Theorem 1.1.

Consider a non-degenerate multi-dimensional, multi-component bracket.

• For D = 1, such a bracket may be reduced, by a change of variable, to a constant, or Darboux, form;

• For D ≥ 2, such a bracket may be reduced, by a change of variable, to a linear form where

  ℋijα(u)={(bkijα+bkijα)ukddXα+bijαuXαk}+{goijαddXα}.

• Furthermore, tensorial conditions exist to determine whether a full reduction to constant form, where all components of all the metrics are constant, may be achieved.

Without the non-degeneracy condition, much less is known.

On substituting this linear structure into the defining condition in Theorem 2.1 one obtains equations for the constants bkijα and goijα. These are best described in terms of an algebraic structure on regarding these constants are structure constants of a multiplication and inner produce, i.e.

These algebraic conditions were defined implicitly in [12] and will be presented explicitly below. In the D = 1 case they define a Novikov algebra. For D ≥ 2 one has multiple Novikov algebras connected by crucial compatibility conditions. Finding a construction that generates examples of multiplications satisfying these compatibility conditions is the raison d’être of this article.

The renaissance in the study of such brackets has come from two directions. Progress has been made in the classification of these brackets, including in the case where the structures are degenerate. Firstly [9, 14]:

• For D = 2, a classification of non-degenerate and degenerate has been completed in the cases when the number of variables is small, with partial results for multi-component systems;

• Full results for D = 3, N = 3 have been obtained, extending the results of Mokhov who studied the D = 3, N = 1, 2.

Secondly, by developing the theory of Poisson Vertex Algebras, a deformation theory has been developed [5]. Using the classification results for D = N = 2, the various cohomology group that govern such deformations have been constructed.

However, there is currently no classification (in the sense of an explicit, exhaustive, list of examples) of such brackets. This is true even for linear Poisson brackets in the D = 1 case: Novikov algebras have only been classified in the cases where the number of components is small, N ≤ 3 [1,2,4]. This lack of examples also hampers the programme of constructing a deformation theory: one requires something to deform.

The aim of this paper is to provide explicit examples of multi-component, multi-dimensional Dubrovin-Novikov brackets. The approach is far from exhaustive – it is certainly not a step in any future classification programme, but it does provide a mechanism for the construction of examples using only linear algebra. The approach is based on a recent result in the study and classification of Novikov algebras [10, 16]. The rest of the paper is laid out as follows: after reviewing the governing equations and known results in Sections 2 and 3, which includes the explicit description of the algebra ℬ introduced implicitly in [12], the main construction in given in Section 4. The material in Section 5 developes the theory of these multi-dimensional brackets in a new direction. This is achieved by showing how the monodromy group associated with one of metrics (under the assumption of non-degeneracy) controls the skeletal structure of the remaining metrics (in the sense that it controls where non-zero terms can appear, but does not fix the value of such terms). This is the least developed part of the paper and clearly could be developed further.

Theorem 2.1.

A bracket (1.1) is a Poisson bracket if and only if the following relations for the coefficients of the bracket are satisfied:

The following Lemma, again from [12, 13], follows from the analysis of the above conditions.

Lemma 2.1.

Given a multidimensional Poisson bracket of the form (1.1), for each α, the corresponding summand on the right-hand side of (1.1) is a one-dimensional Poisson bracket of hydrodynamic type.

Since one may perform linear transformations in the X^α-variables it follows that any linear combination of such one-dimensional Hamiltonian structures must also be Hamiltonian and hence any pair of summands in (1.1) defines a (1 + 1)-dimensional biHamiltonian structure.

Definition 3.1.

1. (a)

   A Novikov algebra is a vector space 𝒜 equipped with a composition (called multiplication) ○ : 𝒜 × 𝒜 → 𝒜 with the properties

   a∘(b∘c)−b∘(a∘c)=(a∘b)∘c−(b∘a)∘c,(a∘b)∘c=(a∘c)∘b

   for all a, b, c ∈ 𝒜.

2. (b)

   A compatible, or quasi-Frobenius inner product, on the algebra 𝒜 is a map 〈, 〉 : 𝒜 × 𝒜 → ℂ with the property

   〈a∘,b,c〉=〈a,c∘b〉

   for all a, b, c ∈ 𝒜.

To find the full set of conditions one just substitutes equation (3.1) into Theorem 2.1 which yields, on writing the resulting conditions algebraically in terms of the multiplications defined by the bkijα and the inner products defined by the goijα:

Note, if α = β then these conditions reduce to the definition of a Novikov algebra with a compatible quasi-Frobenius inner product, in accordance with Lemma 2.1. Conditions (3.2) and (3.4) are equivalent to the requirement that the product

Definition 3.2.

