Factorization of differential operators, quasideterminants, and nonabelian Toda field equations

Mathematical Research Letters 4, 413–425 (1997) FACTORIZATION OF DIFFERENTIAL OPERATORS, QUASIDETERMINANTS, AND NONABELIAN TODA FIELD EQUATIONS Pavel Etingof, Israel Gelfand, and Vladimir Retakh Abstract. We integrate nonabelian Toda field equations [Kr] for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants introduced in [GR1],[GR2], [GR4]. In the appendix we review some results concerning noncommutative versions of other classical integrable equations. Introduction Nonabelian Toda equations are equations with respect to n unknowns φ = (φ1 , ..., φn ) ∈ A[[u, v]], where A is some associative (not necessarily commutative) algebra with unit: (1) ∂ ∂u  ∂φj −1 φ ∂v j   −1 j=1   φ2 φ1 , −1 −1 = φj+1 φj − φj φj−1 , 2 ≤ j ≤ n − 1   j=n −φn φ−1 n−1 , Suppose that A is a ∗-algebra, i.e. it is equipped with an involutive antiautomorphism ∗ : A → A. Then, setting in (1) φn+1−i = (φ∗i )−1 , we obtain a new system of equations. If n = 2k, we get the nonabelian Toda system for root system Ck : (2) ∂ ∂u  ∂φj −1 φ ∂v j   −1 j=1   φ2 φ1 , −1 −1 2≤j ≤k−1 = φj+1 φj − φj φj−1 ,   ∗ −1 −1 −1 (φk ) φk − φk φk−1 , j = k If n = 2k + 1, we get the nonabelian Toda system for root system Bk : (3) ∂ ∂u  ∂φj −1 φ ∂v j  Received January 8, 1997.  −1 j=1   φ2 φ1 , −1 2≤j≤k = φj+1 φ−1 j − φj φj−1 ,   ∗ −1 ∗ −1 (φk ) φk+1 − φk+1 φk , j = k + 1, 413 414 PAVEL ETINGOF, ISRAEL GELFAND, AND VLADIMIR RETAKH where φ∗k+1 = φ−1 k+1 . Quasideterminants were introduced in [GR1], as follows. Let X be an m × mmatrix over A. For any 1 ≤ i, j ≤ m, let ri (X), cj (X) be the i-th row and the j-th column of X. Let X ij be the submatrix of X obtained by removing the i-th row and the j-th column from X. For a row vector r let r(j) be r without the j-th entry. For a column vector c let c(i) be c without the i-th entry. Assume that X ij is invertible. Then the quasideterminant |X|ij ∈ A is defined by the formula (4) |X|ij = xij − ri (X)(j) (X ij )−1 cj (X)(i) , where xij is the ij-th entry of X. In this paper we will use quasideterminants to integrate nonabelian Toda equations for root systems of types A, B, C. We do not know yet how to generalize these results to root systems of types D-G, and to affine root systems. Our method of integration, which is based on interpreting the Toda flow as a flow on the space of factorizations of a fixed ordinary differential operator. This method and explicit solutions of Toda equations are well known in the commutative case (see [LS]; see also [FF] and references therein; for the one variable case see [Ko]). Remark. As it was mentioned in [AP], some version of abelian Toda equations was essentially considered by J.J. Sylvester [S] and G. Darboux [Da]. Nonabelian Toda equations for the root system An−1 were introduced by Polyakov (see [Kr]). Nonabelian Toda lattice for functions of one variable appeared in [PC], [P]. I.M. Krichever [Kr] constructed algebraic-geometric solutions for the periodic twodimensional nonabelian Toda lattice (affine An root system). 1. Factorization of differential operators Let R be an associative algebra over a field k of characteristic zero, and D : R → R be a k-linear derivation. Let f1 , ..., fm be elements of R. By definition, the Wronski matrix W (f1 , ..., fm ) is   f1 ... fm ... Dfm   Df1 W (f1 , ..., fm ) =   .. ... .. Dm−1 f1 ... Dm−1 fm We call a set of elements f1 , ..., fm ∈ R nondegenerate if W (f1 , ..., fm ) is invertible. Denote by R[D] the space of polynomials of the form a0 Dn +a1 Dn−1 +...