Automorphisms of sl(2) and dynamical r-matrices.

A.V. Tsiganov

Department of Mathematical and Computational Physics, Institute of Physics, St.Petersburg University, 198 904, St.Petersburg, Russia

Abstract

arXiv:solv-int/9610003v1 4 Oct 1996

Two outer automorphisms of in?nite-dimensional representations of sl(2) algebra are considered. The similar constructions for the loop algebras and yangians are presented. The corresponding linear and quadratic R-brackets include the dynamical r-matrices.

1

Introduction

We are concerned with the representation theory of loop algebras and yangians and with the method of separation of variables in framework of the inverse scattering method. Our aim is to apply the representation theory of sl(2) and sl(N ) algebras into the r -matrix formalism. Let us start with the following Levi-Civita theorem [6]: if the Hamilton-Jacobi equation associated with hamiltonian H = gij (x1 , . . . , xn )pi pj + hj (x1 , . . . , xn )pj + U (x1 , . . . , xn ) , {pi , xj } = ¦Äij , (1.1)

can be integrated by separation of variables then the equation is integrated with U = 0 (1.1), i.e. in the absence of a force, in the same system of coordinates. This note is devoted to the similar problem. Consider a classical hamiltonian system completely integrable on a 2n-dimensional symplectic manifold D . It means that system possesses n independent integrals I1 , . . . , In in the involution {Ii (x, p), Ij (x, p)} = 0 , where {xj , pj }n j =1 is some coordinate system in D . Introduce the following mapping

n ¡ä Ij ¡ú Ij = i

i, j = 1, . . . , n ,

aij (x, p) ¡¤ [Ii + vi (x, p)] ,

(1.2)

here aij and vj are certain functions on D . Under the suitable conditions for functions aij and vj , the mapping (1.2) could de?ne a ¡ä , . . . , I ¡ä in the involution new integrable system on D with independent integrals I1 n

¡ä {Ii¡ä (x, p), Ij (x, p)} = 0 ,

i, j = 1, . . . , n ,

and could preserve the property of separability in the same coordinate system. Remind, that systems discovered by St¡§ ackel (see review [7]) have the similar properties. Their integrals are

n

Ij =

i=1

aij (x1 , . . . , xn )[p2 i + vi (xi )] , 1

(1.3)

with the special functions aij (x1 , . . . , xn ) and the arbitrary functions vi (xi ). We try to make a ?rst step to understand of the algebraic roots of constraints on the functions aij and vj , which guarantee the properties of integrability and separability for mapping (1.2) in the framework of the inverse scattering method.

2

Outer automorphisms of representations of sl(2)

Let W be an in?nite-dimensional representation of the Lie algebra sl(2) in linear space V de?ned in the Cartan-Weil basis {s3 , s¡À } ¡Ê End(V) equipped with the natural bracket [s3 , s¡À ] = ¡Às¡À , and the single Casimir operator 1 ? = s2 3 + (s+ s? + s? s+ ) . 2 If operator s+ is invertible in End(V) then the mapping s3 ¡ú s¡ä 3 = s3 , s+ ¡ú s¡ä + = s+ , f ¡ÊC (2.6)

?1 s? ¡ú s¡ä ? = s? + f s+ ,

[s+ , s? ] = 2s3

(2.4)

(2.5)

is an outer automorphism of the space of in?nite-dimensional representations of sl(2) in V . The mapping (2.6) shifts the spectrum of ? on the parameter f ? ¡ú ?¡ä = ? + f. (2.7)

Let us call the mapping (2.6) an additive automorphism. Assuming in addition the value of Casimir operator ? is equal to zero ? = 0 in W , then the mapping

?1 sj ¡ú s¡ä j = sj ¡¤ (1 ? f s+ ) , ¡ä 1 2 f s? + ) ,

j = ¡À, 3 ,

¡ä

f ¡ÊC (2.8)

? ¡ú ? = ? ¡¤ (1 ?

? = ? = 0,

is another outer automorphism of representation W . Let us call the mapping (2.8) a multiplicative automorphism. These special in?nite-dimensional representations W could be obtained by using the previous additive automorphism (2.6). Notice, that the similar representations with ? = 0 are well known in the quantum conformal ?eld theory [4]. We can suppose that additive (2.6) and multiplicative (2.8) automorphisms de?ne the one-parametric realizations W (f ) of sl(2). For instance, realization of sl(2) with one free parameter f in the classical mechanics is given by x2 p2 f xp , s+ = , s? = ? + 2 , ?¡ä = f , (2.9) s3 = 2 2 2 x where (x, p) is a pair of canonical coordinate and momenta with the classical Poisson bracket {p, x} = 1. Similar realization of generators sj as the di?erential operators in quantum case is f 2 (2.10) s3 = x?x ? l , s+ = x , s? = x?x ? 2l?x + , ?¡ä = l(l + 1) + f . x 2

For the quantum Calogero system the more sophisticated representation of sl(2) is equal to s3 = 1 N (xk Dk + Dk xk ) , 4 k=1

N N

s+ = 2

k =1

x2 k,

s? = ?2

k =1

2 Dk ,

where xk are coordinates and Dk are the corresponding Dunkl operators [2]. Using the inner automorphisms of sl(2) the one parametric mappings (2.6) and (2.8) can be generalized. Let W be an in?nite-dimensional representation of sl(2) in V de?ned by operators sj ¡Ê End (V ) with the standard bracket and the single quadratic Casimir operator ? 2 2 [si , sj ] = ¦Åijk sk , ? = s2 (2.11) 1 + s2 + s3 . Introduce the general linear operator b ¡Ê End (V ) as b = ¦Á1 s1 + ¦Á2 s2 + ¦Á3 s3 , ¦Áj ¡Ê C , (2.12)

where ¦Áj are three arbitrary parameters. If b is invertible operator, then the mapping

?1 sj ¡ú s¡ä j = sj + ¦Áj b

(2.13)

is an outer automorphism of the space of in?nite-dimensional representations of sl(2) in V , if the parameters ¦Áj lie on the cone

2 2 ¦Á2 1 + ¦Á2 + ¦Á3 = 0 .

