painlevé system (1.1) with (1.2). Furthermore ϕ 3 and ϕ 34 satisfy the completelyintegrable Pfaffian 13

14 equations t(t 1) d dt logϕ 3 = q 1 (q 1 t)p 1 q 2 (q 2 t)p 2 α 1 q 1 α 5 q (1+2α 2 2α 3 2α 4 2α 5 +6η)t 1 4 (1+2α 2 +2α 3 2α 4 2α 5 +2η) t(t 1) d dt logϕ 34 = (q 1 t)p 1 (q 2 t)p 2 η. 6 Derivation of other systems In this section we discuss the derivation of the Painlevé systems for s (22) s (31) and s (41) by a similar manner as in Section For the partition (2 2) The Heisenberg subalgebra s (22) of g(a (1) 3 ) is defined by s (22) = CΛ k 1 CΛ k 2 Cz k H 1 CK with k Z\2Z k Z\2Z k Z\{0} Λ 1 = e 12 +e 30 Λ 2 = e 01 +e 23 H 1 = α 1 +α 3. The subalgebra s (22) admits the gradation of type s = (1010) with the grade operator ( ϑ (22) = 2 z d ) dz +adη (22) η (22) = 1 2 (α 1 +2α 2 +α 3 ). Note that g 0 (1010) = Cf 1 Cf 3 b +. We now assume t 12 = 1 and t 1k = t 2k = 0 (k 3). Then the similarity reduction (4.5) for s (22) is expressed as [ ϑ(22) M 11 B 11 ] = 0 (6.1) with M = ϑ (22) (W)W 1 +W(ρ 1 H 1 +t 11 Λ 1 +Λ 2 )W 1 B 11 = 11 (W)W 1 +WΛ 1 W 1. (6.2) 14

15 Let where W = exp(ω 0 )exp(ω 1 )exp(ω < 1 ) ω 0 = w 1 f 1 w 3 f 3 ω 1 = w 0 f 0 w 2 f 2 w 02 f 02 w 12 f 12 w 23 f 23 w 30 f 30 w 123 f 123 w 301 f 301 and ω < 1 g < 1 (1010). Then the system (6.2) gives explicit formulas of MB 11 as follows: M = κ 0 α 0 +κ 1 α 1 +κ 2 α 2 +κ 3 α 3 +(w 1 t 11 w 3 )e 0 +ϕ 1 e 1 +(w 3 t 11 w 1 )e 2 +ϕ 3 e 3 +t 11 Λ 1 +Λ 2 B 11 = u 0 K +u 1 α 1 +u 2α 2 +w 3x 3 α 3 +w 1e 0 +x 1 e 1 +w 3 e 2 +x 3 e 3 +Λ 1 where and ϕ 1 = t 11 w 2 w 0 ϕ 3 = t 11 w 0 w 2 u 1 = w 1 t 11 x 3 κ 0 κ 1 +ρ 1 t 11 u 2 = w 3ϕ 3 +κ 2 κ 3 +ρ 1 t 11 x 1 = (w 1 t 11 w 3 )ϕ 3 (κ 0 κ 1 +κ 2 κ 3 +2ρ 1 )t 11 (t )w 1 x 3 = (t 11w 1 w 3 )ϕ 3 (κ 0 κ 1 +κ 2 κ 3 +2ρ 1 ) (t )w. 1 Note that κ 0...κ 3 are constants. We also remark that w 1 ϕ 1 +w 3 ϕ 3 +κ 0 κ 1 +κ 2 κ 3 +2ρ 1 = 0. Hence the system (6.1) can be expressed as a system of ordinary differential equations in terms of the variables ϕ 3 w 1 w 3. Let p = w 1ϕ 3 q = t 11w 3 t = t t 11 w. 1 We also set α 0 = 1 2 (1+κ 1 2κ 2 +κ 3 ) α 1 = 1 2 ( κ 1 +κ 3 +2ρ 1 ) α 2 = κ 0 +κ 2 2κ 3 α 3 = 1 2 (1 2κ 0 +κ 1 +κ 3 ) α 4 = 1 2 ( κ 1 +κ 3 2ρ 1 ) 15