+an , ai ∈ R. It is clear that any element of R[D] defines a linear operator on R. Example. R = C ∞ (R), D = operators on the line. d dt . In this case, R[D] is the algebra of differential NONABELIAN TODA FIELD EQUATIONS 415 By analogy with this example, we will call elements of R[D] differential operators. We will consider operators of the form L = Dn + a1 Dn−1 + ... + an . We will call such L an operator of order n with highest coefficient 1. Denote the space of all such operators by Rn (D). Theorem 1.1. (i) Let f1 , ..., fn ∈ R be a nondegenerate set of elements. Then there exists a unique differential operator L ∈ R[D] of order n with highest coefficient 1, such that Lfi = 0 for i = 1, ..., n. It is given by the formula (1.1) Lf = |W (f1 , ..., fn , f )|n+1,n+1 . (ii) Let L be of order n with highest coefficient 1, and f1 , ..., fn be a set of solutions of the equation Lf = 0, such that for any m ≤ n the set of elements f1 , ..., fm is nondegenerate. Then L admits a factorization L = (D − bn )...(D − b1 ), where (1.2) bi = (DWi )Wi−1 , Wi = |W (f1 , ..., fi )|ii . Proof. (i) We look for L in the form L = Dn +a1 Dn−1 +...+an . From the equations Lfi = 0 it follows that (an , ..., a1 ) = −(Dn f1 , ..., Dn fn )W (f1 , ..., fn )−1 . By definition, |W (f1 , ..., fn , f )|n+1,n+1 = Dn f − (Dn f1 , ..., Dn fn )W (f1 , ..., fn )−1 (f, Df, ..., Dn−1 f )T = Dn f + (an , ..., a1 )(f, Df, ..., Dn−1 f )T = Lf. (ii) We will prove the statement by induction in n. For n = 1, the statement is obvious. Suppose it is valid for the differential operator Ln−1 of order n − 1 with highest coefficient 1, which annihilates f1 , ..., fn−1 (by (i), it exists and is unique). Set bn = (DWn )Wn−1 , and consider the operator L̃ = (D − bn )Ln−1 . It is obvious that L̃fi = 0 for i = 1, ..., n − 1. Also, by (i) L̃fn = (D − bn )Ln−1 fn = (D − bn )Wn = 0. Therefore, by (i), L̃ = L.
416 PAVEL ETINGOF, ISRAEL GELFAND, AND VLADIMIR RETAKH Now consider the special case: R = A[[t]], where A is an associative algebra d over k, and D = dt (here t commutes with everything). In this case, it is easy to show that nondegenerate sets of solutions of Lf = 0 exist, and are in 1-1 correspondence with elements of the group GLn (A), via f = (f1 , ..., fn ) → W (f )(0). It is clear that two different sets of solutions of the equation Lf = 0 can define the same factorization of L. However, to each factorization γ of L we can assign a set fγ = (f1 , ..., fn ) of solutions of Lf = 0, which gives back the factorization γ under the correspondence of Theorem 1.1(ii). This set is uniquely defined by the condition that the matrix W (fγ )(0) is lower triangular with 1-s on the diagonal. Here is a formula for computing fγ , which is well known in the commutative case. Proposition 1.2. If γ has the form L = (D − (Dgn )gn−1 )...(D − (Dg1 )g1−1 ), where gi (0) = 1, then fγ = (f1 , ..., fn ), where fj (t) =  t  t1  tj−2 (1.3) ... g1 (t)g1 (t1 )−1 g2 (t1 )g2 (t2 )−1 ...gj (tj−1 )dtj−1 ...dt2 dt1 , 0 0 0 u   i+1 where 0 ( ai ti )dt := ai ui+1 . Proof. It is easy to see that if f = (f1 , ..., fn ), with fj given by (1.3), then W (f )(0) is strictly lower triangular. So it remains to show that fj is a solution of the equation Lj f = 0, where Lj = (D − (Dgj )gj−1 )...(D − (Dg1 )g1−1 ). We prove this by induction in j. The base of induction is clear, since from (1.3) we get f1 = g1 . Let us perform the induction step. By the induction assumption, from (1.3), we have  t fj (t) = g1 (t) g1 (s)−1 h(s)ds, 0 where h obeys the equation (D − (Dgj )gj−1 )...(D − (Dg2 )g2−1 )h = 0. Thus, we get (D − (Dg1 )g1−1 )fj = h. This proves that Lj fj = 0.
