Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann, Kontsevich and Manin \cite{Kaufmann,KMK}. By specializing this construction, using a fixed Frobenius algebra $\mathcal{A},$ one can arrive at such a theory. More generally one can apply the same idea to construct an $\mathcal{A}$-valued Topological Quantum Field Theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed and, as an example, an $\mathcal{A}$-valued modified Camassa-Holm equation is constructed.


Introduction
Of the many ways to generalize the Korteweg-de Vries equation u t = u xxx + 6uu x , the one that will be of most relevance to this paper is the matrix generalization (see, for example, [3,4]) where the two first derivative terms are required due to the non-commutativity of matrix multiplication. If one restricts such an equation to the space of commuting matrices one arrives at the equation U t = U xxx + 6UU x which is identical in form to the original KdV equation but with a matrix-valued, as opposed to a scalar-valued, field (see, for example, [23,42]). The purpose of this paper is to construct A-valued, where A is a Frobenius algebra, generalizations of integrable systems, starting with those associated to an underlying Frobenius manifold M.
The starting point for the study of such A-valued hierarchies is the classical construction of Dubrovin [12] which associates to a Frobenius manifold a bi-Hamiltonian hierarchy of hydrodynamic type. By constructing the tensor product [20,21] of such a manifold with a trivial Frobenius manifold (i.e. a fixed algebra) one automatically obtains a new Frobenius manifold and hence a bi-Hamiltonian hierarchy. The component fields of this new hierarchy can then be reassembled to form an A-valued hierarchy. The important feature of this construction is a simple, explicit, form of the new prepotential that defines the A-valued hierarchies.
More explicitly, given a Frobenius algebra A with basis e i , i = 1 , . . . , n , one can replace the flat coordinates of a Frobenius manifold M with A-valued fields via the mapˆ: t α → t α = t (αi) e i and this action can be extended to functions, at least in the case of analytic Frobenius manifolds. Conversely, an A-valued field can be reduced to a scalar field via the Frobenius one-form (or trace form) ω . This construction is described in Section 2. The main result is the following: Main Theorem. (Theorem 2.9) Let F be the prepotential of a Frobenius manifold M and let A be a Frobenius algebra with 1-form ω . The function F = ω F defines a Frobenius manifold, namely the manifold M ⊗ A .
Normally the prepotential of a tensor product of Frobenius manifolds bears little resemblance to the underlying prepotentials, and in any case is only defined implicitly from the original prepotentials. However when one of the manifolds is trivial, the above closed form of the new prepotential exists and this enables the resulting hierarchies to be constructed explicitly.
In Section 3 the construction of the associated hydrodynamic hierarchies is given. The deformed flat coordinates can be described very simply, and these form the Hamiltonian densities for the new evolution equations. By reassembling the fields these equations can be written as A-valued evolution equations. To write these in Hamiltonian form requires the definition of a functional derivative with respect to an A-valued field, and such a derivative was defined in [26] and with this one can write the flow equations as A-valued bi-Hamiltonian evolution equations.
Such equations are of hydrodynamic type, and while the deformation programme of Dubrovin and Zhang [16] could be applied directly to the equations, a different approach is applied in Section 4 to obtain dispersive bi-Hamiltonian systems. This is achieved via the use of A-valued Lax equations. The construction of these hierarchies -and the A-valued KP hierarchy in particular -follows very closely the scalar construction. The construction of the bi-Hamiltonian structures for these hierarchies is more subtle (Theorem 4.3) and relies on the adoption of the Adler-Gelfand-Dickey (or AGD) scheme [1,10,17] to A-valued fields.
The reduction to the A-valued Gelfand-Dickey (or GD m ) hierarchy is also studied and hence, by taking the appropriate dispersionless limit, the relation to the Frobenius construction given in section 2 and 3 is derived. As an application, we obtain an affirmative answer to Conjecture 5.1 in [42]. The ideas can also be applied to the A-valued Toda lattice hierarchy and its reductions, and this is outlined at the end of Section 4. Section 5 is devoted to various conclusion and suggestions for further study.
2. Frobenius manifolds and their tensor products 2.1. Frobenius algebras and manifolds. We begin with the definition of a Frobenius algebra [12].
Definition 2.1. A Frobenius algebra {A, •, e, ω} over R satisfies the following conditions: (i) • : A × A → A is a commutative, associative algebra with unit e; (ii) ω ∈ A ⋆ defines a non-degenerate inner product a, b = ω(a • b) .
Since ω(a) = e, a the inner product determines the form ω and visa-versa. This linear form ω is often called a trace map. One dimensional Frobenius algebras are trivial: the requirement of an identity and the non-degeneracy of the inner product determines the algebra uniquely and the inner product up to a non-zero constant. Two dimensional algebra are easily classified.
Example 2.3. Let A be an n-dimensional nonsemisimple commutative associative algebra Z n over R with a unity e and a basis e 1 = e, · · · , e n satisfying Taking Λ = (δ i,j+1 ) ∈ gl(m, R), one could obtain a matrix representation of A as e j → Λ j−1 , j = 1, · · · , n.
Similarly, for any a = n k=1 a k e k ∈ A, we introduce n trace-type maps, called "basic" trace-type maps, as follows ω k−1 (a) = a k + a n (1 − δ k,n ), k = 1, · · · , n. (2.5) Every trace map ω k induces a nondegenerate symmetric bilinear form on A given by Thus all of {A, •, e, ω k−1 } are nonsemisimple Frobenius algebras, denoted by Z n,k−1 for k = 1, · · · , n. We remark that if we consider a linear combination of n "basic" trace-type maps as then {A, •, e, tr n } is also a Frobenius algebra which is exactly the algebra {Z n , tr n } used in [42].
A Frobenius manifold has such a structure on each tangent space. (iv) A vector field E exists, linear in the flat-variables, such that the corresponding group of diffeomorphisms acts by conformal transformation on the metric and by rescalings on the algebra on T t M .
These axioms imply the existence of the prepotential F which satifies the WDVVequations of associativity in the flat-coordinates of the metric (strictly speaking only a complex, non-degenerate bilinear form) on M . The multiplication is then defined by the third derivatives of the prepotential: and indices are raised and lowered using the metric η αβ = ∂ ∂t α , ∂ ∂t β .
Example 2.5. Suppose c k ij are the structure constants for the Frobenius algebra A, so e i • e j = c k ij e k and η ij = e i , e j . For such an algebra one obtains a cubic prepotential The Euler vector field takes the form E = i t i ∂ ∂t i and E(F ) = 3F . The notation A will be used for both the algebra and the corresponding manifold.
Motivated by the classical Künneth formula in cohomology, Kaufmann, Kontsevich and Manin [20,21] constructed the tensor product of two Frobenius manifolds M ′ and M ′′ , denoted M ′ ⊗ M ′′ . The following formulation of this construction is taken from [13]. This formulation also gives criteria to check if a particular manifold is the tensor product of two more basic manifolds. For simplicity we use the notation by pairs t (α ′ α ′′ ) , α ′ = 1 , . . . , n ′ , α ′′ = 1 , . . . , n ′′ , and the unit vector field is e = ∂ ∂t (11) and the metric , has the form that is: (iii) If the Euler vector fields of the two manifolds M and M ′′ take the form with scaling dimensions d ′ and d ′′ respectively, then the Euler vector field on M takes the form Such products describe the quantum cohomology of a product of varieties, and within singularity theory it appears when one takes the direct sum of singularities.

