A construction of Multidimensional Dubrovin-Novikov Brackets
aa r X i v : . [ n li n . S I] J un A CONSTRUCTION OF MULTIDIMENSIONALDUBROVIN-NOVIKOV BRACKETS
IAN A.B. STRACHAN
Abstract.
A method for the construction of classes of examples of multi-dimensional,multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This isbased on an extension of the original construction of Gelfand and Dorfman whichgave examples of Novikov algebras in terms of structures defined from commutative,associative algebras. Given such an algebra, the construction involves only linearalgebra.
Contents
1. Introduction 12. Multidimensional Dubrovin-Novikov brackets 33. Linear metrics 34. A construction 55. Monodromy and multi-dimensional Hamiltonian structures 96. Conclusion 11Acknowledgements 11References 111.
Introduction
There has been a recent renaissance in the study of multi-component, multi-dimensionalPoisson brackets of Dubrovin-Novikov type. These brackets, in D -dimensions and with N -components, have the form(1) { I, J } = D X α =1 Z δIδu i H ij δJδu j d D X , where the Hamiltonian operator H ijα is a local, first-order operator of the form H ijα = g ijα ( u ) ∂∂X α + b ijαk ( u ) u kX α . The study of such brackets were initiated by Dubrovin and Novikov in their seminalpapers [7] and [8], and the full conditions (to ensure that the bracket is skew and satisfiesthe Jacobi identity) were derived by Mokhov [12] and are given in Theorem 2 below.However, the classification of such brackets remains an open problem.The most general statement, combining results from [12] and [13] may be stated thus:
Date : October 3, 2018.
Key words and phrases.
Hamiltonian structures, integrable systems.
Theorem 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 ≥ , such a bracket may be reduced, by a change of variable, to a linearform where H ijα ( u ) = (cid:26)(cid:16) b ijαk + b jiαk (cid:17) u k ddX α + b ijα u kX α (cid:27) + (cid:26) g ijαo ddX α (cid:27) . Furthermore, tensorial conditions exist to determine whether a full reduction toconstant form, where all components of all the metrics are constant, may beachieved.
Without the non-degeneracy condition, much less is known.On substituting this linear structure into the defining condition in Theorem 2 oneobtains equations for the constants b ijαk and g ijαo . These are best described in terms ofan algebraic structure on regarding these constants are structure constants of a multi-plication and inner produce, i.e. e i α ◦ e j = b ijαk e k , h e i , e j i α = g ijαo . These algebraic conditions were defined implicitly in [12] and will be presented explicitlybelow. In the D = 1 case they define a Novikov algebra. For D ≥ examples of multiplications satisfying these compatibility conditions isthe raison d’ˆetre of this article.The renaissance in the study of such brackets has come from two directions. Progresshas been made in the classification of these brackets, including in the case where thestructures are degenerate. Firstly [9, 14]: • For D = 2, a classification of non-degenerate and degenerate has been completedin 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 ofMokhov who studied the D = 3 , N = 1 , . Secondly, by developing the theory of Poisson Vertex Algebras, a deformation theoryhas been developed [5]. Using the classification results for D = N = 2 , the variouscohomology 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 ofcomponents is small, N ≤ CONSTRUCTION OF MULTIDIMENSIONAL DUBROVIN-NOVIKOV BRACKETS 3 Multidimensional Dubrovin-Novikov brackets
The full conditions for the bracket (1) to define a Poisson bracket places very strongconditions on the functions g ijα ( u ) and b ijαk ( u ) (see [12, 13]): Theorem 2.