For matrix-valued functions a more general version of proposition 1.2 for the periodic case was proved in [Kr3]. Now consider an application of these results to the noncommutative Vieta theorem [GR3]. Let A be an associative algebra. Call a set of elements x1 , ..., xn ∈ A generic if their Vandermonde matrix V (x1 , ..., xn ) (Vij := xi−1 j ) is invertible. Consider an algebraic equation (1.4) xn + a1 xn−1 + ... + an = 0. with ai ∈ A. Let x1 , ..., xn ∈ A be solutions of (1.4) such that x1 , ..., xi form a generic set for each i. Let V (i) = V (x1 , ..., xi ), and yi = |V (i)|ii xi |V (i)|−1 ii . NONABELIAN TODA FIELD EQUATIONS 417 Theorem 1.3. [GR3] (Noncommutative Vieta theorem) (1.5) ar = (−1)r  yir ...yi1 . i1 <...<ir Proof. (Using differential equations.) Consider the differential operator with constant coefficients in R = A[[t]]: L = Dn + a1 Dn−1 + ... + an . We have solutions fi = etxi of the equation Lf = 0, and for any i the set f1 , ..., fi is nondegenerate, since its Wronski matrix W (i) is of the form (1.6) W (i) = V (i)diag(etx1 , ..., etxi ). Thus, by Theorem 1(ii), the operator L admits a factorization L = (D − bn )...(D − b1 ), where bi = (D|W (i)|ii )|W (i)|−1 ii . Substituting (1.6) into this equation, we get (1.7) bi = |V (i)|ii xi |V (i)|−1 ii = yi . Since [D, bi ] = 0, we obtain the theorem. Note, that the connection between matrix algebraic equations and matrix differential equations appeared in [CS]. 2. Toda equations In this section we will solve equations (1),(2),(3). Let M = {(g, h) ∈ A[[t]] ⊕ A[[t]] : g(0) = h(0) ∈ A∗ }, where A∗ is the set of invertible elements of A. We have the following obvious proposition, which says that solutions of Toda equations are uniquely determined by initial conditions. Proposition 2.1. The assignment φ(u, v) → (φ(u, 0), φ(0, v)) is a bijection between the set of solutions of the Toda equations (1) and the set M . Let B be an algebra over k, R = B[[v]], D = φi ∈ B[[v]] are invertible, define Lφi ∈ R[D] by (2.1) d dv . For φ = (φ1 , ..., φn ), where −1 Lφi = (D − (Dφi )φ−1 i )...(D − (Dφ1 )φ1 ). Now set B = A[[u]]. Then B[[v]] = A[[u, v]], so for any φ = (φ1 , ..., φn ), with all φi ∈ A[[u, v]] invertible, we can define Lφi . 418 PAVEL ETINGOF, ISRAEL GELFAND, AND VLADIMIR RETAKH Proposition 2.2. A vector-function φ(u, v) is a solution of Toda equations (1) if and only if ∂Lφi ∂Lφn φ = −φi+1 φ−1 = 0. i Li−1 , i ≤ n − 1; ∂u ∂u (2.2) Proof. Let φ be a solution of the Toda equations. Set bi = (Dφi )φ−1 i . We have φ Li = Li = (D − bi )...(D − b1 ). Therefore, for i ≤ n − 1 we have i  ∂Li (D − bi )...(D − bj+1 )φj+1 φ−1 =− j (D − bj−1 )...(D − b1 ) ∂u j=1 (2.3) + i−1  (D − bi )...(D − bj+2 )φj+1 φ−1 j (D − bj )...(D − b1 ). j=1 But −1 (D − bj+1 ) ◦ φj+1 φ−1 j = φj+1 φj (D − bj ). Therefore, the right hand side of (2.3) equals −φi+1 φ−1 i Li−1 . The same argu∂Ln ment shows that for i = n the derivative ∂u vanishes. Conversely, it is easy to see that equations (2.2) imply (1).