2.2.
Tensor products with trivial algebras. We now take the tensor product of a Frobenius manifold M with a trivial manifold A defined by a Frobenius algebra (Example 2.5). To emphasize the different roles played by M and A we alter the general notation for tensor products as described above. The tensor product will be written as M A , (so M A = M ⊗ A). The basis e i for A will be retained and the unit denoted by e 1 . Thus notation such as e = ∂ 1 will not be used. Latin indices will be reserved for A-related objects, and Greek indices will be reserved for M-related objects. Thus c γ αβ will denote the structure functions for the multiplication on M and c k ij will denote the structure constants for the multiplication on A . Coordinates on M A are denoted No confusion should arise with this notation.
We begin by constructing a lift of a scalar valued function to an A-valued function and visa-versa.
Definition 2.7. Let f be an analytic function on M (that is, analytic in the flat coordinates for M). The A-valued functionf is defined to be: Since the function is analytic and the algebra A is commutative and associative the above construction is well-defined.
Remark 2.8. This definition requires the existence of a distinguished coordinate system on M in which the function f is analytic. In the case of analytic Frobenius manifolds one automatically has such a distinguished system of coordinates, namely the flat coordinates of the metric.
With these definitions one may construct a new prepotential from the original one. Theorem 2.9. Let F be the prepotential of a Frobenius manifold M and let A be a Frobenius algebra with 1-form ω . The function However such a map is not unique and the tensor structure is lost.
Proof. The proof is in two parts: we first show that the prepotential F defines a Frobenius manifold, and then identify this with the tensor product M ⊗ A .
By construction we have an nm-dimensional manifold with coordinates t (αi) , α = 1 , . . . , m = dimM , i = 1 , . . . , n = dimA . We begin with two simple results: More generally, using the properties of the multiplication on A , (2.7) • The fundamental result that will be used in the following is: This will be used to separate out the A-valued part of various expressions.
With these,