16 and Then we have a = α 0 b = α 3 c = α 4 d = α 2 (α 1 +α 2 ). Theorem 6.1. The system (6.1) with (6.2) gives the sixth Painlevé equation. Furthermore w 1 satisfies the completely integrable Pfaffian equation t(t 1) d dt logw 1 = (q 1)(q t)p α 2 q (1+2α 1 2α 3 4α 4 )t 1 4 (1 2α 1 4α 2 2α 3 ). 6.2 For the partition (3 1) The Heisenberg subalgebra s (31) of g(a (1) 3 ) is defined by s (31) = CΛ k 1 Cz k H 1 CK with k Z\3Z k Z\{0} Λ 1 = e 0 +e 1 +e 23 H 1 = α 1 +2α 2 α 3. The subalgebra s (31) admits the gradation of type s = (1101) with the grade operator ( ) d ϑ (31) = 3z dz +adη (31) η (31) = 1 3 (α 1 +α2 +α3). Note that g 0 (1101) = Cf 2 b +. We now assume t 12 = 1 and t 1k = 0 (k 3). Then the similarity reduction (4.5) for s (31) is expressed as [ ϑ(31) M 11 B 11 ] = 0 (6.3) with M = ϑ (31) (W)W 1 +W(ρ 1 H 1 +t 11 Λ 1 +2Λ 2 1)W 1 B 11 = 11 (W)W 1 +WΛ 1 W 1. (6.4) Let W = exp( w 2 f 2 )exp(ω 1 )exp(ω 2 )exp(ω < 2 ) 16

17 where ω 1 = w 0 f 0 w 1 f 1 w 3 f 3 w 12 f 12 w 23 f 23 ω 2 = w 01 f 01 w 30 f 30 w 012 f 012 w 123 f 123 w 230 f 230 and ω < 2 g < 2 (1101). Then the system (6.4) gives explicit formulas of MB 11 as follows: M = κ 0 α 0 +κ 1 α 1 +κ 2 α 2 +κ 3 α 3 +ϕ 0 e 0 +(ϕ 1 +w 2 ϕ 12 )e 1 +ϕ 2 e 2 +(ϕ 3 w 2 ϕ 23 )e 3 +ϕ 12 e 12 +ϕ 23 e 23 2w 2 e 30 +2Λ 2 1 B 11 = u 3 K ϕ 1 t 11 2 where α 0 + ϕ 0 t 11 2 α1 + w 2ϕ 12 α2 + ϕ e 2 w 2 e 3 +Λ 1 ϕ 0 = 2w 1 +2w 23 +t 11 ϕ 1 = 2w 0 2w 23 +t 11 ϕ 2 = (w 0 2w 1 +t 11 )w 3 2w 30 ϕ 3 = 2w 12 ϕ 12 = 2w 3 ϕ 23 = 2w 0 2w 1 +t 11. Note that κ 0...κ 4 are constants. We also remark that 2w 2 ϕ 2 ϕ 3 ϕ 12 = 2(κ 2 κ 3 3ρ 1 ) ϕ 0 +ϕ 1 +ϕ 23 = 3t 11. Hence the system (6.3) can be expressed as a system of ordinary differential equations in terms of the variables ϕ 0 ϕ 1 ϕ 2 ϕ 12 w 2. Let q 1 = w 2ϕ 12 6 p 1 = 2ϕ 2 6ϕ12 q 2 = ϕ 1 6 p 2 = ϕ 0 6 t = We also set Then we have α 1 = 1 3 (κ 2 κ 3 3ρ 1 ) α 2 = 1 3 (κ 1 2κ 2 +κ 3 ) α 3 = 1 3 (1+κ 0 2κ 1 +κ 2 ) α 4 = 1 3 (1 2κ 0 +κ 1 +κ 3 ). 6t11 Theorem 6.2. The system (6.3) with (6.4) gives the Painlevé system H A(1) 4. Furthermore ϕ 12 satisfies the completely integrable Pfaffian equation d dt logϕ 12 = p 1 +p t