Remark. This proposition is just a reformulation of the well-known statement that the two-dimensional Toda lattice is equivalent to a compatibility condition for the linear system ∂v ξj = ξj+1 + (∂v φj )φ−1 j ξj ∂u ξj = −(φj φ−1 j−1 )ξj−1 with ξn+1 = ξ0 = 0. (Here ξj = Lj−1 f , where Lf = 0). Now we will compute the solutions of Toda equations explicitly. Let η1 , ..., ηn , ψ1 , ..., ψn ∈ A[[t]] be such that ηi (0) = ψi (0) ∈ A∗ . We will find the solution of the following initial value problem for Toda equations: (2.4) φi (u, 0) = ψi (u), φi (0, v) = ηi (v) Proposition 2.1 states that this solution exists and is unique. Let gi (v) = ηi (v)ηi (0)−1 . Let f = (f1 , ..., fn ) be given by formula (1.3) in terms of gi . Define the lower triangular matrix ∆(u) whose entries are given by the formula ∆ij (u) =  u  t1  ti−j−1 −1 ... ψi (ti−j )ψi−1 (ti−j )ψi−1 (ti−j−1 )...ψj−1 (t1 )ψj (u)dti−j ...dt1 . 0 0 0 Let f u = (f1u , ..., fnu ) be defined by the formula f u = f ∆(u). Then we have NONABELIAN TODA FIELD EQUATIONS 419 Theorem 2.3. The solution of the problem (2.3) is given by the formula φi (u, v) = |W (f1u , ..., fiu )|ii . (2.5) φ(u,v) , where φ(u, v) is defined by (2.5). By Theorem 1.1(ii), Proof. Let Lui = Li we have Lui fju = 0 for j ≤ i. Differentiating this equation with respect to u, we get ∂fju ∂Lui u + f = 0. (2.6) Lui ∂u ∂u j On the other hand, it is easy to see that ∆′ (u) = ∆(u)Θ(u), where This implies that Therefore, Lui  −1 ′   ψi (u)ψi (u) i = j Θij (u) = 1 i=j+1   0 otherwise ∂fiu u + fiu ψi−1 ψi′ . = fi+1 ∂u ∂fju ∂u = 0 for i ≥ j + 1, and ∂fju u u = |W (f1u , ..., fi+1 )|i+1i+1 = φi+1 . Lui = Lui fi+1 ∂u Thus, from (2.6) we get ∂Lui u u f = −φi+1 φ−1 i Li−1 fj , i ≥ j. ∂u j By Theorem 1.1 (i), this implies equations (2.2), which are equivalent to Toda equations. It is obvious that φ(u, v) satisfies the required initial conditions. The theorem is proved.