Normalization
We define η (αi)(βj) by , and e 1 is the unit for the multiplication on A . This is non-degenerate (since by assumption η αβ and η ij are non-degenerate) and this will be taken to be the metric and used to raise and lower indices. In particular, η (αi)(βj) = η αβ η ij . Associativity Using the metric to raise an index one obtains and this defines a multiplication on M A . The structure of this multiplication may be made more transparent if one writes the basis for T M A as a tensor product: With this, the multiplication may be written as: By construction this multiplication defines a commutative multiplication with unit e = ∂ ∂t (11) To prove associativity we first rewrite the equation that has to be satisfied by F, namely the WDVV equation: which becomes, on using equation (2.7), (2.10) Since the prepotential F for the Frobenius manifold M satisfies the WDVV equation where η αβ = η αβ e 1 . This reduces to (2.11) Thus we have, by multiplying by e q • e s • e p • e k , and evaluating the function with ω , gives the identity (2.10). Hence F satisfies the WDVV equation in the flat coordinates of the metric η (αi)(βj) .

Quasi-homogeneity
This follows immediately from the definition of F, but one can also derive the result by direct computation. The quasi-homogeneity of F is expressed by the equation where quadratic terms will be ignored. On lifting this and using the evaluation map defined by ω one obtains These show that F defines a Frobenius manifold. It remains to show that this is the tensor product M ⊗ A . In fact this is straightforward. Parts (i) and (iii) of Proposition 2.6 are immediate from above (since for the trivial Frobenius manifold A, q i = r i = d = 0), so it just remains to verify condition (ii). Since c γ αβ is independent of t 1 it follows that at points t (αi) = 0 , α > 1 , i > 1 that c γ αβ = c γ αβ t (σ1) e 1 and the result follows from equation (2.9).
Hence the prepotential F = ω(F ) defines the Frobenius manifold structure on the tensor product M A = M ⊗ A . If the multiplications on M and A are semisimple then the multiplication on M A is also semisimple [20,21].
Remark 2.10. Note the existence of such a prepotential F for such a tensor product follows from the original work of Kaufmann, Kontsevich and Manin. However the explicit form for such an F is not immediate from their construction. The above result gives an explicit and easily computable prepotential in the case when one of the manifolds is trivial.
Example 2.11. Let M be a one-dimensional Frobenius manifold Example 2.12. Suppose A is a Frobenius algebra Z ε,0 2,2 defined in Example 2.2. When ε = 0, A is semisimple. When ε = 0, A is nonsemisimple and exactly the algebra Z 2,2 given in Example 2.3. Let M be a 2-dimensional Frobenius manifold with the flat coordinate (t 1 , t 2 ). We denote The unit vector field and the Euler vector field of M A are given by, respectively, and the potential of M A is given by We remark that when ε = 0, M A is a polynomial semisimple Frobenius manifold. By a result of Hertling [18], the manifold M A decomposes into a product of A 2 -Frobenius manifolds. The algebra A can be seen as controlling this decomposition.
Case 2. M = QH * (CP 1 ), i.e., The unit vector field and the Euler vector field of M A are given by, respectively, ∂ ∂v 4 and the potential of M A is given by, for ε = 0