18 6.3 For the partition (4 1) The Heisenberg subalgebra s (41) of g(a (1) 4 ) is defined by s (41) = CΛ k 1 Cz k H 1 CK with k Z\4Z k Z\{0} Λ 1 = e 0 +e 1 +e 4 +e 23 H 1 = α 1 +2α 2 2α 3 α 4. The subalgebra s (41) admits the gradation of type s = (22112) with the grade operator ( ϑ (41) = 8 z d ) dz +adη (41) η (41) = 1 8 (3α 1 +4α 2 +4α 3 +3α 4 ). Note that g 0 (22112) = b +. We now assume t 12 = 1 and t 1k = 0 (k 3). Then the similarity reduction (4.5) for s (41) is expressed as [ ϑ(41) M 11 B 11 ] = 0 (6.5) with Let where M = ϑ (41) (W)W 1 +W(ρ 1 H 1 +2t 11 Λ 1 +4Λ 2 1 )W 1 B 11 = 11 (W)W 1 +WΛ 1 W 1. W = exp(ω 1 )exp(ω 2 )exp(ω 3 )exp(ω 4 )exp(ω < 4 ) ω 1 = w 2 f 2 w 3 f 3 ω 2 = w 0 f 0 w 1 f 1 w 4 f 4 w 23 f 23 ω 3 = w 12 f 12 w 34 f 34 ω 4 = w 01 f 01 w 40 f 40 w 123 f 123 w 234 f 234 (6.6) and ω < 4 g < 4 (22112). Then the system (6.6) gives explicit formulas of MB 11 as follows: M = κ 0 α 0 +κ 1α 1 +κ 2α 2 +κ 3α 3 +κ 4α 4 +ϕ 0e 0 +ϕ 1 e 1 +ϕ 2 e 2 +ϕ 3 e 3 +ϕ 4 e 4 +ϕ 12 e 12 +ϕ 23 e 23 +ϕ 34 e 34 +4Λ 2 1 B 11 = u 4 K +u 0 α0 + ϕ 0 2t 11 α1 +u 2 α2 +u 3 α3 + ϕ e 2 + ϕ 34 4 e 3 +Λ 1 18

19 where ϕ 0 = 4w 1 4w 4 +2t 11 ϕ 1 = 4w 0 +2w 2 w 3 4w 23 +2t 11 ϕ 2 = 2(2w 1 w 4 t 11 )w 3 4w 34 ϕ 3 = 2(w 1 2w 4 t 11 )w 2 +4w 12 ϕ 12 = 4w 3 ϕ 23 = 4w 1 +4w 4 +2t 11 ϕ 34 = 4w 2 and 64t 11 u 0 = (ϕ 0 4t 11 )(4ϕ 1 +ϕ 12 ϕ 34 )+4ϕ 2 ϕ t (κ 0 κ 1 +κ 2 κ 4 2ρ 1 ) 64t 11 u 2 = ϕ 0 (4ϕ 1 +ϕ 12 ϕ 34 )+4(ϕ 2 t 11 ϕ 12 )ϕ 34 16t (κ 0 κ 1 +κ 2 κ 4 2ρ 1 ) 64t 11 u 3 = ϕ 0 (4ϕ 1 +ϕ 12 ϕ 34 )+4ϕ 2 ϕ 34 16t (κ 0 κ 1 +κ 2 κ 4 2ρ 1 ). Note that κ 0...κ 4 are constants. We also remark that (ϕ 0 4t 11 )ϕ 12 ϕ 34 +4ϕ 3 ϕ 12 +4ϕ 2 ϕ 34 = 16( κ 2 +κ 3 +4ρ 1 ) 4ϕ 1 +4ϕ 4 +ϕ 12 ϕ 34 = 16t 11 ϕ 0 +ϕ 23 = 4t 11. Hence the system (6.5) can be described as a system of ordinary differential equations in terms of the variables ϕ 0 ϕ 1 ϕ 2 ϕ 12 ϕ 34. Let We also set Then we have q 1 = ϕ 0 p 1 = t 11ϕ 1 4t 11 8 q 2 = ϕ 0 4t 11 + ϕ 2 p 2 = t 11ϕ 12 ϕ 34 t 11 ϕ t = t α 1 = 1 8 (2 2κ 0 +κ 1 +κ 4 ) α 2 = 1 8 (2+κ 0 2κ 1 +κ 2 ) α 3 = 1 8 (1+κ 1 2κ 2 +κ 3 ) α 4 = 1 8 (κ 2 κ 3 4ρ 1 ) α 5 = 1 8 (1 κ 3 +κ 4 +4ρ 1 ). Theorem 6.3. The system (6.5) with (6.6) gives the Painlevé system H A(1) 5. Furthermore ϕ 12 satisfies the completely integrable Pfaffian equation t d dt logϕ 12 = q 1 p 1 q 2 p 2 +tq t 1+2α 1 +2α 3 +2α

