Bi-Hamiltonian Systems in (2+1) and Higher Dimensions Defined by Novikov Algebras

The results from the article [Strachan I.A.B., Szablikowski B.M., Stud. Appl. Math. 133 (2014), 84–117] are extended over consideration of central extensions allowing the introducing of additional independent variables. Algebraic conditions associated to the first-order central extension with respect to additional independent variables are derived. As result (2 + 1)and, in principle, higher-dimensional multicomponent bi-Hamiltonian systems are constructed. Necessary classification of the central extensions for low-dimensional Novikov algebras is performed and the theory is illustrated by significant (2+1)and (3+1)dimensional examples.


Introduction
In the article [16] we have presented construction of (1+1)-dimensional integrable bi-Hamiltonian systems associated with Novikov algebras. These systems are multicomponent generalizations of the Camassa-Holm equation [8] and can be interpreted as Euler equations on the respective centrally extended Lie algebras. On the other hand, the central extension procedure is one of the most effective methods allowing for the construction of (2 + 1) analogs of (1 + 1)-dimensional systems. However, such procedure is not always possible. All the more, systematic construction of (3 + 1) and higher-dimensional integrable systems is often quite difficult, see the recent survey [6] and references therein.
Novikov algebras naturally appear in the context of the homogeneous first-order Hamiltonian operators [9,4]. These operators define Lie-Poisson structures associated with the so-called translationally invariant Lie algebras that are in one to one correspondence with Novikov algebras. For more information about this and directly related topics see [16] and the recent works [14,15,17].
The key result of this paper is the derivation of the algebraic conditions that must be satisfied on a Novikov algebra by the first order central extension defined with respect to an additional independent variable. Further, we show that the Killing forms must practically satisfy the same algebraic conditions. Subsequently, we classify such central extensions with respect to low dimensional Novikov algebras, that were originally classified in the articles [2,3] and [7]. Consequently, we show how to construct (2 + 1) and, in principle, higher dimensional bi-Hamiltonian hierarchies on the centrally extended Lie algebras associated to Novikov algebras. This construction is formulated with respect to a Killing form. Finally, we illustrate the above theory by explicit examples of (2+1) and (3+1)-dimensional integrable systems and related bi-Hamiltonian structures.

Novikov algebras
Let (A, ·) be a Novikov algebra 1 , this means that it is right-commutative: (a · b) · c = (a · c) · b, and left-symmetric (quasi-associative): The structure constants of a Novikov algebra A, with basis vectors e 1 , . . . , e n , are given by b i jk such that where B is a characteristic matrix. By L A we understand the algebra of A-valued (smooth) functions of (independent) spatial variables x, y, ... belonging to the domain Ω = S 1 × S 1 × . . . , As the regular dual algebra L * A to L A we choose the vector space of A * -valued functions, The related duality pairing L * A × L A → C is given by where the integral is over all spatial variables, dx ≡ dxdy... . By ( , ) : A * × A → C we mean the natural pairing between A and its dual A * . The Lie algebra structure on L A is defined, with respect to the distinguished variable x, by the bracket In fact, (1) is a Lie bracket on L A iff (A, ·) is a Novikov algebra.