A-valued principal hierarchies on M A
It was shown by Dubrovin that, given a Frobenius manifold M, one can construct an associated bi-Hamiltonian hierarchy of hydrodynamic type, known as the principal hierarchy, with the geometry of the manifold encoding the various components required in its construction. This hierarchy may be written as with (compatible) Hamiltonian operators where g αβ = c αβ γ E γ is the intersection form on M (and Γ αβ γ = −g αµ Γ β µγ ). The Hamiltonian densities h (N,σ) come from the coefficients in the expansion of the deformed flat connection for the Dubrovin connection, and these satisfy the recursion relation (together with certain normalization conditions). The Frobenius manifold M A will automatically inherit such a hierarchy by the very nature of it being a Frobenius manifold. However such a hierarchy is best written as an A-valued system, with m-A-valued dependent fields rather than mn-scalar-valued dependent fields.
We begin by showing how the deformed flat variables on M A may be constructed from those on M . This is achieved by lifting and evaluation the Hamiltonian densities for M . Proof. This is a straightforward calculation (we drop the σ-label on the various h's for clarity): We have Thus using ω to evaluate this A-valued expression gives If N = 0, then, since t µ = t (µs) e s , which is, as required, a Casimir function on M A .
In the obvious way, one can lift the operators P 1 , P 2 to A-valued operators and obtain the following theorem: Theorem 3.2. The principal hierarchy on M A may be written in terms of A-valued fields, densities and operators, as Since t α = t (αi) e i by definition, one obatins since as the components of η are constants, η αβ = η αβ e 1 .

Second Hamiltonian Structure
The second Hamiltonian operator P αβ 2 on M takes the form 1 Note, since the Euler vector field on A is trivial (q i = r i = d A = 0) it follows that q (βj) = q β and d is the same on both M and M A . Also, by definition, For simplicity we will consider the first term in (3.4) only, the corresponding proof of the second term follows practically verbatim the proof of the first. Thus 1 We ignore the precise normalization of the second Hamiltonian structure. We also assume here that the manifold M is non-resonant. It is easy to show that if M is non-resonant, then so is M A .
since g αβ = c αβ γ •(1−q γ )t (γq) e q . On using the associative and commutative properties of the multiplication, and on contracting with e i one obtains . Note that these flows on M A simplify if r = 1 .
The form of the flows in Theorem 3.2 is somewhat hybrid in nature and to rewrite them as a genuine A-valued bi-Hamiltonian system one must introduce the variational derivative with respect to an A-valued field. Such a derivative was introduced in [26] and is defined by the equation With this the flows may be written as an A-valued bi-Hamiltonian system.
Proof. From (3.5) , With this, the result follows immediately.
At this point one could develop the construction of A-valued deformations of these Hamiltonian operators, along the lines of Dubrovin and Zhang [16], and hence obtain dispersive A-valued bi-Hamiltonian systems. We turn instead to the direct construction of such hierarchies via A-valued Lax operators.

Frobenius algebra-valued integrable systems
Let {A, •, e := 1 n , ω := tr} be an n-dimensional Frobenius algebra with the basis e 1 = 1 n , e 2 · · · , e n . In this section, we will introduce some A-valued integrable systems via A-valued Lax operators including the A-valued KP hierarchy and the A-valued Toda lattice hierarchy. For simplicity, here we mainly study the properties of the A-valued KP hierarchy and its dispersionless analogue.

4.1.
The A-valued KP hierarchy and its dispersionless analogue.

The A-valued KP hierarchy via the A-valued Lax operator. Let
be an A-valued pseudo-differential operator (ΨDO) with coefficients U 1 , U 2 , · · · being smooth A-valued functions of an infinite many variables t = (t 1 , t 2 , · · · ) and t 1 = x.  where B r is the pure differential part of the operator L r = L • · · · • L r terms .

Remark 4.2.
By analogy with the discussion on the Z n -KP hierarchy [42], it is easy to generalize the Sato theory and the symmetry theory of the KP hierarchy to the A-KP hierarchy [28,10]. For instance, we could show that there is an A-valued function τ = τ (t 1 , t 2 , · · · ) describing the whole system such that res L i = ∂ ∂t i (τ x • τ −1 ), i = 1, 2, · · · .