If initial conditions (2.4) satisfy the symmetry property φn+1−i = (φ∗i )−1 , then we obtain a solution of the initial value problem for equations (2) (for even n), and (3) (for odd n). This gives a complete description of solutions of systems of equations (1),(2),(3). In the case φ(u, 0) = 1, the Toda flow can be interpreted as a flow on the space of factorizations of a differential operator. Namely, let L be a differential operator of order n with highest coefficient 1. Let F (L) be the space of factorizations of L. Let Nn− be the group of strictly lower triangular matrices over A. It is easy to see that the map π : F (L) → Nn− , given by π(γ) = W (fγ )(0), is a bijection. We will identify F (L) with Nn− using π. Let γ(u) = γ(0)euJn be a curve on Nn− generated by the 1-parameter subgroup uJn e , where Jn is the lower triangular nilpotent Jordan matrix ((Jn )ij = δi,j+1 ). Let L = (D − bun )...(D − bu1 ) be the factorization of L corresponding to the point ∂ ). γ(u). Let φi (u, v) be such that (Dφi )φ−1 = bui , and φi (u, 0) = 1. (here D = ∂v i Then we have the following Corollary from Theorem 2.3. 420 PAVEL ETINGOF, ISRAEL GELFAND, AND VLADIMIR RETAKH Corollary 2.4. φ = (φ1 , ..., φn ) is a solution of Toda equations (1), and all solutions with φ(u, 0) = 1 are obtained in this way. Proof. For the proof it is enough to observe that if ψi (u) = 1 then Θ(u) = Jn , and ∆(u) = euJn .
Remark. An analogous statement can be made for equations (2) and (3). In this case, instead of an arbitrary differential operator L ∈ Rn (D) one should consider a selfadjoint (respectively, skew-adjoint) operators, instead of the group Nn− – a maximal nilpotent subgroup in Spn (A) (respectively, On (A)), and instead of Jn the sum of simple root elements in the Lie algebra of this group. In the commutative case, such a picture of the Toda flow is known for all simple Lie groups [FF]. Consider now the statement of Theorem 2.3 for i = 1. In this case we have φ1 (u, v) = f1u (v) = (2.7) n  fi0 (v)∆i1 (u). i=1 Thus, we get Corollary 2.5. [RS] If φ is a solution of the Toda equations then φ1 (u, v) =  n i=1 pi (u)qi (v), where pi , qi are some formal series. Now let us discuss infinite Toda equations. These are equations (1) with n = ∞. These equations allow to express φi recursively in terms of f = φ1 , which can be done using quasideterminants:  ∂ i−1 f ∂f , ..., i−1 , φi = |Yi (f )|ii , Yi (f ) := W f, ∂u ∂u  (2.8) i ∂ f where ∂u i are regarded as functions of v for a fixed u. This formula is easily proved by induction. It appears in [GR2], section 4.5 in the case u = v, and in [RS] in the 2-variable case. Formula (2.8) can be used to give another expression for the general solution of finite Toda equations, which appears in [RS]. Indeed, we have the following easy proposition. Proposition 2.6. Let f ∈ A[[u, v]] be such that Y1 (f ), ..., Yn (f ) are invertible matrices. Then |Yn+1 (f )|n+1n+1 = 0 if and only if f is “kernel of rank n”, i.e. (2.9) f= n  pi (u)qi (v). i=1 Thus, taking f = φ1 of the form (2.9), with Y1 , ..., Yn invertible, and using formula (2.6) we will get a solution (φ1 , ..., φn ) of the finite Toda system of length n. It is not difficult to show that in this way one gets all possible solutions. NONABELIAN TODA FIELD EQUATIONS 421 Example. Consider the nonabelian Liouville equation   ∂ ∂φ −1 (2.10) φ = (φ∗ )−1 φ−1 , ∂u ∂v which is a special case of (2) for k = 1. Consider the initial value problem (2.11) φ(0, v) = η(v), φ(u, 