Central extensions
Consider a bilinear form g : A × A → C, then we will say that the form g: • satisfies the quasi-Frobenius condition if • satisfies the cyclic condition if • is totally symmetric if the trilinear form is symmetric with respect to all arguments.
We are interested in the central extensions of the Lie algebra L A . This means that on the direct sum L A ≡ L A ⊕ C there is a Lie bracket of the form: where ω : L A × L A → C is a 2-cocycle. This means that ω is skew-symmetric, and it satisfies the cyclic condition: The differential 2-cocycles, yielding central extensions of the Lie algebras L A associated with Novikov algebras, are generated by appropriate bilinear forms satisfying various algebraic conditions that were originally derived in [4], see also [16]: iff g is symmetric and satisfies the quasi-Frobenius condition.
• A bilinear form f : A × A → C defines the second order 2-cocycle iff f is skew-symmetric and satisfies the quasi-Frobenius and cyclic conditions.
• A bilinear form h : A × A → C defines the third order 2-cocycle iff h is symmetric and the related trilinear form c(a, b, c) ≡ h(a · b, c) is totally symmetric.
There are no 2-cocycles of higher order. Our aim is to complete the above theory with 2-cocycles by which one can introduce a new independent variable. Theorem 1. A bilinear form η : A × A → C generates on L A the first-order 2-cocycle: defined with respect to an (additional) independent variable y ∈ S 1 , if and only if the form η is symmetric, satisfies the quasi-Frobenius and cyclic conditions.
Proof . Integrating by parts Then, Collecting terms with respect to the functionally independent variables, for instance {u xy , v, w}, we obtain the required conditions on the bilinear form η.
In the article [2] the three and less dimensional Novikov algebras over complex numbers were fully classified. In arbitrary dimension the classification is far from being complete. For instance, in dimension four only transitive Novikov algebras were classified [7,3]. From the point of view of the construction of integrable hierarchies from Novikov algebras the transitive 2 algebras are not of interest as they result in 'degenerate' systems of evolution equations, see [16].
For a given Novikov algebra classification of the bilinear forms generating differential central extensions is rather straightforward as it involves only solving the systems of linear equations, that can be done without much difficulty using any software for symbolic computations. The classification of differential 2-cocycles, with respect to variable x, associated with the low dimensional Novikov algebras was presented in the work [16]. Here, we extend this classification over the case including differential 2-cocycles of the form (3), defined with respect to an additional independent variable. The results are summarized in the following theorem and Table 1. We in fact are only interested with the cases of (3) defined by means of nondegenerate η.
Theorem 2. We assume that A is a Novikov algebra that is not transitive and cannot be decomposed into direct sum of lower dimensional Novikov algebras.
• In dimension 1 there is only one relevant Novikov algebra, A = C, and in this case there is no central extension of the form (3).
• In dimension 2 there is only one Novikov algebra, (N 6) with κ = −2, for which there exists the central extension (3) defined by a nondegenerate bilinear form η.
• In dimension 3 there is also only one Novikov algebra, this time (D6) with κ = −2, for which there exists the central extension (3) defined by a nondegenerate η.
• In dimension 4 there is at least one such Novikov algebra, which, for instance, is the Novikov algebra (N 3) ⊗ (N 6) with κ = −2.
Notice that linear combination of 2-cocyles is a 2-cocycle. This means, that each linearly independent 2-cocycle of the form (3) can be defined with respect to a different additional independent variable.

Killing form
The Killing form on L A is a symmetric nondegenerate bilinear form K : Assume that K is defined by a bilinear form κ : A × A → C: whereκ : A → A * . If there exists a Killing form then the dual Lie algebra L A * can be identified with L A and the co-adjoint action can be identified with the adjoint action: Proof . We have Collecting terms with respect to functionally independent variables we obtain two conditions: Taking into account the symmetry of the form κ we get the conditions from the theorem.
We see that the algebraic conditions for the Killing forms (4) are the same as for the 2cocycles (3) with nondegenerate η. This means that on a Novikov algebra, on which there exists a 'nondegenerate' 2-cocycle (3), one can always define the Killing form (4).
In the following lemma we rewrite the algebraic conditions that must be satisfied by differential central extensions with respect to a given Killing form.
Lemma 1. We assume that the Novikov algebra A is such that there on L A exists the associated Killing form (4). Then: • a linear formĝ : A → A generates the two-cocycle of first order, iff the following (equivalent) relations hold on A: • a linear formf : A → A generates the two-cocycle of second order, iff on A the following two conditions are satisfied, • a linear formĥ : A → A generates the two-cocycle of third order, are valid on A; • a linear formη : A → A generates the two-cocycle of first order with respect to an additional independent variable y ∈ S 1 , We skip the proof as it is rather straightforward and it does not add anything significant for further considerations.

Bi-Hamiltonian structure
Consider the centrally extended Lie-Poisson bracket, associated with the Lie algebra L A , and defined with respect to the Killing form (4): where u ∈ L A and H, F ∈ F L A . Here, F L A is the space of functionals on the Lie algebra L A : where the densities H[u] are smooth functions with respect to all variables: The respective (variational) differentials δ u H are defined through the standard formula: The second Poisson bracket compatible with (5) is Naturally, the Poisson brackets (5) and (7) are compatible since any linear composition of 2cocyles is a 2-cocycle. The related Poisson tensors P i : L A → L A , defined with respect to the Killing form (4), and The linear operators I i : L A → L A , such that have the form: and The formsĝ i ,f i ,ĥ i andη i,k generate the respective 2-cocycles, see Lemma (1), and n is the dimension of the linear space of the 2-cocycles of type (3) introducing additional spatial variables. The related bi-Hamiltonian chain has the form where H i ∈ F (L A ) and H 0 is a Casimir of P 0 . The evolution equations from the hierarchy are of (n + 2) dimension, 1 with respect an evolution variable t i and n + 1 with respect to spatial variables: x, y 1 , . . . , y n . Assuming invertibility of I 0 : L A → L A we can introduce the auxiliary dependent variableū such that Here, we understand the invertibility of the operator I 0 as of the pseudo-differential operators.
Theorem 4. We assume that the Novikov algebra A possesses the right unity e. Then, the first two nontrivial evolution equations from the hierarchy (11) takes the form and the respective Hamiltonians are Proof . The kernel of P 0 is spanned by x-independent elements of L A . For simplicity, we choose that δ u H 0 = e. Thus, and solving for δ u H 1 we find that δ u H 1 =ū. Hence, and we have The Hamiltonians H i related to the cosymmetries δ u H i can be constructed using the homotopy formula, see [11,5] and proof of Theorem 1 in [16]. Alternatively, one can verify the Hamiltonians (14) using the formula (6), but this way is less straightforward as one needs to use the particular properties of the respective 2-cocycles.
The evolution equations (13) and the Hamiltonians (14) can be written, in a more explicit form, using the auxiliary variableū, see Appendix A.