(2.14)

For this mapping the spectrum of the Casimir operator ? (2.11) is additively shifted on the parameter f = ¦Áj

3

? ¡ú ?¡ä = ? + f = ? +

j =1

¦Áj .

(2.15)

If the value of Casimir operator ? is equal to zero ? = 0 and the general linear operator b ¡Ê End (V ) (2.12) is invertible, then the mapping

?1 sj ¡ú s¡ä j = sj (1 ? b ) ,

? ¡ú ?¡ä = ? ¡¤ (1 ? b?1 )2

(2.16)

is an outer multiplicative automorphism of representation W . Automorphisms (2.13) and (2.16) de?ne the general two and three-parametric realizations W (¦Áj ) of sl(2). For the sl(N ) algebra the similar additive automorphism shifted the highest central element of N -th order can be taken in the form sij ¡ú s¡ä ij = sij + Aij

(m?1)

B (m)

,

sij ¡Ê End(V ) ,

m = N (N ? 1)/2 .

(2.17)

In operators sij functions B (m) and Aij are certain polynomials of degrees m and m ? 1. Here we shall not go into details of this problem, one example related to sl(3) will be presented in Section 4. Motivated by realization (2.9) we present one application of automorphism (2.6) in the theory of integrable systems. Consider a classical Hamiltonian system completely integrable on phase space D = R2n with the natural hamiltonian

n

(m?1)

H=

j =1

p2 j + V (x1 , . . . , xn ) .

3

Let the phase space be identi?ed completely or partially with the m coadjoint orbits in sl(2)? as (2.9). Then the mapping H ¡úH =H+

¡ä

fj , x2 j =1 j

m

fj ¡Ê R

(2.18)

preserves the properties of integrability and separability. The list of such systems can be found in [7]. The main our aim is in developing similar constructions for the loop algebras and yangians. Some examples of the integrable systems related with this approach have been considered in [3, 10, 11].

3

Dynamical r-matrices associated to sl(2)

All details of the general r -matrix scheme can be found in review [8] and in references therein. We recall brie?y necessary elements of r -matrix scheme for loop algebra g = sl(2) to assume the standard identi?cation of the dual spaces. k The loop algebra g = sl(2) consists of Laurent polynomials sj (¦Ë) = k sj ¦Ë , j = l 1, 2, 3 of spectral parameter ¦Ë with coe?cients in a = sl(2) and commutator [si ¦Ë , sj ¦Ëm ] = ¦Åijk sk ¦Ël+m . The standard R-bracket associated with loop algebra g is de?ned by the following decomposition of g into a linear sum of two subalgebras

g = g++ g? ,

.

g+ = ¨’i¡Ý0 a¦Ëi ,

g? = ¨’i<0 a¦Ëi ,

R = P+ ? P? .

(3.1)

Here R is a standard r -matrix and P¡À means the projection operators onto The Lax equation may be presented in the form dL(¦Ë) = ?ad? gA ¡¤ L , dt A= 1 R(dP (L)) , 2

g¡À parallel g? .

(3.2)

where Lax matrix L(¦Ë) belongs to g? , P is an ad? -invariant polynomial on a? . For algebra 2 sl(2) polynomial P (L) is a function of the unique invariant polynomial ?(¦Ë) = 3 j =1 sj (¦Ë). Let P (L) = ¦Õ(¦Ë)?(¦Ë), where ¦Õ is a functions of ¦Ë. The integrals of motion Ik related to ?ow (3.2) are Ik (L) = Res¦Ë=0 (¦Õk (¦Ë)?(¦Ë)) , (3.3) where ¦Õk (¦Ë) are various functions of spectral parameter de?ning a complete set of integrals of motion [8]. The Lax matrix L(¦Ë) (3.2) is de?ned on the whole in?nite-dimensional phase space g? . It is well known that the standard R-bracket associated with (3.1) has also a large collection of ?nite-dimensional Poisson (ad? R -invariant) subspaces [8]

? j LM,N = ¨’N j =?M a ¦Ë ,

provided M ¡Ý 0 ; N ¡Ý 1 .