0) = ψ(u), η(0) = ψ(0) = a. By Theorem 2.3, we get the following formula for the solution: (2.12)    v u −1 ∗ −1 ∗ ∗ −1 −1 −1 η (t)(η ) (t)a (ψ ) (s)ψ (s)dsdt ψ(u). φ(u, v) = η(v) a + 0 0 For example, if ψ(u) = 1, we get   (2.13) φ(u, v) = η(v) 1 + u v η −1 ∗ −1 (t)(η ) 0  (t)dt . In the commutative case, these formulas coincide with the standard formulas for solutions of the Liouville equation. Appendix: Noncommutative soliton equations In this appendix we will review some results about noncommutative versions of classical soliton hierarchies (KdV,KP). These results are mostly known or can be obtained by a trivial generalization of the corresponding commutative results, but they have never been exposed systematically. We will follow Dickey’s book [D]. A1. Nonabelian KP hierarchy. Nonabelian KP hierarchy is defined in the same way as the usual KP hierarchy [D]. Let (A1) L = ∂ + w0 ∂ −1 + w1 ∂ −2 + .... d , wi ∈ A[[x, t1 , t2 , ...]], where be a formal pseudodifferential operator. Here ∂ = dx A is an associative (not necessarily commutative) algebra with 1. Consider the following infinite system of differential equations: (A2) ∂L = [Bm , L], Bm = (Lm )+ , ∂tm where for a pseudodifferential operator M , we denote by M+ , M− the differential and the integral parts of M (i.e. all terms with nonnegative, respectively m negative, powers of ∂). For brevity we will write Lm ± instead of (L )± . Each of the differential equations (A2) defines a formal flow on the space P m of pseudodifferential operators of the form (A1). Indeed, [Lm + , L] = −[L− , L], m and the order of [L− , L] is at most zero. 422 PAVEL ETINGOF, ISRAEL GELFAND, AND VLADIMIR RETAKH Proposition A1. The flows defined by equations (A2) commute with each other. Proof. The proof is the same as the proof of Proposition (5.2.3) in [D]. Namely, one needs to check the zero curvature condition (A3) ∂Bm ∂Bn − − [Bm , Bn ] = 0, ∂tn ∂tm which is done by a direct calculation given in [D]. The hierarchy of flows defined by (A2) for m = 1, 2, 3, ... is called the noncommutative KP hierarchy. A2. Noncommutative KdV and nKdV hierarchies. As in the commutative case, we can restrict the KP hierarchy to the subspace Pn ⊂ P of all operators L such that Ln is a differential operator, i.e. Ln− = 0 ∂Ln n [D]. This subspace is invariant under the KP flows, as ∂tm− = [Lm + , L− ]− . The space Pn can be identified with the space Dn of differential operators of the form (A4) M = ∂ n + u2 ∂ n−2 + ... + un , by the map L → M = Ln . The KP hierarchy induces a hierarchy of flows on Dn called the nKdV hierarchy: (A5) ∂M m/n = [M+ , M ]. ∂tm Among these flows, the flows corresponding to m = nl, l ∈ Z+ , are trivial, but the other flows are nontrivial. For n = 2 the nKdV hierarchy is the usual KdV hierarchy. The first two nontrivial equations of this hierarchy are (A6) ut1 = ux , ut3 = 1 (uxxx + 3ux u + 3uux ). 4 The second equation is the noncommutative KdV equation. A3. Finite-zone solutions of the nKdV hierarchy. Denote the vector p fields of the noncommutative nKdV hierarchy by Vm , m = nl. Let V = i=1 ai Vi , ap = 0, be a finite linear combination of Vi with coefficients ai ∈ l. Let S(a), a = (a1 , ..., ap ) be the set of stationary points of V in Dn = A[[x]]n−1 . The set S(a) can be identified with A(n−1)p . Indeed, S(a) is the subset of Dn defined by the differential equations V (M ) = 0, which have the form (A7) dp ui (p−1) = Fi (uj , u′j , ..., uj ). dxp NONABELIAN TODA FIELD EQUATIONS 423 So ui are uniquely determined by their first p derivatives at 0. Since the vector field V commutes with the nKdV hierarchy, S(a) is invariant under the flows of this hierarchy. Using the identification S(a) → A(n−1)p , we can rewrite each of the nKdV flows as a system of ordinary differential equations on A(n−1)p . Thus, solutions of nKdV equations belonging to S(a) can be computed by solving ordinary differential equations. Such solutions are called finite-zone solutions.  