20 A Lax pair In the previous section we have derived several Painlevé systems. Each of them can be regarded as the compatibility condition of a Lax pair (see Remark 4.1) dψ dt = BΨ ϑ n(ψ) = MΨ. In this section we give an explicit description of M and B by means of a bases of sl n+1 [zz 1 ]. A.1 For the partition (2 2) The matrix M is described as follows: ε 1 2(qp+α 1+α 2 ) w 1 t 0 w 0 ε 1 (q t) 2 M = t 1 tz 0 2 tp ε3 w 1 w 1 (1 q)z z 0 ε 4 where ε 1...ε 4 are linear conbinations of α 0...α 3. The matrix B is expressed as follows: u 1 u 0 x B = u 2 u 1 x 2 0 t z 0 u 3 u 2 x 3. x 0 z 0 0 u 0 u 3 EachcomponentofB isrationalinqpw 1 ; seesection6.1. Thecompatibility condition of this Lax pair gives the sixth Painlevé equation. Remark A.1. It is known that P VI arises from the Lax pairs of two types 2 2 matrix system [IKSY] and 8 8 matrix system [NY3]. The result of this section means that we derive a new Lax pair for P VI. A.2 For the partition (3 1) The matrix M is described as follows: ε 1 6(q2 q 1 ) ϕ z ε 2 6ϕ12 p 1 6(p2 q M = 2 2 t) 2 6q 1 6{q ϕ 12 z 0 ε 1 (p 1 +p 2 q 2 t) α 1 } 3 ϕ 12 6p 2 z 2z 0 ε 4 20

21 where ε 1...ε 4 are linear conbinations of α 0...α 3. The matrix B is expressed as follows: u 1 u B = 2 0 u 2 u 1 x u 3 u 2 x 3. z 0 0 u 0 u 3 Each component of B is rational in q 1 p 1 q 2 p 2 ϕ 12 ; see Section 6.2. The compatibility condition of this Lax pair gives the Painlevé system H A(1) 4. Note that the system H A(1) 4 also arise from the Lax pair by means of 5 5 matricies [NY1]. A.3 For the partition (4 1) The matrix M is described as follows: M = ε 1 8p 1 2t ϕ ε 2 2tϕ12 (q 2 q 1 ) 4 2t(1 q 1 ) ε 3 32{(1 q 2 )p 2 α 4 } ϕ 12 32p 2 2tϕ12 4z 0 0 ε 4 8(p 1+p 2 +t) 2t 4 2tq 1 z 4z 0 0 ε 5 where ε 1...ε 5 are linear conbinations of α 0...α 4. The matrix B is expressed as follows: u 1 u B = 1 0 u 2 u 1 x t 0 0 u 3 u 2 x u 4 u 3 1. z u 0 u 4 Each component of B is rational in q 1 p 1 q 2 p 2 ϕ 12 ; see Section 6.3. The compatibility condition of this Lax pair gives the Painlevé system H A(1) 5. Note that the system H A(1) 5 also arise from the Lax pair by means of 6 6 matricies [NY1]. 21

22 A.4 For the partition (221) The matrix M is described as follows: 0 4 tϕ 34 p 2 8 t(q1 p 1 +q 2 p 2 +η) ϕ 3 ϕ 3 2 t 0 2ϕ 0 0 ϕ 3 (tq 2 q 1 ) 2 tϕ 34 2 M = ϕ (t q 1 ) ϕ s 2 tz tϕ 34p 1 ϕ 3 2ϕ 3 (q 1 q 2 ) tϕ34 z 2z where ε 1...ε 5 are linear conbinations of α 0...α 4 and ϕ 2 = 8{(q 2 1)(q 1 p 1 +q 2 p 2 +η)+α 3 } ϕ 34. The matrix B is expressed as follows: u 1 u 0 x 1 x B = u 2 u 1 x 2 x 23 0 t 0 0 u 3 u 2 x 3 0 z 0 0 u 4 u 3 x 4. x 0 z u 0 u 4 Each component of B is rational in q 1 p 1 q 2 p 2 ϕ 3 ϕ 34 ; see Section 5.2. The compatibility condition of this Lax pair gives the system (1.1) with (1.2). A.5 For the partition (3 3) The matrix M is described as follows: 3t ε 2/3 p w t 1/3 w 0 ε 3 (t q 1 ) t M = 0 0 ε 3 3(q 1p 1 +q 2 p 2 +η) 1 w 3 0 t 1/3 w ε 3 (q 2 1) 4 1 t 1/3 1 3t z ε 1/3 p 2 t 1/3 5 w 3 w 3 (q 1 q 2 ) z z ε t 2/3 6 where ε 1...ε 6 are linear conbinations of α 0...α 5. The matrix B is expressed as follows: u 1 u 0 x B = 1 0 u 2 u 1 x u 3 u 2 x t 4/ u 4 u 3 x 4 0. z u 5 u 4 x 5 x 0 z u 0 u 5 22