(3.4)

and, as a rule, the concrete physical systems are related to the restrictions of the ?ow (3.2) to certain low-dimensional Poisson submanifolds LM,N (3.4). The r -matrix scheme is extended easily to the twisted subalgebras of loop algebra g and corresponding r -matrices have rational, trigonometric and elliptic dependence on spectral parameter. We shall work with a tensor form of the Lax equations and R-bracket on sl(2), that allows us to consider all the r -matrices simultaneously. In addition, for brevity, we shall consider the sl(2, R) only. 4

The Lax matrix L(¦Ë) (3.2) associated to the hamiltonian (3.3) is given by

3

L(¦Ë) =

k =1

sk (¦Ë)¦Òk ,

?(¦Ë) =

1 trL2 (¦Ë) = ?detL(¦Ë) , 2

(3.5)

where ¦Òj are Pauli matrices. The R-bracket related to decomposition (3.1) take the following form {L(¦Ë),L(?)} = [r12 (¦Ë, ?),L(¦Ë)] ? [r21 (¦Ë, ?),L(?) ] , where the standard notations are introduced: L(¦Ë) = L(¦Ë) ? I ,

3 1 1 2 1 2

(3.6)

L(?) = I ? L(?) , (3.7)

2

r12 (¦Ë, ?) =

k =1

wk (¦Ë, ?) ¡¤ ¦Òk ? ¦Òk

r21 (¦Ë, ?) = ¦°r12 (?, ¦Ë)¦° .

Here ¦° is the permutation operator of auxiliary spaces and wk (¦Ë, ?) are certain functions of spectral parameters only. Their explicit dependence on ¦Ë, ? is not important for the moment. Returning now to the additive automorphism of representations of sl(2) (2.13) we introduce the related mappings on the loop algebras g = sl(2, R) or g = ¨’n sl(2, R). The entries of the Lax matrix are Laurent polynomials with coe?cients in sl(2). Let us consider the mapping transforming these coe?cients. If parameters aj lie on the cone (2.14), then the ?rst mapping similar to the automorphism (2.13)

3

sj (¦Ë) =

k

s j ¦Ëk ¡ú s ¡ä j (¦Ë) =

k

[sj + ¦Áj b?1 ]¦Ëk ,

b=

j =1

¦Áj sj ,

(3.8)

is a canonical change of variables on sl(2). This mapping is de?ned for the in?nite-dimensional representations of sl(2) and mapping (3.8) is a Poisson map with respect to the ?rst natural Lie-Poisson bracket and to the second linear Lie-Poisson bracket associated with the standard R-matrix (3.1) on sl(2). For the certain Lax representation L(¦Ë) mapping (3.8) transforms the integrals of motion Ik similar to (2.18). Let us introduce the general linear operator b(¦Ë) ¡Ê sl(2)

3

b(¦Ë) =

j =1

¦Áj (¦Ë)sj (¦Ë) ,

(3.9)

where ¦Áj (¦Ë) are functions of spectral parameter and of central elements on sl(2). If functions ¦Áj (¦Ë) lie on the cone 2 2 ¦Á2 (3.10) 1 (¦Ë) + ¦Á2 (¦Ë) + ¦Á3 (¦Ë) = 0 , then the second mapping similar to the automorphism (2.13)

?1 sj (¦Ë) ¡ú s¡ä j (¦Ë) = sj (¦Ë) + ¦Áj (¦Ë)b (¦Ë) .

(3.11)

is a Poisson map with respect to the ?rst natural Lie-Poisson bracket. For the mapping (3.11) the Lax matrix (3.5), the invariant polynomial ?(¦Ë) and integrals of motion Ik (3.3) are

5

additively shifted

3

L(¦Ë) ¡ú L¡ä (¦Ë) = L(¦Ë) +

k =1

¦Ák ¦Òk ¡¤ b?1 (¦Ë) ,

3

(3.12)

?(¦Ë) ¡ú ?¡ä (¦Ë) = ?(¦Ë) + V (¦Ë) = ?(¦Ë) +

k =1 ¡ä Ik ¡ú Ik = Ik + Uk ,

¦Ák (¦Ë) .

(3.13) (3.14)

Uk = Res¦Ë=0 (¦Õk (¦Ë)V (¦Ë)) ¡Ê C .

Restriction (3.10) guarantees that initial integrals of motion Ik (3.3) transform by some constants Uk ¡Ê C (1.2). Hence, two Lax matrices L(¦Ë) and L¡ä (¦Ë) (original and the image of mapping (3.11)) correspond to the same integrable system. The entries of the initial matrix L(¦Ë) belong to g? . The entries of the matrix L¡ä (¦Ë) are the Laurent polynomials of spectral parameter ¦Ë with coe?cients from the universal enveloping algebra of a? . Nevertheless, the second Poisson bracket {L¡ä (¦Ë), L¡ä (?)} can be directly calculated, because of all the necessary Poisson brackets between sj (¦Ë) and b(¦Ë) are preassigned by (3.6). Further we consider a special class of the linear dynamical R-bracket related to sl(2) and de?ned by the following second restriction on coe?cients ¦Áj (¦Ë) {b(¦Ë), b(?)} = g(¦Ë, ?)b(¦Ë) ? g(?, ¦Ë)b(?) , or, that is equivalent, wj (¦Ë, ?)¦Áj (?)¦Ái (¦Ë) ? wi (¦Ë, ?)¦Ái (?)¦Áj (¦Ë) = g(¦Ë, ?) ¡¤ ¦Ák (¦Ë) , (3.16) ¦Ë, ? ¡Ê C , (3.15)

where (j, i, k) are cyclic permutations of indices (1, 2, 3) and the scale function g(¦Ë, ?) depends of the spectral parameters only. This restriction is closely related with the separation of variables method. It guarantees that all zeroes of b(¦Ë) are mutually commuting. Below we assume that both conditions (3.10) and (3.15) are always ful?lled for the mapping (3.11). In this case we get the Lax matrix L¡ä (¦Ë) (3.12) obeys the linear R-bracket (3.6), where ¡ä -matrices depending on dynamical variables constant rij -matrices substituted by rij

3

r12 (¦Ë, ?) ¡ú with coe?cients ¦Áij being ¦Áij (¦Ë, ?) = g(¦Ë, ?)

