Spectral parameterization for the power sums of quantum supermatrix

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SPECTRAL PARAMETERIZATION FOR THE POWER SUMS OF QUANTUM SUPERMATRIX DIMITRI GUREVICH,PAVEL PYATOV,AND PAVEL SAPONOV Abstract.A parameterization for the power sums of GL (m |n )type quantum (super)matrix is obtained in terms of it’s spectral values.

POWER SUMS OF QUANTUM SUPERMATRIX1

1.Introduction

This paper is a complement to our previous works[GPS1,GPS2,GPS3]devoted to the quantum matrix algebras(QMA)of GL(m|n)type.Here we continue investigation of the commutative characteristic subalgebra of the QMA.To be more precise,for the set of‘quan-tum’traces of‘powers’of the quantum(super)matrix we?nd out a parameterization in terms of spectral values which are the quantum analogs of the set of(super)matrix eigen-values.Note,that the abovementioned set of quantum traces generates the characteristic subalgebra.To illustrate the statements let us brie?y recall the corresponding facts from the classical matrix algebra.

As is well known,any N×N complex matrix M∈Mat N(C)satis?es a polynomial Cayley-Hamilton(or,characteristic)identity,which can be presented in a factorized form

N

i=1(M?μi I)=0,

where I is a unit matrix,andμi,i=1,...,N,are the eigenvalues of M.Opening the brackets one can rewrite the identity in a form

N

k=0(?1)k e k(μ)M N?k=0,

where e k(μ),k=0,1,...,N,are elementary symmetric polynomials1in variables{μi}1≤i≤N. The elementary symmetric polynomials generate the whole algebra of symmetric polynomials in the eigenvaluesμi.Another well-known generating set for symmetric polynomials is given by the power sums

p k(μ):=

N

i=1μk i≡Tr(M k).

A relation between two sets of generators is provided by the Newton’s recurrence

k e k+

k

r=1(?1)r p r e k?r=0?k≥1.

In papers[GPS1,GPS2]analogues of the above classical results were found for a family of Hecke type QMAs,which includes the q-generalizations of the GL(m|n)type supermatrices for all integer m≥0and n≥0(see the next section for de?nitions).In particular,the Cayley-Hamilton identities for these algebras were derived,what clari?ed the way of intro-ducing the spectral values for quantum matrices.It is remarkable that the coe?cients of the Cayley-Hamilton polynomials commute among themselves.They generate a commuta-tive characteristic subalgebra in the QMA,which serves as an analogue of the algebra of symmetric polynomials in the eigenvalues of matrix M.It is the Cayley-Hamilton identity which allows one to present the elements of the characteristic subalgebra as(super)symmetric polynomials in the spectral values of the quantum matrix.

As in the classical case,the quantum analogs of the power sums can be de?ned as some speci?c traces of‘powers’of the quantum matrix.By their construction the power sums belong to the characteristic subalgebra,but their explicit expression in terms of the spectral values was not known yet.The main goal of the present paper is to derive such an expression.

2DIMITRI GUREVICH,PAVEL PYATOV,AND PAVEL SAPONOV

Our presentation is strongly based on the previous works cited above.In the next section we give a list of notations,de?nitions and main results which will be used below.For more detailed exposition,proofs and a short overview the reader is referred to[GPS1,GPS2].

2.Some basic results and definitions

Let V be a?nite dimensional linear space over the?eld of complex numbers C,dim V=N. Let I denote the identity matrix(its dimension being clear from the context,if not explicitly speci?ed),and P∈Aut(V?2)be the permutation automorphism:P(u?v)=v?u.

With any element X∈End(V?p),p=1,2,...,we associate a sequence of endomorphisms X i∈End(V?k),k≥p,i=1,...,k?p+1,according to the rule

X i=I?(i?1)?X?I?(k?p?i+1),1≤i≤k?p+1,

where I is the identical automorphism of V.

Consider a pair of invertible operators R,F∈Aut(V?2)subject to the following conditions:

(1)The operators R and F satisfy the Yang-Baxter equations

R1R2R1=R2R1R2,F1F2F1=F2F1F2.(2.1) Such operators are called R-matrices.