Hamiltonian structures of the A-valued KP hierarchy.
In what follows we denote AB = A • B for A, B ∈ A. Let P = i P i ∂ i be an A-valued ΨDO, P + the pure differential part of the operator P and We will use the AGD-scheme (e.g. [1,17,10]) to construct Hamiltonian structures of the A-KP hierarchy. For our purpose, let L = 1 n ∂ + U 0 + U 1 ∂ −1 + U 2 ∂ −2 + · · · be an A-valued ΨDO with an additional term U 0 . Denoting (4.7) In the following our Hamiltonian structures will be established in terms of the "dynamical coordinates" {v [i]q }. More precisely, Then: (1) two compatible Poisson brackets of the A-KP hierarchy associated with L m are given by wheref ,g are two functionals and δf δL will be defined below (4.15) depending on the trace-type map tr .
(2) The Hamiltonians of the A-KP hierarchy corresponding to two Poisson brackets (4.10) and (4.11) in the n th pair arẽ h r = − m r + m tr res L m+r dx andg r = m r tr res L r dx .
Proof. The proof is a straightforward generalization of the scalar case (e.g., Chapters 2,3,5 in [10]) by using the lemma given in [10,42] Lemma 4.4. Suppose P and Q are two A-valued ΨDOs, then there exists an Avalued function h(x, t) such that res [P, Q] = ∂h(x, t) ∂x .
We denote byD the differential algebra of polynomials in formal symbols v (j) where v (j) [i]q = ∂ j v [i]q ∂x j for q = 1, · · · , n and j = 0, 1, · · · . We consider a subalgebra D ofD with the element of the form tr f(V ), where f(V ) is an A-valued differential polynomial w.r.t. its arguments V i . We denote the space of functionals by The variational derivative with respect to an A-valued field [26] is defined by equation (3.5). In the present context, for V = n q=1 v q e q , the variational derivative δf δV is defined bỹ . For clarity we use the notation δf δV instead of δf δV .
Suppose a = (a m−1 , a m−2 , · · · ) with elements We define a vector field associated to a by the formula (4.13) Obviously, ∂ a and ∂ commute, i.e., (4.14) The set of all vector fields ∂ a will be denoted by V. Owing to the formula (4.12) and (4.14), the action of V on D can be transferred to D: If we set and identify the vector a = (a n−1 , a n−2 , · · · ) with the A-valued ΨDO a = we then have ∂ af = tr res a δf δL dx. (4.16) Furthermore, it is easy to verify that (V, [ , ]) is a Lie algebra. Let Ω 1 be the dual space of V consisting of formal A-valued integral operators with the pairing ∂ a , X = a, X = tr res aXdx. So by using the formula (4.16), we see that which is equivalent to the condition where Corollary 4.5. The Z n -KP hierarchy defined in [42] has at least n "basic" different local bi-Hamiltonian structures.
Proof. According to Example 2.3, the algebra Z n has at least n "basic" different ways to be realized as the Frobenius algebra. Thus this corollary follows from Example 2.3 and Theorem 4.3.
We next consider an example to illustrate our construction.
In particular, if one chooses the algebra A to be the algebra Z 2 defined in Example 2.3 one obtains the Z 2 -KdV equation for V = ve 1 + we 2 given by (4.21) According to Corollary 4.5, the system (4.21) can be written as v w with Hamiltonians with Hamiltonians where J 0 = 1 4 ∂ 3 + v∂ + ∂v and J 1 = w∂ + ∂w.