¡ä r12

= r12 +

i,j =1

¦Áij (¦Ë, ?) ¦Òi ? ¦Òj ,

(3.17)

¦Ái (¦Ë)¦Áj (¦Ë)wj (¦Ë, ?) ¦Áj (?)¦Ái (?)wi (¦Ë, ?) ? g(?, ¦Ë) . 2 b (?) b2 (¦Ë)

(3.18)

The proof see in [5]. ¡ä (¦Ë, ?) (3.17) obey the classical dynamical Yang-Baxter equation Dynamical matrices rjk

¡ä ¡ä ¡ä ¡ä ¡ä ¡ä [r12 (¦Ë, ?), r13 (¦Ë, ¦Í )] + [r12 (¦Ë, ?), r23 (?, ¦Í )] + [r32 (¦Í, ?), r13 (¦Ë, ¦Í )] +

(3.19)

¡ä +[L¡ä 2 (?), r13 (¦Ë, ¦Í )]

?

¡ä [L¡ä 3 (¦Í ), r12 (¦Ë, ?)]

+

[X123 (¦Ë, ?, ¦Í ), L¡ä 2 (?) ?

L¡ä 3 (¦Í )]

= 0,

where an explicit expression of the tensor X (¦Ë, ?, ¦Í ) is not important for the moment (see [3, 5, 12]). This equation is the image of a standard 2-cocycle in the new object, which consists of 6

Laurent polynomials of spectral parameter ¦Ë with coe?cients from the corresponding universal enveloping algebra. We can suppose that mapping (3.11) de?nes family of Lax matrices L¡ä (¦Ë) and dynamical r -matrices (3.17) with a ?xed set of parameters ¦Áj (¦Ë) for certain integrable system. Thus, the Lax matrix L(¦Ë) (3.2), the constant r -matrices (3.7) and the standard classical Yang-Baxter equation on sl(2) are the limits of the dynamical ones at ¦Áj (¦Ë) = 0. The mapping (3.11) allows us to construct an in?nite set of the Lax matrices associated to loop algebra sl(2), which correspond to di?erent integrable systems. According to [5] we introduce an in?nite set of mappings

?1 sj (¦Ë) ¡ú s¡ä j (¦Ë) = sj (¦Ë) + ¦Áj (¦Ë)b (¦Ë) M j Nj

,

(3.20)

where [z ]M N means restriction of z onto the certain Poisson subspase LM N (3.4) of the standard R-bracket (3.1). As an example, we can use the linear combinations of the following Laurent and Fourier projections [z ]M N = ?

?

+¡Þ

k =?¡Þ

zk ¦Ëk ?

?

MN

N

¡Ô

k =?M

zk ¦Ëk ,

?

[z ]M N

if the corresponding r -matrices have rational or trigonometric dependence on spectral parameter [5]. Generally speaking, we can not describe now all the possible restrictions of the mapping (3.11) to certain low-dimensional submanifolds, which allow us to de?ne the second Poisson bracket on sl(2). However, for a fairly large class of properly de?ned projections (3.21) the mappings (3.20) are the Poisson maps with respect to the second R-bracket and the corresponding dynamical r -matrices obey the dynamical Yang-Baxter equation (3.19). For instance, an application of projections (3.21) yields dynamical r -matrices

3 ¡ä r12 (¦Ë, ?) ¡ú r12 = r12 + i,j =1

= ?

?

(3.21)

N

+¡Þ

k =?¡Þ

zk exp(k ¡¤ ¦Ë) ?

¡Ô

k =?M

zk exp(k ¡¤ ¦Ë) ,

MN

¦Áij (¦Ë, ?) ¦Òi ? ¦Òj ,

(3.22)

with the following coe?cients ¦Áij (¦Ë, ?) = g(¦Ë, ?)wi (¦Ë, ?) ¦Áj (?)¦Ái (?) b2 (?) ? g(?, ¦Ë)wj (¦Ë, ?)

MN

¦Ái (¦Ë)¦Áj (¦Ë) b2 (¦Ë)

.

MN

(3.23)

The essential feature of the restricting mappings (3.20) is, in comparison with the mapping (3.11), that the invariant polynomial ?(¦Ë) and all integrals of motion Ik are shifted now on the items depending on dynamical variables ?(¦Ë) ¡ú ?M N (¦Ë) = ?(¦Ë) + VM N (sj , ¦Áj , ¦Ë) , (3.24) Ik ¡ú

¡ä Ik

= Ik + Res¦Ë=0 (¦Õk (¦Ë)VM N (sj , ¦Áj , ¦Ë))

7

where VM N being

3

VM N (sj , ¦Áj , ¦Ë) =

k =1

2sk (¦Ë) + [aj (¦Ë)b?1 (¦Ë)]M N [aj (¦Ë)b?1 (¦Ë)]M N .