(2)The pair of R-matrices{R,F}is compatible,that is

R1F2F1=F2F1R2,F1F2R1=R2F1F2.(2.2)

(3)The matrices of both operators R and F are strictly skew invertible.Taking the

operator R as an example,this requirement means the following:

a)R is skew invertible if there exists an operatorΨR∈End(V?2)such that

Tr(2)R12ΨR23=P13,

where the subscript in the notation of the trace shows the number of the space V, where the trace is evaluated(the enumeration of the component spaces in the tensor product is taken as follows V?k:=V1?V2?···?V k).

b)The strictness condition implies additionally that the operator D R1:=Tr(2)ΨR12is

invertible.

With the matrix D R one de?nes the R-trace operation2Tr R:Mat N(W)→W

Tr R(X):=

N

i,j=1D R j i X i j,X∈Mat N(W),

where W is any linear space.

Given a compatible pair{R,F}of strictly skew invertible R-matrices the quantum matrix algebra M(R,F)is a unital associative algebra generated by N2components of the matrix M i j N i=1subject to the relations

R1M2=M2R1.(2.3) Here we have introduced a notation

M

k+1

:=F k M

2In a literature on quantum groups the R-trace is usually named the quantum trace or,shortly,the q-trace. Giving the di?erent name to this operation we hope to avoid misleading associations with a parameter q of the Hecke algebra(see below).

POWER SUMS OF QUANTUM SUPERMATRIX3 for the copies M

k M

k

M

q?q?1

=0?k=2,3,....(2.5) Given any Hecke R-matrix R,one can construct a series of R-matrix representationsρR of the A type Hecke algebras H k(q)ρR

?→End(V?k),k=2,3,....The characteristic properties of these representations are used for a classi?cation of the Hecke R-matrices4.Not going into details of the construction we only mention that under conditions(2.5)the Hecke algebra H k(q)is isomorphic to the group algebra of the symmetric group C[S k]and its irreducible representations are labelled by a set of partitionsλ?k,the corresponding central idempotents in H k(q)are further denoted as eλ.We?x some decomposition of eλinto the sum of primitive idempotents eλa∈H k(q):eλ= dλa=1eλa,where dλis the dimension of the representation with labelλ.It is also suitable to introduce the following notations:

–Given two arbitrary integers m≥0and n≥0,an in?nite set of partitionsλ= (λ1,λ2,...),satisfying restrictionλm+1≤n is denoted as H(m,n).

–The partition((n+1)m+1)?(m+1)(n+1)is shortly denoted asλm,n.The corre-sponding Young diagram is a rectangle with m+1rows of the length n+1.Note thatλm,n is a minimal partition not belonging to the set H(m,n).

Now we are ready to formulate the classi?cation of the Hecke R-matrices.

Proposition1.([H,GPS3])For a generic value of q the set of the Hecke R-matrices is separated into subsets labelled by an ordered pair of non-negative integers{m,n}.R-matrices belonging to the subset with label{m,n}are called GL(m|n)type ones(alternatively,they are assigned a bi-rank(m|n)).R-matrix representationsρR generated by a GL(m|n)type R-matrix R ful?ll the following criterion:for all integer k≥2and for any partitionν?k the images of the idempotents eν∈H k(q)satisfy the relations

ρR(eν)=0i?ν∈H(m,n),

or,equivalently,i?λm,n?ν,where the inclusionμ=(μ1,μ2,...)?ν=(ν1,ν2,...)means thatμi≤νi?i.

The algebra M(R,F)de?ned by a Hecke(GL(m|n)type)R-matrix R is further referred to as the Hecke(GL(m|n)type)quantum matrix algebra.

4DIMITRI GUREVICH,PAVEL PYATOV,AND PAVEL SAPONOV

For the Hecke type QMA M(R,F)we consider a set of its elements sλ(M)called the Schur functions

s0(M):=1,sλ(M):=Tr

R(1...p)

(M pρR(eλa)),λ?p,p=1,2,..., where the latter formula does not depend on a particular choice of the primitive idempotent eλa (actually,one can substitute it by d?1λeλ).As was shown in[IOP1],a linear span of the Schur functions sλ(M)?λ,is an abelian subalgebra in M(R,F).We further call it the characteristic subalgebra of M(R,F).It follows that the characteristic subalgebra of the GL(m|n)type QMA is spanned by the Schur functions sλ(M),λ∈H(m,n).The multiplication table for the elements sλ(M)∈M(R,F)coincides with that for the basis of Schur functions in the ring of symmetric functions(see[Mac])thus justifying the notation.One has[GPS2]

sλ(M)sμ(M)= νCνλμsν(M),(2.6)

where Cνλμare the Littlewood-Richardson coe?https://www.sodocs.net/doc/ab6539197.html,ter on we shall need the information about generating sets of the characteristic subalgebra.