A bracket (1) is a Poisson bracket if and only if the following relationsfor the coefficients of the bracket are satisfied: g ijα = g jiα ,∂g ijα ∂u k = b ijαk + b jiαk , X ( α,β ) (cid:16) g siα b jrβs − g sjβ b irαs (cid:17) = 0 , X ( i,j,k ) (cid:16) g siα b jrβs − g sjβ b irαs (cid:17) = 0 , X ( α,β ) g siα ∂b jrβs ∂u q − ∂b jrβq ∂u s ! + b ijα s b srβq − b irαs b sjβq = 0 ,g siβ ∂b jrα ∂u s − b ijβs b srαq − b irβs b jsαq = g sjα ∂b irβ ∂u s − b jiαs b srβq − b jrαs b isβq ,∂∂u k " g siα ∂b jrβs ∂u q − ∂b jrβq ∂u s ! b ijαs b srβq − b irαs b sjβq + X ( i,j,r ) " b siβq ∂b jrαk ∂u s − ∂b jrαs ∂u k ! + ∂∂u q " g siβ ∂b jrαs ∂u k − ∂b jrαk ∂u s ! b ijβs b srαk − b irβs b sjαk + X ( i,j,r ) " b siαk ∂b jrβq ∂u s − ∂b jrβs ∂u q ! = 0 . The following Lemma, again from [12, 13], follows from the analysis of the above con-ditions.
Lemma 3.
Given a multidimensional Poisson bracket of the form (1), for each α , the corresponding summand on the right-hand side of (1) is a one-dimensional Poissonbracket of hydrodynamic type.
Since one may perform linear transformations in the X α -variables it follows that anylinear combination of such one-dimensional Hamiltonian structures must also be Hamil-tonian and hence any pair of summands in (1) defines a (1 + 1)-dimensional biHamilto-nian structure. 3. Linear metrics
In this section we consider families of metrics g ijα , α = 1 , . . . , D that are of the form(2) g ijα = (cid:16) b ijαk + b jiαk (cid:17) u k + g ijαo . By Lemma 3, this, for each separate α , must define a (1 + 1)-dimensional Poissonstructure of hydrodynamic type, and the conditions for this were derived by Gelfandand Dorfman [11] and further studied by Balinskii and Novikov [3]. These are bestdescribed in terms of a product e i α ◦ e j = b ijαk e k IAN A.B. STRACHAN and inner product h e i , e j i α = g ijαo and the required conditions become the statement that this product defines a Novikovalgebra with a compatible quasi-Frobenius inner product: Definition 4. (a)
A Novikov algebra is a vector space A equipped with a composi-tion (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 . (b) A compatible, or quasi-Frobenius inner product, on the algebra A is a map h , i : A × A → C with the property h a ◦ b, c i = h a, c ◦ b i for all a , b , c ∈ A . To find the full set of conditions one just substitutes equation (2) into Theorem 2which yields, on writing the resulting conditions algebraically in terms of the multipli-cations defined by the b ijαk and the inner products defined by the g ijαo : X ( α,β ) ( u β ◦ v ) α ◦ w + w α ◦ ( u β ◦ v ) − ( w α ◦ v ) β ◦ u − u β ◦ ( w α ◦ v ) = 0 , (3) X ( u,v,w ) ( u β ◦ v ) α ◦ w + w α ◦ ( u β ◦ v ) − ( w α ◦ v ) β ◦ u − u β ◦ ( w α ◦ v ) = 0 , (4) X ( α,β ) ( u α ◦ v ) β ◦ w − ( u α ◦ w ) β ◦ v = 0 , (5) ( u β ◦ v ) α ◦ w − u β ◦ ( v α ◦ w ) − ( v α ◦ u ) β ◦ w + v α ◦ ( u β ◦ w ) = 0 , (6) X ( α,β ) h u, v β ◦ w i α − h v, u α ◦ w i β = 0 , (7) X ( u,v,w ) h u, v β ◦ w i α − h v, u α ◦ w i β = 0 , (8)Note, if α = β then these conditions reduce to the definition of a Novikov algebra witha compatible quasi-Frobenius inner product, in accordance with Lemma 3. Conditions(3) and (5) are equivalent to the requirement that the product u ◦ v = u α ◦ v + u β ◦ v defines a Novikov algebra for arbitrary α and β , and a simple calculation shows thatcondition (4) follows from conditions (3,5) and (6) . Condition (7) is equivalent to therequirement that inner product h u, v i = h u, v i α + h u, v i β is quasi-Frobenius with respect to the product u ◦ v = u α ◦ v + u β ◦ v . We thus obtain the definition of the algebra B : CONSTRUCTION OF MULTIDIMENSIONAL DUBROVIN-NOVIKOV BRACKETS 5
Definition 5.