(3.25)

¡ä of the mapping (3.20) are functionally di?erent from Hence, images of integrals of motion Ik the original ones Ik . The second feature of the mapping (3.20) is that it can be also applied to the ?nitedimensional representations of sl(2). Consider the multipole Lax matrices related to the rational r -matrix [8]. Let g = ¨’n sl(2, R) and elements of mapping (3.11) are

¦Á1 = 1 ,

¦Á2 = i ,

¦Á3 = 0 ,

b(¦Ë) =

s+ , ¦Ë ? ek k =1

n

(k )

ej = ek ¡Ê R .

(3.26)

The Taylor projection of b?1 (¦Ë) is determined by the following recurrence relations [3]

N n

b?1 (¦Ë)

M N M =0

=

k =0

V k ¦ËN ? k ,

Vk =

i=1

?

?s(i)

+

k ?1 j =0

?, Vk?1?j ej i

?

V0 = 1 .

(3.27)

(k )

The Taylor projection (3.27) is the well-de?ned polynomial of the nilpotent operators s+ without the negative powers, which can be also used for ?nite-dimensional representations of sl(2). Once again, the mappings (3.20) are not just isomorphisms of the Lax matrices L(¦Ë) and L¡ä (¦Ë), in contrast with mapping (3.11), but they also preserve the second Poisson R¡ä , i.e. preserve the properties bracket together with all the commuting integrals of motion Ik of integrability and separability. Thus, the mappings (3.20) play the role of a dressing procedure allowing to construct the Lax matrices L¡ä M N (¦Ë) for an in?nite set of new integrable systems starting from the single known Lax matrix L(¦Ë) associated to one integrable model. Now we shall brie?y discuss the second multiplicative automorphism (2.16). Let us start ? with some Lax matrix L(¦Ë) ¡Ê sl(2) . This matrix relates to the integrable system with the Lax representation (3.2) and with the following integrals of motion (3.3) Ik = Res¦Ë=0 (¦Õk (¦Ë)?(¦Ë)) . Recall, that the multiplicative automorphism (2.16) of sl(2) transforms the Casimir operator by the rule ? ¡ú ?¡ä = ? ¡¤ ?(b) = ? ¡¤ (1 ? b?1 )2 .

¡ä (see 1.2) de?ned Motivated by this transformation we consider the new set of integrals Ik

by

¡ä Ik = Res¦Ë=0 (?k (b, ¦Ë) ¡¤ ?(¦Ë)) ,

(3.28)

where ?k (b, ¦Ë) are certain functions of b(¦Ë) (3.9) and of the spectral parameter ¦Ë. The involution conditions ¡ä {Ii¡ä , Ij } = 0, i, j = 1, . . . , n , yield the set of equations in ?k (b, ¦Ë) Res¦Ë?=0 {?(¦Ë), b(?)}?j (¦Ë)?k (?) ? ln ?k ? ln ?j (¦Ë)?(¦Ë) ? (?)?(?) = 0 . ?b ?b 8

¡ä could be functionally di?erent from the original ones I . New integrals of motion Ij j In the quantum case the Poisson brackets should be replaced by the standard commutator relations on sl(2) and the non-dynamical linear R-matrix bracket (3.6) becomes the following commutator relations

L(¦Ë),L(?) = [r12 (¦Ë, ?),L(¦Ë)] ? [r21 (¦Ë, ?),L(?) ] ,

1

2

1

2

r(¦Ë, ?) = ?i? hr (¦Ë, ?) .

(3.29)

Theory of the general quantum linear R-bracket with dynamical r -matrices is not well developed yet, but a nice feature of the presented Poisson mappings (3.11) and (3.20) is that it admits a natural quantization. All the necessary commutator relations between operators sj (¦Ë) and linear invertible operator b(¦Ë) (3.9) are preassigned by (3.29) and, therefore, the introduction of the quantum dynamical r -matrices (3.17) is a straightforward calculation. A similar calculation yields the direct quantum counterpart of the dynamical Yang-Baxter equation.

4

Dynamical r-matrices and separation of variables

The presented algebraic construction are intimately connected with the method of separation of variables in classical mechanics. We shall use technique developed by Sklyanin in framework of r -matrix formalism and based on the application of the Baker-Akhiezer function [9]. Recall, that the Baker-Akhiezer function ¦·(¦Ë) is the eigenvector of the Lax matrix L(¦Ë)¦·(¦Ë) = z (¦Ë)¦·(¦Ë) , {H, ¦·} = ¦·t (¦Ë) = A¦·(¦Ë) , (4.1)

corresponding to the eigenvalue z (¦Ë), which has certain analyticity properties. Here A is a second matrix in the Lax representation (3.2). Since an eigenvector is de?ned up to a scalar factor, to exclude the ambiguity in the de?nition of ¦·(¦Ë) one has to ?x a normalization of ¦·(¦Ë) imposing a linear constraint

N

¦Âj (¦Ë)¦·j (¦Ë) = 1 .

j =1

The main purpose of this Section is to ?nd a correspondence between the outer automorphisms of in?nite-dimensional representations of sl(2) and the normalization of the Baker-Akhiezer function. According to the Sklyanin recipe the variables of separation (xj , pj ) are de?ned by the poles of the properly normalized Baker-Akhiezer function ¦·(¦Ë) and the corresponding eigen? values z (¦Ë) of the Lax matrix [9]. For the Lax matrix L(¦Ë) ¡Ê sl(2) the poles xj of ¦·(¦Ë) are zeroes of the following function

2 2 B (¦Ë, ¦Â1 , ¦Â2 ) = ¦Â1 (¦Ë)s+ (¦Ë) ? ¦Â2 (¦Ë)s? (¦Ë) ? 2¦Â1 (¦Ë)¦Â2 (¦Ë)s3 (¦Ë) .