The dispersionless
A-KP hierarchy. Because of the similarities (see [40]), here we list the analogous results for the A-dKP hierarchy without proofs. We will use the following notation in this part. For an A-valued Laurent series of the form A = i A i p i , we denote by A + the polynomial part of the Laurent series A and be an A-valued Laurent series. Let us assume that L m , m ∈ N, is of the form wheref ,g ∈D are two functionals. The variational derivative δf δL is given by where δf δV i is defined in (4.12). When we restrict these to the submanifold V m−1 = 0, the first Hamiltonian structure automatically reduces to this submanifold, but the second one is reducible if and only if res L, δf δL = 0.  .24), the A-dKP hierarchy (4.23) reduces to the A-dGD m hierarchy. Similarly, one could get the bi-Hamiltonian structure of the A-dGD m hierarchy, which is of hydrodynamics-type. It is well known (see, for example, [14,36,11,22,12]) that from this hydrodynamic bi-Hamiltonian structure, one could obtain a semisimple polynomial Frobenius manifold, denoted by M dGD A . When A = R, this Frobenius manifold is a semisimple Frobenius manifold, denoted by M dGD , which coincides [12] with that on the orbit space of Coxeter group of type A. Furthermore, it follows from Theorem 2.9 and Theorem 3.2 that  If one imposes a constraint (by analogy to the constructions in [8,6]) we could define the A-valued extended (M, N)-bigraded Toda hierarchy and take a dispersionless limit to obtain the Avalued extended (M, N)-bigraded dispersionless Toda lattice hierarchy (A-dETL in brief). Following the method in [35,8], one could show the existence of A-valued τ functions about the A-Toda lattice hierarchy and its reductions. Furthermore, using R-matrix methods [5,6,37] one could construct their local bi-Hamiltonian structures of the A-Toda lattice hierarchy, its dispersionless analogue and their reductions.
In particular, the A-dETL hierarchy has a hydrodynamic bi-Hamiltonian structure which will produce a Frobenius manifold denoted by M dETL A . When A = R, this Frobenius manifold is a semisimple Frobenius manifold, denoted by M dETL , which coincides with that on the orbit space of extended affine Weyl group of type A defined by Dubrovin and Zhang in [15]. Analogous to Corollary 4.8, we have the following: Remark 4.11. Besides the above two canonical hierarchies, we could also define an A-2dBKP hierarchy via A-valued Lax operators [24,37], and study its reduction and the relation to finite-dimensional Frobenius manifolds [41]. A more challenging problem is to study the relation between the A-dKP (or A-2dBKP or A-dToda lattice) hierarchy and infinite-dimensional Frobenius mainfolds for dimA > 1. The case A = R has been studied in [7,27,38,39]. A natural step is to introduce certain tensor product for an infinite-dimensional Frobenius manifold and a finite-dimensional Frobenius manifold. Another problem is to study how to embed a finite-dimensional Frobenius manifold into an infinite finite-dimensional Frobenius manifold as its Frobenius submanifold in the sense of Strachan [29,32]. We will study these in a separate publication.

Conclusions
Central to the results of this paper is the use of a distinguished coordinate system, namely the flat coordinates of the Frobenius manifold M . But the lifting procedure may be applied to any geometric structure which is analytic in some fixed coordinate system. However, such results loose some of their coordinate free character: one is using a specific coordinate system to define new objects then relying in their tensorial properties to define then properly in an arbitrary system of coordinates. As an example of this, one can apply the idea to F -manifolds defined by Hertling and Manin [19]. The proof is straightforward and will be omitted. The link between F -manifolds and equations of hydrodynamic type has been explored by a number of authors [32,25] so one should be able to apply the idea of this paper to construct their A-valued counterparts.
In quantum cohomology, the tensor product of Frobenius manifolds generalizes the classical Künneth product formula. In singularity theory it corresponds to the direct sum of singularities. If one of the manifolds is trivial then this descriptions degenerates -there is no parameter space of versal deformations. However, one could try to construct an A-valued singularity theory. This is purely speculative, but Arnold has constructed a theory of versal deformations of matrices [2] but it remains to see if this is what would be required.
As remarked earlier, since M A is a Frobenius manifold in its own right, one can apply the deformation theory developed by Dubrovin and Zhang [16] directly to the hydrodynamic flows given in Theorem 3.2. But central to this approach is the existence of a single τ -function. However the deformations/dispersive systems constructed in Section 4 have A-valued τ -functions (see Remark 4.2). Thus we have two distinct deformation procedures, unless they are connected by some set of transformations. It may be possible to construct a deformation theory along the lines of [16] but with an A-valued τ -function. This paper has concentrated on Frobenius algebra-valued integrable systems. But there are many other algebra valued generalizations of KdV equation, from Jordan algebra to Novikov algebra-valued systems [33,34,30,31]. Whether such algebravalued systems can be combined with the theory of Frobenius manifolds remains an open questions. In connection with this problem, the bi-Hamiltonian systems in this paper are all local, and this can be traced to the commutativity on the Amultiplication. Developing a theory which encompasses the non-commutative/nonlocal hierarchies, such as the original matrix KdV equation (1.1) would be of considerable interest.