ai xi . Thus, Remark. If M ∈ S(a) then [Q(M 1/n )+ , M ] = 0, where Q(x) = we have two commuting differential operators M and N = Q(M 1/n )+ in 1 variable. If the algebra A is finite dimensional over its center then there exists a nonzero polynomial R(x, y) with coefficients in the center of A, such that R(M, N ) = 0. This polynomial defines an algebraic curve, called the spectral curve. The operator M can then be computed explicitly using the method of Krichever [Kr2]. Similarly, all nKdV equations resticted to the space of such operators can be solved explicitly in quadratures. However, if A is infinitedimensional over its center, the polynomial R does not necessarily exist, and we do not know any way of computing M explicitly. A4. Multisoliton solutions of the noncommutative KP hierarchy. Here we will construct N-soliton solutions of the KP and KdV hierarchies in the noncommutative case. We will use the dressing method. In the exposition we will closely follow Dickey’s book [D]. We will now construct a solution of the KP hierarchy. Let t = (t1 , t2 , ...) Consider the formal series (A8) ξ(x, t, α) = (x + t1 )α + t2 α2 + ... + tr αr + ..., α ∈ A. Fix α1 , ..., αN , β1 , ..., βN , a1 , ..., aN ∈ A, and set (A9) ys (x, t) = eξ(x,t,αs ) + as eξ(x,t,βs ) . Define the differential operator of order N , with highest coefficient 1, by (A10) Φf (x) = |W (y1 , ..., yN , f )|N +1,N +1 (x), where the derivatives in the Wronski matrix are taken with respect to x, and we assume that the functions y1 , ..., yN are a generic set (in the sense of Chapter 1). Set (A11) L = Φ∂Φ−1 . Proposition A2. The operator-valued series L(x, t) is a solution of the KP hierarchy. Proof. The proof is the same as the proof of Proposition 5.3.6 in [D]. Such solutions are called N-soliton solutions. 424 PAVEL ETINGOF, ISRAEL GELFAND, AND VLADIMIR RETAKH Proposition A2 can be used to construct N-soliton solutions of the nKdV hierarchy. Namely, we should restrict the above construction to the case when βk = εk αk , where εk is an n-th root of unity. In this case it can be shown as in [D] that the operator M = Ln = Φ∂ n Φ−1 is a differential operator of order n, and it is a solution of the nKdV hierarchy. As an example, let us consider the N-soliton solutions of the KdV hierarchy. In this case, we may set t2k = 0, and we have ys = eξ(x,t,αs ) + as e−ξ(x,t,αs ) . (A12) Proposition A3. Let (A13) bi = (∂Wi )Wi−1 , Wi := |W (y1 , ..., yi )|ii . Then the function (A14a) N  u(x, t) = 2∂( bi ). i=1 is a solution of the noncommutative KdV hierarchy. 