(4.2)

Introduce functions ¦Áj (¦Ë) as

2 2 ¦Á1 (¦Ë) = [(¦Â1 ? ¦Â2 )f ](¦Ë) , 2 2 ¦Á2 (¦Ë) = i[(¦Â1 + ¦Â2 )f ](¦Ë) ,

¦Á3 = ?2[¦Â1 ¦Â2 f ](¦Ë) ,

(4.3)

where f (¦Ë) is a certain common function of a spectral parameter only. The coe?cients ¦Áj (¦Ë) lie on the cone (3.10) and de?ne the linear operator b(¦Ë, ¦Áj ) = B (¦Ë, ¦Â1 , ¦Â2 ) (3.9) for the mappings (3.11) and (3.20). The second restriction (3.15) on ¦Áj (¦Ë) guarantees that all zeroes xj of b(¦Ë) are mutually commuting {xj , xk } = 0. 9

For the linear R-bracket the eigenvalues z (¦Ë) of the Lax matrix L(¦Ë) corresponding to zeroes xj are canonically conjugated momenta pj . The pairs of variables (xj , pj ) lie on the spectral curve W (pj , xj ) = 0 , W (z, ¦Ë) = det(z ? L(¦Ë)) = z 2 ? ?(¦Ë) , (4.4)

which ?ts exactly to separated equations [9]. The equations (4.1) and divisor of poles ¦·(¦Ë) are covariant with respect to the mappings (3.11-3.20) L ¡ú L¡ä = L + ? LM N , H ¡ú H ¡ä = H + VM N . These mappings could be considered as an analogous of a standard Darboux transformation. The mappings (3.20) change the separated equations (4.4) as

2 ¡ä ¡ä p2 j ? ?(xj , Ik ) = 0 ¡ú pj ? ? (xj , Ik ) = 0 ,

(4.5)

¡ä are given by (3.24). Thus, by taking the single where ?¡ä (¦Ë) and new integrals of motion Ik Lax matrix L(¦Ë) associated to some integrable system, which can be integrated by separation of variables in coordinates {xj , pj }, we get an in?nite set of completely integrable systems determined by the mappings (3.20), which are separable in the same variables {xj , pj }. The linear operator b(¦Ë) (3.9) is a symmetric function of its zeroes {xj }n j =1 and dynamical r -matrices (3.17) depends of the spectral parameters and only half of the dynamical variables {xj }n j =1 . Moreover, if normalization ¦Â (¦Ë) of ¦·(¦Ë) is given by arbitrary constant numeric vector, then three functions ¦Áj (¦Ë) (4.3) di?er by certain numeric constants ¦Âj

¦Áj (¦Ë) = ¦Âj f (¦Ë) ,

¦Âj ¡Ê C,

2 = 0. ¦Âj

(4.6)

Then by (4.6) the invariant polynomial ?(¦Ë) and all the integrals of motion Ik (3.3) are shifted on items said to be potentials depending of separated coordinates xj (1.2): ?¡ä = ? + VM N (x1 , . . . , xn ) = ? + 2b(xj , ¦Ë) f (¦Ë) b(xj , ¦Ë) ,

MN

(4.7)

¡ä Ik

= Ik + Uk (x1 , . . . , xn ) = Ik + Res¦Ë=0 (¦Õk (¦Ë)VM N (x1 , . . . , xn )) .

Thus, all integrable systems related to the mappings (3.20) with the constant normalization of the corresponding Baker-Akhiezer function obey to the Levi-Civita theorem (1.1) [6]. ¡ä (3.28) associated to the multiplicative automorphism (2.16) could Integrals of motion Ij be di?er from original ones on certain functions aij (x1 . . . , xn ) of only half of the dynamical ackel systems (1.3)). Here functions aij are de?ned by variables {xj }n j =1 (see (1.2) and the St¡§ (3.28) for any concrete representation. ? Method of separation of variables for the Lax matrices L(¦Ë) ¡Ê sl(N ) was studied [9]. The separated coordinates are obtained as zeroes of the certain polynomial B (¦Ë) of degree N (N ? 1)/2 in components of L(¦Ë). The spectral curve W (z, ¦Ë) = det(z ? L(¦Ë)) is a nonhyperelliptic algebraic curve for N > 2 and the Levi-Civita theorem can not be applied directly to this case. Still the polynomial B (¦Ë) depending of separated coordinates can be used to construct a counterpart of the mappings (3.11) and (3.20) for sl(N ). ? Consider, for instance, the loop algebra sl(3). Let the Lax matrix L(¦Ë) ¡Ê sl(3) obeys the standard linear R-bracket (3.6) with rational r -matrix r12 = (¦Ë ? ?)?1 ¦°, where ¦° is the permutation operator: ¦°x ? y = y ? x, x, y ¡Ê C3 . The entries sij (¦Ë) of the Lax matrix L(¦Ë) 10

are constructed in variables sij (i, j = 1, 2, 3. sii = 0) with the following standard Poisson brackets {sij , skm } = sim ¦Äjk ? skj ¦Äim , which de?ne the natural Lie-Poisson bracket on sl(3) . According to [9], the simplest choice of normalization ¦Â (¦Ë) of ¦·(¦Ë) is ¦Â1 = ¦Â2 = 0 , then the polynomial B (¦Ë) is given by B (¦Ë) = s32 (¦Ë)u13 (¦Ë) ? s31 (¦Ë)u23 (¦Ë) , (4.9) ¦Â3 = 1 , (4.8)