The algebra B consists of the vector space C N equipped with multiplica-tions α ◦ and inner-products h , i α , α = 1 , . . . , D such that: (i) the multiplication α ◦ defines Novikov algebra with a compatible, quasi-Frobeniusinner-product h , i α for all α = 1 , . . . , D ;(ii) the sum u ◦ v = u α ◦ v + u β ◦ v defines a Novikov algebra with a compatible,quasi-Frobenius inner-product h u, v i = h u, v i α + h u, v i β for all α , β = 1 , . . . , D ;(iii) the following compatibility condition holds: (a) ( u β ◦ v ) α ◦ w − u β ◦ ( v α ◦ w ) = ( v α ◦ u ) β ◦ w − v α ◦ ( u β ◦ w )(b) X ( u,v,w ) h u, v β ◦ w i α − h v, u α ◦ w i 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 willconcentrate on the explicit construction of algebras B in the following two cases:Class A: homogeneous structures, i.e. g ijαo = 0 for all α ;Class B: one of the terms is constant and the remaining terms homogeneous, i.e. g ijαo = 0 , α ◦ 6 = 0 , α = 2 , . . . , D ; g ij,α =1 o = 0 , i =1 ◦ = 0 , with, in both cases, no requirement on the non-degeneracy of the structures. Notethough, if one of the metrics, say g ij,α =1 is non-degenerate, and hence a flat metric,one may introduce flat-coordinates in which i =1 ◦ = 0 , and det( g ij,α =1 o ) = 0 , and so thesecond case above is, if one of the individual structures is non-degenerate, a subclassof the first. In case (b) the conditions linking the multiplications and innner-productsreduce to the following: h u β ◦ v, w i α =1 = h u, w β ◦ v i α =1 , i.e. the single inner product is quasi-Frobenius with respect to each of the multiplica-tions, and X ( u,v,w ) h u, v β ◦ w i α =1 = 0 , as derived in [12, 13]. 4. A construction
Examples of Novikov algebras are easy to construct. In the original paper of Gelfandand Dorfman [11] the following Novikov product was found a ◦ b = a · δb . IAN A.B. STRACHAN
Here · is an associative, commutative product and δ a derivation on this algebra. Thisresult was extended by [10] to a product a ◦ b = v a · b + a · δb where v is a constant element of the underlying field. In [16] this was further extendedto a product a ◦ b = v · a · b + a · δb where v is a constant element in the algebra. Note that if an inner product h , i isFrobenius with respect to the product · then it is automatically quasi-Frobenius withrespect to the above Novikov product.Since v is a constant and derivations form a vector space one can easily constructpencils of Novikov algebras, a α ◦ b = v α · a · b + a · δ α b ,a β ◦ b = v β · a · b + a · δ β b , (here we concentrate on the case where v is a constant vector rather than a scalar. Anear identical result holds if one ignores the first product and regard v as a constantscalar). Note, the labels α , β on the derivation just denotes its pairing with the v α , v β in the first term of these products. Clearly u ◦ v = u α ◦ v + u β ◦ v , = ( v α + v β ) · u · v + u · ( δ α + δ β ) v defines a Novikov algebra.Note, without further restrictions on the constant vectors and derivations, this resultprovides a construction of pencils of flat homogeneous metrics, and hence examplesof bi-Hamiltonian structures in (1 + 1)-dimensions. To find Hamiltonian structures in(1 + D )-dimensions within this class of pencils of Novikov algebras places restrictionson the data { v α , δ α } . Proposition 6.
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 constantvectors and derivations with the properties [ δ α , δ β ] = α · δ β − β · δ α ,δ α ( v β ) − δ β ( v α ) = 0 . Then the set { C N ; α ◦ : α = 1 , . . . , M } is a B -algebra where the products α ◦ are definedby u α ◦ v = v α · u · v + u · δ α v . This defines an M -dimensional, N -component Hamiltonian structure. The following results are immediate:
Corollary 7.
