Lax Operator Algebras and Applications to τ -Symmetries for Multilayer Integrable Couplings

: The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As illustrative examples, isospectral and non-isospectral integrable couplings and the corresponding commutator relations of their Lax operators are computed explicitly in the cases of the Volterra lattice hierarchy and the AKNS hierarchy, along with their τ -symmetry algebras

To the best of our knowledge, there has been very little work done on N × N integrable couplings, because they are extremely complex [38,39].In this paper, based on a kind of N × N non-semisimple Lie algebra, we explore the Lie algebraic structures of zero curvature representations for general multilayer integrable couplings and apply them to the construction of τ-symmetries (a type of time-dependent symmetry).
Let us first describe some notations in the discrete case, as follows [10,19].Let E be the shift operator, where f : Z → R, m, n ∈ Z, and the inverse operator of E − E −1 are defined as Thus, the corresponding inverses of the forward and backward difference operators can be determined by which are normally used to generate discrete integrable hierarchies (both isospectral and non-isospectral hierarchies).We assume that a pair of discrete matrix spectral problems reads where u = (u 1 , u 2 , • • • , u q ) T , u i = u i (n, t), i = 1, 2, • • • , q, are complex or real functions defined over Z × R. The Lax pair of U ≡ U 0 and V ≡ V 0 , involving the spectral parameter λ, comes from a certain matrix loop algebra g and determines a discrete integrable equation through the discrete zero curvature equation This means that a triple (U, V, K) satisfies where λ t = f (λ), U λ = ∂U/∂λ, and U (u)[K] denotes the Gateaux derivative To generate a multilayer (N-layer) integrable coupling we use a matrix loop algebra ḡ(λ) consisting of square matrices of the following block form [9]: where P ≡ P 0 , P i , i = 1, 2, • • • , N, are square submatrices of the same size as U and V.This loop algebra ḡ(λ) has two sub-loop algebras which form a semi-direct sum: ḡ(λ) = g gc .The notion of semi-direct sums means that the two subalgebras g and gc satisfy [g, gc ] ⊆ gc .We also require the closure property between g and gc under the matrix multiplication gg c , gc g ⊆ gc .Now, a multilayer discrete integrable coupling is determined by the following enlarged zero curvature representation: where This implies that an enlarged triple ( Ū, V, K) with This precisely presents Similarly, we can start with continuous matrix spectral problems: where The Lax pair of U and V, involving the spectral parameter λ, determines a continuous integrable equation through the continuous zero curvature equation This means that a triple (U, V, K) satisfies Thus, the N-coupled integrable coupling can be derived from the enlarged zero curvature representation where Ū and V are defined by (13).This implies that the enlarged triple ( Ū, V, K) with The rest of the paper is structured as follows.In Section 2, we give some definitions and the Lie algebra of enlarged vector fields for the above multilayer integrable couplings, both discrete and continuous.In Section 3, we establish Lie algebraic structures of zero curvature representations for the general integrable couplings presented above.In Section 4, we illustrate our general theory by taking the Volterra lattice hierarchy and the AKNS hierarchy as two specific examples and present the corresponding τ-symmetry algebras of the resulting integrable couplings.Section 5 is devoted to conclusions and discussion.

The Lie Algebra of Enlarged Vector Fields
Firstly, we denote by B all complex or real functions which are C ∞ -differentiable with respect to {n, t} and C ∞ -Gateaux differentiable with respect to {u, v 1 , v 2 , • • • , v N }, and we set B r = (P 1 , P 2 , • • • , P r ) T P i ∈ B .Moreover, by V r , we denote all r × r matrix linear difference operators: Then, we define We now set where K, S ∈ B q , K i , S i ∈ B q i .The Gateaux derivative is defined as follows: where R ∈ V r or V r (0) .In particular, we have Theorem 1.
Then, we have the relation Proof.By the definition of the Gateaux derivative, we have ) At the same time, we similarly have These two equalities engender our required equality.
From the above theorem, we can easily deduce the following corollary.
Here, we note that U i (v i , λ) have nothing to do with the original potential vector u.
Evidently, we can also compute the commutator of two enlarged vector fields K, S ∈ B q+∑ N i=1 q i as follows: where Clearly, we can show that the above product forms a Lie algebra in B q+∑ N i=1 q i .Theorem 2. For enlarged vector fields K, S ∈ B q+∑ N i=1 q i , the product (29) defines a Lie algebra in ∈ B q+∑ N i=1 q i and using the definition of the product (29), we have Thus, by direct calculation, we have This implies that (29) defines a Lie algebra in B q+∑ N i=1 q i .