Remark 1.
Observe that H 1 from (14) is a quadratic Hamiltonian functional: The operator I 0 (10a), when it is invertible, can be interpreted as the inertia operator. The variables u andū play the role of the momentum and the velocity. As result, the second nontrivial flow in the hierarchy (14) can be interpreted as the integrable Euler equation on the centrally extended Lie algebra L A corresponding to the geodesic flow on the space with metric defined by means of the inertia operator I 0 . Compare this with Remark 1 and Appendix A.3 in the article [16]. For more information on the subject we refer the reader to the books [1, 10]. This Novikov algebra is 2-dimensional, non-abelian and non-associative, it is also non-transitive. The associated Lie algebra L A with the Lie bracket (1) is isomorphic to the Lie algebra Vect(S 1 )⋉ Vect * (S 1 ), that is the semidirect sum of the algebra of smooth vector fields on the circle S 1 , Vect(S 1 ), with its dual Vect * (S 1 ), see [12] and [13]. The structure matrix and the multiplication of the algebra A are: The right unity is e = (1, 0) T . Taking the advantage of the classification of the form η in Table 1, we can define the Killing form (4) bŷ For u ∈ L A and the related differential of H we choose: as then we get the usual Euclidean duality: where Ω = S 1 × S 1 and dx ≡ dxdy.
Using the classification of central extensions in Table (1) the operators (10) are In this case the first order cocycle is trivial, this means that in the Poisson tensor (8a)ĝ 1 can be obtained/removed through a shift of the dependent variables. So, we do not consider g 1 in I 1 . The relation (12) between the momentum variable u and the velocity variableū = (ū(x, y),v(x, y)) T is The second flow from the hierarchy (11) takes the form The related Poisson tensors (8) are Notice that the above Poisson tensors are defined with respect to the standard duality pairing. The Hamiltonian functionals (14) are The particular cases η 1 = η 2 = h = 0 and g 1 = g 2 = β = 0 of the above bi-Hamiltonian structure were considered in the work [13], see the two examples therein.
Then, we find the Poisson tensors (8): the second system from the hierarchy (13): and related Hamiltonians (14): Taking w =w = 0 and g 3 = 0 the above evolution equation and the related bi-Hamiltonian structure reduce to the case from the previous section.
In this case there are two linearly independent central extensions of the form (3), see Table 1. Hence, we introduce two additional independent variables y and z. To simplify this example, we further consider only the central extension of the type (3).
As result, the operators (10) are Next, forū = (ū(x, y),v(x, y),w(x, y),s(x, y, z)) T the relation (12) is The Poisson tensors (8) are and the second nontrivial system from the hierarchy (13) is Notice that this is a (3 + 1)-dimensional integrable bi-Hamiltonian evolution equation and it can be explicitly written either by means of the momentum u or velocityū variables. The respective Hamiltonians (14) are 2 Ω us +ūs +vw + vw dx, and Notice that in the example from this section one can reduce dimension taking y = z or y = x or z = x.

A Explicit form of the Bi-Hamiltonian hierarchy
In the variableū the bi-Hamiltonian chain (11) takes the form and H 0 = K c, I 0ū , Proposition 1. The explicit form, in the variableū, of the second flow from (15) iŝ 1,kūy k , and the explicit form of Hamiltonians is Proof . We will skip the details. The proposition is a consequence of the following relations which can be proven using all the properties of the linear forms generating respective 2-cocycles obtained in Lemma (1).
The above proposition is the higher dimensional extension of Theorem 1 from the article [16], see also the equation (26) therein. Table 1. The classification of the bilinear forms generating respective differential 2-cocycles of the form (2) and (3) for all non-decomposable 1 and 2-dimensional Novikov algebras and the significant cases from Theorem 2.
type B g f h η comments