The algebra ℬ consists of the vector space ℂ^N equipped with multiplications ∘α and inner-products 〈, 〉_α, α = 1, ..., D such that:

1. (i)

   the multiplication α∘ defines Novikov algebra with a compatible, quasi-Frobenius inner-product 〈, 〉_α for all α = 1, ..., D;

2. (ii)

   the sum u∘v=u∘αv+u∘βv defines a Novikov algebra with a compatible, quasi-Frobenius inner-product 〈u, v〉 = 〈u, v〉_α + 〈u, v〉_β for all α, β = 1,..., D;

3. (iii)

   the following compatibility condition holds:

   1. (a)

      (u∘βv)∘αw−u∘β(v∘αw)=(v∘αu)∘βw−v∘α(u∘βw)

   2. (b)

      ∑(u,v,w)(u,v∘βw)α−(v,u∘αw)β=0 for all α, β = 1, ..., D and for all u, v, w ∈ ℂ^N.

This algebra was defined implicitly in [12] – here we just write it out explicitly. We will concentrate on the explicit construction of algebras ℬ in the following two cases:

• Class A: homogeneous structures, i.e. goijα=0 for all α;

• Class B: one of the terms is constant and the remaining terms homogeneous, i.e.

  goijα=0,   ∘α≠0,   α=2,…,D;
  goij,α=1≠0,   ∘i=1≠0,
  with, in both cases, no requirement on the non-degeneracy of the structures. Note though, if one of the metrics, say g^i,j,α=1 is non-degenerate, and hence a flat metric, one may introduce flat-coordinates in which ∘i=1=0, and det(goij,α=1)≠0, and so the second case above is, if one of the individual structures is non-degenerate, a subclass of the first. In case (b) the conditions linking the multiplications and innner-products reduce to the following:
  〈u∘βv,w〉α=1=〈u,w∘βv〉α=1,
  i.e. the single inner product is quasi-Frobenius with respect to each of the multiplications, and
  ∑(u,v,w)(u,v∘βw)α=1=0,
  as derived in [12, 13].

Proposition 4.1.

Let 𝒜 be a commutative, associative algebra with product · : 𝒜 × 𝒜 → 𝒜 and space of derivations der(𝒜). Consider the set {v_α, δ_α : α = 1,...,M} of constant vectors and derivations with the properties

Corollary 4.1.

Let 𝒜 be an associative, commutative algebra with a two-dimensional space of derivations der(𝒜). Then the above construction gives a D = 2, dim(𝒜)-component Hamiltonian structure.

Proof.

Since derivations form a Lie algebra and der(𝒜) is two dimensional, then automatically

Lemma 4.1.

Consider the space der˜(𝒜)⊂der(𝒜) of derivations with the properties

Proof.

Purely computational

We illustrate this construction with the following example.

Example 4.1.

Consider the commutative, associative algebra

The derivations on this algebra are

Thus, by the above Corollary, one obtains pairs of compatible products, namely

Thus the construction yields (N − 2) pairs of compatible multiplications, and hence (N − 2) examples of 2-dimensional, N-component Hamiltonian structures. The corresponding metrics are:

Solving the associated Gauss-Manin equations (for a systematic approach to solving these equations for this class of metrics, see [15]) gives the coordinate transformation

Thus this examples furnishes examples in both of the classes of structures defined above. We end this section with another example, this giving a 4-component, 3-dimensional example.

Example 4.2.

Consider the commutative, associative algebra with multiplication table

The derivations of this algebra are easily constructed:

Since the first metric is non-degenerate one may easily solve the Gauss-Manin equations to find its flat coordinates. Applying this transformation to the full set of structures gives

This example may be easily generalized.

Example 5.1.

For the algebra 𝒜 defined by the first metric in Example 4.1 the monodomy group is 𝒲(𝒜) ≅ 𝕑₄[1, 2, 3], which acts on the v-variables via

This monodromy group also controls the structure of the remaining metrics g^ijα for α ≥ 2. On applying the transformation u = u(v) to the entire pencil of metrics

This, coupled to the triangular structure of the transformation u = u(v) which implies the the metrics g^ijα(v) for α ≥ 2 must have entries on or above the antidiagonal, constrains these constants [15].

Example 5.2.

In the above Example 5.1, the monodromy group 𝒲(𝒜) ≅ 𝕑₄[1, 2, 3] and the only non-zero entries are d211, d312 and hence the pencil of metric takes the skeletal form

This methods only fixes the skeletal forms of the pencil: it does note fix the values of the constants. These have to be found by ensuring the resulting structures satisfy the ℬ-algebra conditions.