Proposition2.([IOP1,IOP2])For generic values of q the characteristic subalgebra of the Hecke QMA M(R,F)is generated by any one of the following three sets

(1)the single column Schur functions:a k(M):=s(1k)(M),k=0,1,2,...;

(2)the single row Schur functions:s k(M):=s(k)(M),k=0,1,2,...;

(3)the set of power sums:

p0(M):=(Tr R I)1,p k(M):=Tr

R(1...k)(M

k

R k?1...R1),k≥1.(2.7)

These sets are connected by a series of recursive Newton and Wronski relations

(?1)k k q a k(M)+ k?1r=0(?q)r a r(M)p k?r(M)=0,(2.8) k q s k(M)? k?1r=0q?r s r(M)p k?r(M)=0,(2.9)

k r=0(?1)r a r(M)s k?r(M)=0?k≥1.(2.10) On introducing the generating functions for these sets of generators

A(t):= k≥0a k(M)t k,S(t):= k≥0s k(M)t k,P(t):=1+(q?q?1) k≥1p k(M)t k,(2.11) one can rewrite relations(2.8)–(2.10)in a compact form[Mac,I]5

P(?t)A(qt)=A(q?1t),P(t)S(q?1t)=S(qt),A(t)S(?t)=1.(2.12) For the GL(m|n)type QMA the zeroth power sum equals[GPS3]

p0(M)=q n?m(m?n)q1.(2.13) One of the remarkable properties of the Hecke QMA M(R,F)is the existence of the char-acteristic identity for the matrix M of it’s generators.To formulate the result we introduce a notion of the matrix?-product of quantum matrices(for a detailed exposition see[OP2], section4.4).Namely,starting with the quantum matrix of generators M and the scalar quantum matrices sλ(M)I?λ?k,k≥0,we construct the whole set of quantum matrices

POWER SUMS OF QUANTUM SUPERMATRIX 5

by the following recursive procedure:given any quantum matrix N ,its ?-multiplication by s λ(M )I and left ?-multiplication by M are also quantum matrices de?ned as

M ?(s λ(M )I )=(s λ(M )I )?M :=M ·s λ(M ),

M ?N :=M ·φ(N ),where φ(N )1:=Tr R (2)N

0:=I,M k

:=M ?···?M k times =Tr R (2...k )(M k R k ?1...R 1)?k >1.

We note that for the family of the so-called re?ection equation algebras —these are the QMAs of the form M (R,R )—the ?-product is identical to the usual matrix product.

The characteristic identity depends essentially on a type of the quantum matrix algebra.For the GL (m |n )type QMA it is an (m +n )-th order polynomial identity in ?-powers of the matrix M with coe?cients in the characteristic subalgebra.One has the following q -analogue of the classical Cayley-Hamilton theorem.

Theorem 3.([GPS1,GPS2])The characteristic identity for the matrix of generators of the GL (m |n )type QMA M (R,F )reads

m k =0(?q )k M

n ?r s [m |n ]r (M ) ≡0,(2.14)where we used a shorthand notation for the partitions [m |n ]k := (n +1)k ,n m ?k ,[m |n ]r := n m ,r .

Remarkably enough,for generic type QMA (i.e.,if mn >0)the characteristic identity (2.14)factorizes in two parts.Therefore,when setting factorization problem for the charac-teristic polynomial one is forced to separate all the m +n roots into two parts of sizes m and n .

Let C [μ,ν]be an algebra of polynomials in two sets of mutually commuting and alge-braically independent variables μ:={μi }1≤i ≤m and ν:={νj }1≤j ≤n .Consider a map of the coe?cients of the characteristic polynomial into C [μ,ν]

s [m |n ]k (M )→s [m |n ]k (μ,ν):=s [m |n ](μ,ν)e k (q ?1μ),

1≤k ≤m ,(2.15)s [m |n ]r (M )→s [m |n ]r (μ,ν):=s [m |n ](μ,ν)e r (?qν),1≤r ≤n ,(2.16)where e k (·)are the elementary symmetric polynomials in their arguments (e.g.,e k (μ)≡e k (μ1,...,μm )= 1≤i 1<···

expression for the polynomial s [m |n ](μ,ν).We now de?ne a central extension of the ?-product algebra of the quantum matrices by the scalar matrices of the form p (μ,ν)I ,p (μ,ν)∈C [μ,ν],such that s λ(M )I ≡s λ(μ,ν)I .In the extended algebra the characteristic identity (2.14)takes a completely factorized form

m i =1(M ?μi I )?

n j =1(M ?νj I )·(s [m |n ](μ,ν))2≡0.