?

where uij (¦Ë) is (ij )-cofactor of the determinant of L(¦Ë). The polynomial B (¦Ë) (4.9) allows to introduce the following mapping sij ¡ú s¡ä ij = sij , s13 ¡ú s23 ¡ú s¡ä 13 ¡ä s23 (ij ) = (13), (23) , (4.10)

?1

= s13 + s32 f (¦Ë)B ?1 (¦Ë) , = s23 ? s31 f (¦Ë)B (¦Ë) ,

as an analog of the mapping (3.11) with the normalization (4.8). This mapping leaves ?xed the spectral invariants ¦Ó1 and ¦Ó2 and shifts the third invariant polynomial ¦Ó3

¡ä ¦Ó1 (¦Ë) = trL(¦Ë) = 0 ¡ú ¦Ó1 (¦Ë) = ¦Ó1 (¦Ë) = 0 ¡ä ¦Ó2 (¦Ë) = trL2 (¦Ë) ¡ú ¦Ó2 (¦Ë) = ¦Ó2 (¦Ë) , ¡ä ¦Ó3 (¦Ë) = detL(¦Ë) ¡ú ¦Ó3 (¦Ë) = ¦Ó3 (¦Ë) + f (¦Ë) ,

The general Lax matrix L¡ä (¦Ë) de?ned by (4.10) obeys the linear R-bracket with the dynamical r -matrices. As an example, the Lax representations for the Henon-Heiles system and a system with quartic potential [1] can be embedded into the proposed scheme by using the more sophisticated normalization. So, we can apply the known polynomials B (¦Ë) and method of separation of variables to construct the analogs of mappings (3.11) and (3.20) for the loop algebra sl(N ). It would be interesting to ?nd the general counterparts of outer automorphisms (2.13) and (2.16) for in?nite-dimensional representations of sl(N ) and to consider the inverse problem of a choosing the correct normalization of Baker-Akhiezer function by using these automorphisms. A similar, but more di?cult problem arises for the simple Lie algebras other than sl(N ).

5

Dynamical r-matrices and quadratic R-bracket

Now we consider analogs of the additive and multiplicative automorphisms of sl(2) (2.6)-(2.8) for the quadratic R-bracket in classical mechanics. Introduce the formal particular mapping T (u) = A B C D (u) ¡ú T ¡ä (u) = A + f D ?1 B C D (u). (5.1)

If the initial matrix T (u) obeys the standard quadratic R-bracket {T (u),T (v )} = [r (u ? v ),T (u)T (v ) ] , 11

1 2 1 2

with

r = (u ? v )?1 ¦° ,

(5.2)

then the image T ¡ä (u) of mapping (5.1) obeys the dynamical quadratic R-bracket {T ¡ä (u),T ¡ä (v )} = [r (u ? v ),T ¡ä (u)T ¡ä (v ) ] + T ¡ä (v )s21 T ¡ä (u) ? T ¡ä (u)s12 T ¡ä (v ) . Here dynamical matrices sjk are given by s12 (u, v ) = 1 f (v ) f (u) ? 2 ¡¤ ¦Ò? ? ¦Ò+ , 2 u ? v D (u) D (v ) (5.4) s21 (u, v ) = ¦°s12 (v, u)¦° . In contrast to the linear case, the mapping (5.1) related to additive automorphism (2.6) changes the form of R-bracket. However, if the functions D (u) and f (u) (5.1) are independent on spectral parameter u, then the dynamical matrices sjk in (5.3) go to zero [10]. Moreover, in this case, we can use the more general mapping (see (5.1)) A(u) ¡ú A¡ä (u) = A(u) + g(D ) , where g(D ) is an arbitrary function on entry D . This mapping changes the standard R-bracket (5.2) to dynamical bracket (5.3), but it preserves the property of integrability {tr T ¡ä (u), tr T ¡ä (v )} = 0 . We have to emphasize that additive and multiplicative automorphisms of sl(2) (2.6)-(2.8) give rise to the integrable systems associated to the two root systems BCn and Dn [10], respectively. Consider the special solutions TA (u) = of the classical re?ection equation T (u),T (v )

1 2 1 2 1 2 2 1 1 2

(5.3)

A(u) B (u) C (u) ?A(?u)

,

(5.5)

=

1

r (u ? v ),T (u)T (v ) +

2 2 1

1

2

(5.6)

+ T (u)r (u + v )T (v ) ? T (v )r (u + v )T (u). with the rational r = (u ? v )?1 ¦° r -matrix. An application of the additive and multiplicative automorphisms of sl(2) (2.6)-(2.8) is more tricky in the quadratic case. If the entry B (u) is independent of spectral parameter u, i.e. ?B/?u = 0, we can construct the new solution of the re?ection equation (5.6) TC = TA + 0 0 ¦ÃB ?1 0 , ¦Ã ¡Ê R. (5.7)