23 Each component of B is rational in q 1 p 1 q 2 p 2 w 3 ; see Section 5.1. The compatibility condition of this Lax pair gives the system (1.1) with (1.2). B Affine Weyl group symmetry The system (1.1) with (1.2) admits affine Weyl group symmetry of type A (1) 5. In this section we describe its action on the dependent variables and parameters. Let r i (i = 0...5) be birational canonical transformations defined by for i = 0; for i = 1; for i = 2; for i = 3; for i = 4; α 0 α 0 α 1 α 0 +α 1 α 5 α 0 +α 5 p 1 p 1 α 0 q 1 q 2 p 2 p 2 α 0 q 2 q 1 α 0 α 0 +α 1 α 1 α 1 α 2 α 1 +α 2 q 1 q 1 + α 1 p 1 α 1 α 1 +α 2 α 2 α 2 α 3 α 2 +α 3 p 1 p 1 α 2 q 1 t α 2 α 2 +α 3 α 3 α 3 α 4 α 3 +α 4 α 3 q 1 q 1 q 1 + q 1 p 1 +q 2 p 2 α 3 +η p α 3 p 1 1 p 1 q 1 p 1 +q 2 p 2 +η α 3 q 2 q 2 q 2 + q 1 p 1 +q 2 p 2 α 3 +η p α 3 p 2 2 p 2 q 1 p 1 +q 2 p 2 +η α 3 α 3 +α 4 α 4 α 4 α 5 α 4 +α 5 p 2 p 2 α 4 q 2 1 α 0 α 0 +α 5 α 4 α 4 +α 5 α 5 α 5 q 2 q 2 + α 5 p 2 for i = 5. Then the system (1.1) with (1.2) is invariant under the action of them. Furthermore a group of symmetries r 0...r 5 is isomorphic to the affine Weyl group of type A (1) 5. 23

24 The group of symmetries defined above arises from the gauge transformations ( ) αi r i (Ψ) = exp f i Ψ (i = 0...5) ϕ i where ϕ 0 = w 3(q 2 q 1 ) ϕ 3t 2/3 1 = t2/3 p 1 ϕ 2 = w 3(q 1 t) w 3 3t ϕ 3 = q 1p 1 +q 2 p 2 +η ϕ 4 = w 3(1 q 2 ) ϕ w 3 3t 1/3 5 = t1/3 p 2 w 3 for the Lax pair of Appendix A.5. Note that those transformations are derived from the following ones [NY2]: r i (G) = Gexp( e i )exp(f i )exp( e i ) (i = 0...5) where G is an N B + -valued function given in Section 4. Acknowledgement The authers are grateful to Professors Kenji Kajiwara Masatoshi Noumi Masahiko Saito and Yasuhiko Yamada for valuable discussions and advices. References [AS] M. J. Ablowitz and H. Segur Exact linearization of a Painlevé transcendent Phys. Rev. Lett. 38 (1977) [C] R. Carter Conjugacy classes in the Weyl group Compositio Math. 25 (1972) [DF] F. Delduc and L. Fehér Regular conjugacy classes in the Weyl group and integral hierarchies J. Phys. A: Math. Gen. 28 (1995) [DS] V. G. Drinfel d and V. V. Sokolov Lie algebras and equations of Korteweg-de Vries type J. Sov. Math. 30 (1985) [FS] K. Fuji and T. Suzuki Higher order Painlevé system of type D (1) 2n+2 arising from integrable hierarchy Int. Math. Res. Not. 1 (2008) [GHM] M. F. de Groot T. J. Hollowood and J. L. Miramontes Generalized Drinfeld-Sokolov hierarchies Comm. Math. Phys. 145 (1992)

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