6DIMITRI GUREVICH,PAVEL PYATOV,AND PAVEL SAPONOV

Assuming that s[m|n](μ,ν)=0we can interpret the variablesμi,i=1,...,m,andνj,j= 1,...,n,as eigenvalues of the quantum matrix M.They are called,respectively,“even”and “odd”spectral values of M.

The map(2.15),(2.16)admits a unique extension to a homomorphic map of the charac-teristic subalgebra into the algebra C[μ,ν]of polynomials in spectral valuesμi andνhttps://www.sodocs.net/doc/ab6539197.html,ing the Littlewood-Richardson multiplication rule(2.6)we obtain(see[GPS2])

a k(M)≡s[k|1](M)→a k(μ,ν):=

k

r=0e r(q?1μ)h k?r(?qν),(2.17)

s k(M)≡s[1|k](M)→s k(μ,ν):=

k

r=0e r(?qν)h k?r(q?1μ),(2.18)

where h k(...)stands for the complete symmetric polynomial in its variables:h k(μ)≡h k(μ1,...,μm)= 1≤i1≤···≤i k≤mμi1...μi k.Since each of the sets{a k(M)}k≥0,{s k(M)}k≥0 generates the characteristic subalgebra,the homomorphism is completely de?ned by(2.17) or by(2.18).In particular,formulas(2.17),(2.18)prescribe an explicit expression for the unspeci?ed polynomial s[m|n](μ,ν)in(2.15),(2.16),which is the image of s

[m|n]

(M):

s

[m|n](M)→s[m|n](μ,ν)=

m

i=1n j=1 q?1μi?qνj .

This homomorphic map induced by(2.17),or(2.18)is called the spectral parameterization of the characteristic subalgebra.In the next section we derive the spectral parameterization for the third generating set of the characteristic subalgebra,that is the set of power sums {p k(M)}k≥0.

3.Spectral parameterization of power sums

In this section we are working with the GL(m|n)type QMA M(R,F)de?ned by rela-tions(2.3)with an R-matrix R satisfying the criterion of the proposition1.During the considerations we assume that the parameter q is generic(see(2.5)),although afterwards this restriction can be waived out:unlike the cases of a k(M)and s k(M)the power sums p k(M)are consistently de?ned for all q∈C\0.

For the particular case of the GL(m)≡GL(m|0)type re?ection equation algebra M(R,R) the spectral parameterization of the power sums was found in[GS].Taking into account that in the GL(m)case the quantum matrix has only“even”eigenvalues{μi}1≤i≤m,the result

reads

p k(M)→

m

i=1d iμk i,where d i:=q?1m j=iμi?q?2μj

POWER SUMS OF QUANTUM SUPERMATRIX7 Lemma4.The sets of polynomials{a k(μ,ν)}k≥0,{s k(μ,ν)}k≥0(see(2.17),(2.18))and {πk(μ,ν)}k≥1satisfy the Newton’s recurrent relations

(?1)k k a k(μ,ν)+k?1

r=0(?1)r a r(μ,ν)πk?r(μ,ν)=0,(3.2)

k s k(μ,ν)?k?1

r=0s r(μ,ν)πk?r(μ,ν)=0?k≥1.(3.3)

Proof.First,we recall relations among the generating functions for the power sums and the elementary symmetric and complete symmetric polynomials in a?nite set of variables x:={x i}1≤i≤p(see[Mac],section I.2)

E(x|t):=

p

k=0e k(x)t k=p i=1(1+x i t),H(x|t):= k≥0h k(x)t k=p i=1(1?x i t)?1,

P(x|t):= k≥1p k(x)t k?1=?d dt log H(x|t).(3.4) Consider three functions depending on two sets of variables x:={x i}1≤i≤m and y:= {y i}1≤i≤n:

A(x,y|t):=E(x|t)H(?y|t),S(x,y|t):=H(x|t)E(?y|t),

Π(x,y|t):=P(x|t)?P(y|t).(3.5) These functions serve as super-matrix analogues of,respectively,generating functions of the elementary and complete symmetric polynomials and the power sums(see[Mac],section I.5, exercise27and bibliographic references for it).Indeed,using(3.4)it is easy to check that these functions satisfy relations similar to(3.4)

Π(x,y|t)=?d

dt

log S(x,y|t).(3.6)

Now the assertion of the lemma follows from an observation that A(q?1μ,qν|t),S(q?1μ,qν|t) andΠ(q?1μ,qν|t)are,respectively,generating functions for the sets of polynomials{a k(μ,ν)}k≥0, {s k(μ,ν)}k≥0and{πk(μ,ν)}k≥1.The relations(3.2)and(3.3)are just expansions of(3.6)in powers of t.

8DIMITRI GUREVICH,PAVEL PYATOV,AND PAVEL SAPONOV where the“weight”coe?cients d i and?d j explicitly read

d i:=q?1

m

p=1

p=i

μi?q?2μp

μi?νj

,(3.8)

?d

j:=?q

n

i=1νj?q?2μiνj?νp.(3.9)

Recall,that the spectral values{μi}and{νj}are supposed to be algebraically independent, and therefore all the coe?cients d i and?d j are nonzero and well de?ned.

Proof.For the proof we need yet another recursive set of formulas for the power sums {p k(μ,ν)}k≥1(3.7)and the polynomials{πk(μ,ν)}k≥1(3.1).

Lemma6.The following relations hold true

k p k(μ,ν)=k qπk(μ,ν)+(q?q?1)k?1

r=1r qπr(μ,ν)p k?r(μ,ν)?k≥1.(3.10)

In terms of the generating functions

P(μ,ν|t):=1+(q?q?1) k≥1p k(μ,ν)t kΠ(q?1μ,qν|t):= k≥1πk(μ,ν)t k?1

(P(μ,ν|t)is the spectral parameterization of P(t),see(2.11);Π(μ,ν|t)was?rst de?ned in (3.5))the relations(3.10)shortly read

P(μ,ν|t) qΠ(q?1μ,qν|qt)?q?1Π(q?1μ,qν|q?1t) =d

(z?μi)

n

j=1(z?q2νj)

z?μi Res f(z)

z?νj

Res f(z)

z=μi

=(q?q?1)μi d i,Res f(z)

POWER SUMS OF QUANTUM SUPERMATRIX9 and therefore,

p0(μ,ν)=q n?m(m?n)q1.

So,we have veri?ed the consistency of(3.7)with

our

previous result(2.13)for p0(M).

In order to prove relations(3.10)we expand functions z k f(z),k≥1,into the sum of simple fractions.Besides the simple poles inμi andνj,the function z k f(z)possesses the k-th order pole at z=∞,or,introducing a new variable y=z?1,at the point y=0.Taking into account that

Res z k f(z)

z=μi ,Res z k f(z)

z=νj

,

we come to the corresponding expansion

z k f(z)=

k

r=0z k?r z?μi+n j=1?d iνk+1j

dy r

(q?q?1)k!

?k≥1.(3.14)

Let us turn to the calculation of f k.Recalling that y=z?1we write the?rst order derivative f′(y)in the form

d f(y)

(1?μi y)(1?q?2μi y)?

n

j=1qνj

dy k

=k! m i=1k a=0q?1?2(k?a)μk+1i

(1?νj y)1+a(1?q2νj y)1+k?a .

The above relation leads immediately to

u k(0)=k!(k+1)q m i=1(q?1μi)k+1?n j=1(qνj)k+1 :=k!(k+1)qπk+1(μ,ν)?k≥0.(3.16) Now,di?erentiating the relation(3.15)at y=0gives rise to

f k=(q?q?1)k?1

r=0 r k?1 u r(0)f k?r?1,k≥1,

10DIMITRI GUREVICH,PAVEL PYATOV,AND PAVEL SAPONOV

where we use an obvious condition f 0=1.On taking into account relations (3.14)and (3.16),

we

can

easily prove

the

assertion

of

the

lemma.Indeed,

kp k

(

μ,ν)=f k (k ?1)!k ?1 r =0 r k ?1

u r (0)f k ?r ?1=k ?1 r =0(r +1)q (k ?r )!