Assuming in addition that entry A(u) is a linear function of u, i.e. A(u) = ua1 + a2 , let us introduce the second bounary matrix ? ? ¦Á +¦Â 0 ? ? u TBC = TC + ? ¦Á ?, ¦Á ? 1 (A(u) ? A(?u)) + ¦Â (A(u) + A(?u)) B ?¦Â u u (5.8) ¦Á, ¦Â, ¡Ê R 12

which is new solution of re?ection equation (5.6). These solution TC (5.7) and TBC (5.8) could be associated with the additive automorphism (2.6). If the central element ? = det TA (u) = 0 is equal to zero and ?B/?u = 0 the third boundary matrix TD = A(u) ? A(?u)B ?1 B ¡¤ (1 ? B ?1 )2 C (u) ?A(?u) + A(u)B ?1 . (5.9)

is a solution of re?ection equation (5.6). This solution could be associated with the multiplicative automorphism (2.8). As an example, we consider the Toda lattices. Let the initial boundary matrix TA (u) is given by (u ? p) exp(q ) exp(2q ) TA (u) = , (5.10) u2 ? p 2 (u + p) exp(q ) where (q, p) is a pair of canonically conjugate variables. According to [10] matrices TA , TBC and TD correspond to the Toda lattices associated with the Lie algebras of An , Bn (¦Â = ¦Ã = 0), Cn (¦Á = ¦Â = 0) and Dn series, respectively. Among the hamiltonians, in comparison with (1.2), there are HA HBC HD = 1 n 2 n?1 p + exp(xj +1 ? xj ) , 2 j =1 j j =1

= HA + ¦Ã exp(?2x1 ) + (2¦Á + 2¦Âp1 ) exp(?x1 ) , = HA + exp(?x1 ? x2 ) .

In classical and quantum mechanics the boundary matrices (5.8) and (5.9) have been used for the relativistic Toda lattices and for the Heisenberg XXX and XXZ models in [10]. The more complicated dynamical quadratic R-bracket has been introduced in [11] for the Neumann top, Kowalewski top and Toda lattice associated to the Lie algebra G2 .

6

Conclusions

The outer automorphisms of in?nite-dimensional representation of sl(2) give possibility to construct new Lax matrices. The corresponding linear and quadratic R-brackets include the dynamical r -matrices, which obey the dynamical Yang-Baxter equations. For the loop algebras as a second step we applied the certain projection of general Lax matrix onto the low-dimensional subspaces, which preserve the R-bracket. Thus, the set of the Lax matrices associated to the di?erent integrable system can be obtained. The similar dynamical deformations of quadratic R-bracket has been applied for the construction of Lax matrix for integrable systems associated to the root systems BCn and Dn . Among the known and possible examples there are: a wide class of the St¡§ ackel systems; the integrable extensions of the classical tops - Euler top, Manakov and Steklov tops, Lagrange and Goryachev-Chaplygin top; generalizations of the Heisenberg and Gaudin magnets; the Toda and the Calogero-Moser systems. The following problems, however, remain open. We have not an exhaustive description of outer automorphisms of in?nite-dimensional representations of simple Lie algebras and of the all admissible low-dimensional submanifolds, which allow a well-de?ned restriction to the corresponding linear classical and quantum R-brackets. 13

7

Acknowledgments

We would like to thank I.V. Komarov for reading the manuscript and making valuable comments. This research has been partially supported by RFBR grant 96-0100537.

References

[1] S. Baker, V.Z. Enolskii, and A. P. Fordy. Phys.Lett, A201:167¨C174, 1995. M. Blaszak and S. Rauch-Wojciechowski. J.Math.Phys., 35:1693¨C1709, 1994. [2] C.F. Dunkl Trans.Amer.Math.Soc., 311:167,1989. [3] J.C. Eilbeck, V.Z. Enolskii, V.B. Kuznetsov, and A.V. Tsiganov. J.Phys., A27:567¨C578, 1994. [4] L.N.Lipatov. JETP Lett., 59:596-599, 1994. L.D. Faddeev and G.P. Korchemsky. High energy QCD as a completely integrable model. In Preprint ITP-SB-94-14 hep-th/9404173, 1994. [5] P.P Kulish, S. Rauch-Wojciechowski, and A.V. Tsiganov. Mod.Phys.Lett., A9:2063¨C2073, 1994. Stationary problems for equations of the KdV type and dynamical r -matrices. Preprint LiTH-MAT-R-95-12, hep-th/9503066. [6] T. Levi-Civita. Math.Ann., 59:383¨C397, 1904. [7] A.M. Perelomov. Integrable system of classical mechanics and Lie algebras. Birkhauser, Basel, 1991. [8] A.G. Reyman and M.A. Semenov-Tian Shansky. Group Theoretical Methods in the theory of Finite-Dimensional Integrable systems. In V.I. Arnold and S.P. Novikov, editors, Dynamical systems VII, volume EMS 16, Berlin, 1993. Springer. [9] E.K. Sklyanin. Progr.Theor.Phys.Suppl., 118:35¨C61, 1995. [10] A.V. Tsiganov. J.Phys., A27:6759¨C6780, 1994. [11] A.V. Tsiganov. The Kowalewski top: a new Lax representation. Preprint ISRN-LiTHMAT-R-95-27, 1995. [12] Y. Zeng and J. Hietarinta. Classical Poisson structure and r -matrices from constrained ?ows. J.Phys., A29:5241-5252, 1996.

14