The Algebraic Structure of Lax Operators
Next, we aim to discuss the algebraic structure of discrete zero curvature equations for N-coupled integrable couplings.First, the commutator of two smooth functions f

The Discrete Case
In what follows, we always assume that the enlarged spectral operator Ū ∈ V has an injective Gateaux derivative operator Ū : . We assume that and for f (λ) ∈ C ∞ (C), we set and For can be computed as follows: where Here, we set V 0 ≡ V and W 0 ≡ W for the convenience of writing.This shows a special structure of the commutator of enlarged Lax operators and plays a crucial role in our computation.
which is equivalent to the following N + 1 equations: Proof.The proof of (38) can be found in [10].We consider (39) directly.From Equation ( 15), we have On the other hand, by means of we immediately have Thus, we have For further calculation, by using we can obtain Proof.From the above theorem and the assumption, we have Since Ū is injective, it follows that [K, S] = L and [K, S] i = L i , i.e., [ K, S] = L.
It follows from the above theorem that if two enlarged evolution equations ūt = K, ūt = S, K, S ∈ B q+∑ N i=1 q i are the compatibility conditions of the spectral problems where a, b are constants and m, n ≥ 0, respectively, then the product equation ūt = [ K, S] is the compatibility condition of the following spectral problem: where V, W is defined by (35).

The Continuous Case
For the continuous case, we denote by B all complex or real functions Moreover, we denote all r × r matrix integro-differential operators as Assuming that we have the following theorem, which is similar to Theorem 3.
That is to say, which is equivalent to the following N + 1 equations: Proof.
On the other hand, by means of we can immediately obtain Thus, we can obtain It follows that if two enlarged equations ūt = K, ūt = S, K, S ∈ B q+∑ N i=1 q i are the compatibility conditions of the continuous spectral problems where a, b are constants and m, n ≥ 0, respectively, then the product equation ūt = [ K, S] is the compatibility condition of the spectral problem where V, W is defined by (35).We point out that all triples of ( V, K, f ) satisfying either ( 14) or ( 19) constitute a Lie algebra with the algebraic structures established above.

The Case of the Volterra Lattice Hierarchy
Next, we establish the τ-symmetry algebra by using the above construction process for the 3-coupled Volterra lattice integrable coupling.The enlarged spectral problem of this integrable coupling reads where u ≡ u(n, t), v 1 ≡ v 1 (n, t), and v 2 ≡ v 2 (n, t) are dependent variables.The associated enlarged temporal spectral problem is as follows: The compatibility condition U t − (EV)U + UV = 0 in this case gives, equivalently, In other words, we have a number of formulas as follows: 1 − e (−1) 1 1 − e 2 − e To consider these further, set and thus Equation (54) becomes 1j − e 1j − e (−1) 1j 2j − e (1) (2) (2) (2) Hence, we can derive the isospectral and non-isospectral 3-coupled integrable couplings as follows.

The Case of the AKNS Hierarchy
In the following, we present the enlarged spectral problem of the AKNS integrable coupling by semi-direct sums of Lie algebras, as follows: where u, v and v i , w i , i = 1, 2, • • • , N, are dependent variables.The associated enlarged temporal spectral problem is assumed to be φt can also be calculated, which automatically gives rise to the τ-symmetry algebra.Because the process of calculation is completely similar, we recommend that the interested reader refers to the discrete case above to see all hierarchies of τ-symmetries.In addition, similar τ-symmetry algebras could be generated through the above scheme for the vector AKNS soliton hierarchy [32,33].

Conclusions and Discussion
In this paper, we applied a type of non-semisimple matrix loop algebra to construct the algebraic structures of zero curvature representations for multilayer integrable couplings, both continuous and discrete.We furnished the commutator relations of Lax operators for isospectral (λ t = 0) and non-isospectral (λ t = λ m , m ≥ 0) hierarchies.Finally, τ-symmetry algebras for the multilayer Volterra lattice and AKNS integrable couplings were presented.Our theories supplement the existing theories and the corresponding results, particularly those from [19,23,25].
Inspired by the research related to the Frobenius algebra, the authors in [39] constructed the following non-semisimple Lie algebra, which consists of an N × N square matrix of the form and obtained a Z N isospectral mKdV integrable coupling and a Z N non-isospectral ANKS integrable coupling.We believe that the method presented above can be also applied to the above non-semisimple matrix loop algebra (75).Considering that the method is similar, we omit the construction process for convenience.For integrable couplings, Hamiltonian structures could be presented by applying the variational identities associated with non-semisimple Lie algebras.Moreover, symmetries of non-isospectral hierarchies could be constructed in terms of series of vector fields [36].An interesting question for us is how to construct new non-semisimple Lie algebras and how to classify these types of Lie algebras.Addressing this question will lead to a classification theory of integrable couplings.