=k q πk (μ,ν)+(q ?q ?1)k ?1 r =1r q πr (μ,ν)p k ?r (μ,ν).

At last,the equivalence of (3.10)and (3.11)is veri?ed by a direct calculation.

dt log P (μ,ν|t )A (q ?1μ,qν|?qt ) =

d References

[GPS1]“Cayley-Hamilton Theorem for Quantum Matrix Algebras of GL (m |n )type”,Algebra i Analiz,vol.

17no.1(2005),pp.160–182(in Russian).English translation in:St.Petersburg Math.J.,vol.17,no.1(2006)pp.119–135;arXiv:math.QA/0412192.

[GPS2] D.I.Gurevich,P.N.Pyatov,P.A.Saponov,“Quantum matrix algebras of the GL(m—n)type:the structure and spectral parameterization of the characteristic subalgebra”,Teor.Matem.Fiz.,vol.147,no.1(2006)pp.14–46(in Russian).English translation in:Theoretical and Mathematical Physics,vol.147,no.1(2006)pp.460–485;arXiv:math.QA/0508506.

[GPS3] D.I.Gurevich,P.N.Pyatov,P.A.Saponov,“Representation theory of (modi?ed)re?ection equation algebra of GL (m |n )type”,Algebra &Analiz,vol.20,no.2(2008)pp.70–133;arXiv:math.QA/0612815.

[GS] D.Gurevich,P.Saponov,“Geometry of non-commtative orbits related to Hecke symmetries”,Con-temporary Mathematics,vol.433,(2007)pp.209-250;arXiv:math/0411579.

[H]Hai P.H.,“Poincar′e Series of Quantum spaces Associated to Hecke Operators”,Acta Math.Vietnam 24(1999)235–246.

[I]Isaev A.P.,private communication .See remark 3.13in:Isaev,A.P.,Pyatov,P.N.,“Spectral variables for quantum group cotangent bundle:dynamical R-matrix and evolution of isotropic q -top”,Preprint MPIM 2008(2008).

[IOP1]

Isaev A.,Ogievetsky O.and Pyatov P.,“Generalized Cayley-Hamilton-Newton identities”.Czechoslo-vak J.Phys.,vol.48,no.11(1998)pp.1369–1374;arXiv:math.QA/9809047.

POWER SUMS OF QUANTUM SUPERMATRIX11 [IOP2]Isaev A.P.,Ogievetsky O.V.,Pyatov P.N.,“On quantum matrix algebras satisfying the Cayley-Hamilton-Newton identities”,J.Phys.A:Math.Gen.,vol32(1999)L115-L121;

arXiv:math.QA/9809170.

[Mac]Macdonald I.G.“Symmetric Functions and Hall Polynomials”(Oxford Mathematical Monographs), Oxford Science Publications,1995.

[OP1]Ogievetsky O.and Pyatov P.,“Lecture on Hecke algebras”,in Proc.of the International School ”Symmetries and Integrable Systems”Dubna,Russia,June8-11,1999.JINR,Dubna,D2,5-2000-218,pp.39-88;Preprint MPIM2001-40(2001).

[OP2]Ogievetsky O.V.and Pyatov P.N.,“Orthogonal and symplectic quantum matrix algebras and Cayley-Hamilton theorem for them”;arXiv:math.QA/0511618.

[R]Ram A.,“Seminormal representations of Weyl groups and Iwahori-Hecke algebras”,Proc.of London Math.Soc.vol.75(1997)99-133;arXiv:math.RT/9511223.

[U]Umeda T.,“Newton’s formula for gl n”,Proc.of the AMS,vol.126,no.11(1998)3169–3175. [W]Wenzl H.“Hecke algebras of type A n and subfactors”,Invent.Math.92(1988)349–383.

Dimitri Gurevich,Max Planck Institute for Mathematics,Vivatsgasse7,D-53111Bonn,Ger-many&USTV,Universit′e de Valenciennes,59304Valenciennes,France

E-mail address:gurevich@univ-valenciennes.fr

Pavel Pyatov,Max Planck Institute for Mathematics,Vivatsgasse7,D-53111Bonn,Ger-many&Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, 141980Dubna,Moscow Region,Russia

E-mail address:pyatov@theor.jinr.ru

Pavel Saponov,Division of Theoretical Physics,IHEP,142281Protvino,Moscow region, Russia

E-mail address:Pavel.Saponov@ihep.ru