2 2 Proof. Let Φ = ∂ N + v1 ∂ N −1 + .... Then from the equation  (∂ + u)Φ = Φ∂ we obtain u = −2∂v1 , and from Theorem 1.1(ii) v1 = − i bi , where bi are given by (A13). This implies (A14a). Another formula for u, which is equivalent to (A14a), is the following. Let Y (y1 , ..., yN ) be the matrix which coincides with the Wronski matrix (N ) (N −1) instead of yi . Let W (y1 , ..., yN ) except at the last row, where it has yi YN = |Y (y1 , ..., yN )|N N . Then the function u given by (A14a) can be written as (A14b.) u = 2∂(YN WN−1 ), In the commutative case formulas (A14a),(A14b) reduce to the classical formula (A15) u = 2∂ 2 ln[detW (y1 , ..., yN )]. In particular, we can obtain N-soliton solutions of the noncommutative KdV equation ut = 41 (uxxx + 3ux u + 3uux ). For this purpose set ti = 0, i = 3, and t3 = t. Then we get (A16) 3 3 ys = eαs x+αs t + as e−αs x−αs t , and the solution u(x, t) is given by (A14a),(A14b). For example, consider the 1-soliton solution. According to (A14a), it has the form 3 3 3 3 ∂ (A17) u = 2 [(eαx+α t − ae−αx−α t )α(eαx+α t + ae−αx−α t )−1 ]. ∂x In the commutative case, it reduces to the well known solution 1 2α2 , c = ln a, (A18) u= 2 3 cosh (αx + α t − c) 2 – the solution corresponding to the solitary wave which was observed by J. S. Russell in August of 1834. NONABELIAN TODA FIELD EQUATIONS 425 References [AP] [CS] [Da] [D] [FF] [GR1] [GR2] [GR3] [GR4] [Ko] [Kr] [Kr2] [Kr3] [LS] [P] [PC] [RS] [S] H. Au-Yang and J. H. H. Perk, Critical correlations in a Z-invariant inhomogeneous Ising model, Physica A 144 (1987), 44–104. A. Connes and A. Schwarz, Matrix Vieta theorem revisited, Lett. in Math. Physics (to appear). G. Darboux, Leçons sur la théorie générale des surfaces et les applications géometriques du calcul infinitésimal, deuxiéme partie, livre IV, Ch. VI, GauthiersVillars et fils, Paris, 1889. L. A. Dickey, Soliton equations and Hamiltonian systems, Adv. Ser. Math. Phys., 12, World Scientific Publishing Co., River Edge, NJ, 1991. B. Feigin and E. Frenkel, Integrals of motion and quantum groups, Lecture Notes in Math., 1620, Springer, Berlin, 1996. I. Gelfand and V. Retakh, Determinants of matrices over noncommutative rings, Funct. An. Appl. 25 (1991), 91–102. , A theory of noncommutative determinants and characteristic functions of graphs, Funct. An. Appl. 26 (1992), 1–20. , Noncommutative Vieta theorem and symemtric functions, Gelfand Math. Seminars 1993-95, Birkhauser, Boston, 1996. , A theory of noncommutative determinants and characteristic functions of graphs, I, in: Publ. LACIM, UQAM 14 (1993), 1–26. B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), 195–338. I. M. Krichever, The periodic non-abelian Toda chain and its two-dimensional generalization, Russian Math. Surveys 36 (1981), 82–89. , Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys 32 (1977), 185–213. , Non-linear equations and elliptic curves, J. Soviet Math. 28 (1985), 51–90. A. N. Leznov and M. V. Saveliev, Representation of zero curvature for the system of nonlinear partial differential equations xα,zz̄ = exp(kx)α and its integrability, Lett. Math. Phys 3 (1979), 488–494. J. H. H. Perk, Equations of motions for the transverse correlations of the onedimensional XY Z-model at finite temperature, Phys. Lett. A 92 (1980), 1–5. J. H. H. Perk and H. W. Capel, Transverse correlations in the inhomogeneous XY model at infinite temperature, Phys. A 92 (1978), 163–184. A. V. Razumov and M. V. Saveliev, Maximally nonabelian Toda systems, LPTENS96/69, hep-th 9612081. J. J. Sylvester, Sur une classe nouvelle déquations diffentièlles et déquations aux différences finies d’une intégrable, C. R. Acad. Sci. 54 (1862), 129, 170. Department of Mathematics, Harvard University, Cambridge, MA 02138 E-mail address: etingof@math.harvard.edu, retakh@math.harvard.edu Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 E-mail address: igelfand@math.rutgers.edu