Let A be an associative, commutative algebra with a two-dimensionalspace of derivations der ( A ) . Then the above construction gives a D = 2 , dim ( A ) -component Hamiltonian structure. CONSTRUCTION OF MULTIDIMENSIONAL DUBROVIN-NOVIKOV BRACKETS 7
Proof.
Since derivations form a Lie algebra and der ( A ) is two dimensional, then auto-matically [ δ , δ ] = k δ − k δ for some constants k and k . Hence one obtains compatible products a ◦ b = k a · b + a · δ b ,a ◦ b = k a · b + a · δ b and hence an algebra B = { C N ; ◦ , ◦} and a 2-dimensional, dim ( A )-component, Hamil-tonian structure (cid:3) Lemma 8.
Consider the space g der ( A ) ⊂ der ( A ) of derivations with the properties [ δ α , δ β ] = α · δ β − β · δ α ,δ α ( v β ) − δ β ( v α ) = 0 for some set of vectors { v α , i = 1 , . . . , M } . Then this is a Lie algebra with respect tothe structure [ δ α , δ β ] = δ α δ β − δ β δ α . Proof.
Purely computational (cid:3)
We illustrate this construction with the following example.
Example 9.
Consider the commutative, associative algebra
A ∼ = C [ z ] / h z N i with basis e i = z i − , i = 1 , . . . , N (and with the convention that e i = 0 for i > N ). Inthis basis the multiplication is just e i · e j = e i + j − . The derivations on this algebra are δ k e i = ( i − e i + k − , k = 1 , . . . , N − and these form the Lie algebra [ δ i , δ j ] = ( j − i ) δ i + j − . In particular, [ δ , δ k ] = ( k − δ k , = ( k − δ k − δ . Thus, by the above Corollary, one obtains pairs of compatible products, namely u ◦ v = ( k − u · v + u · δ v ,u k ◦ v = 0 u · v + u · δ k v , or, in the above basis, the products e i ◦ e j = ( j + k − e i + j − ,e i k ◦ e j = ( j − e i + j + k − . Thus the construction yields ( N − pairs of compatible multiplications, and hence ( N − examples of -dimensional, N -component Hamiltonian structures. The correspondingmetrics are: g ijα (cid:12)(cid:12) α =1 = ( i + j − k − u i + j − ,g ijα (cid:12)(cid:12) α = k = ( i + j − u i + j + k − where k ∈ { , . . . , N − } . The first of these metrics is always non-degenerate and somay be, via a change of variable, transformed to a constant form. For example, if N = 3 (so k = 2 ) one obtains the metrics g ijα (cid:12)(cid:12) α =1 = u u u u u u ,g ijα (cid:12)(cid:12) α = k = u u . Solving the associated Gauss-Manin equations (for a systematic approach to solvingthese equations for this class of metrics, see [15] ) gives the coordinate transformation u = v v + 18 ( v ) , u = v ( v ) , u = ( v ) and in the new coordinates one obtains the pair g ijα (cid:12)(cid:12) α =1 = ,g ijα (cid:12)(cid:12) α = k = − v v v . 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-dimensionalexample.
Example 10.
Consider the commutative, associative algebra with multiplication table · e e e e e e e e e e e e e e e e e e The derivations of this algebra are easily constructed: δ e e e e = e e e , δ e e e e = e e e ,δ e e e e = e , δ e e e e = e , and the set { δ , δ , δ } satisfy the conditions [ δ , δ ] = 0 . δ − . δ , [ δ , δ ] = 0 . δ − . δ , [ δ , δ ] = 0 . δ − . δ , CONSTRUCTION OF MULTIDIMENSIONAL DUBROVIN-NOVIKOV BRACKETS 9 and hence the above construction may be implemented. This gives the metrics g ij,α =1 = u u u u u u u u u u g ij,α =3 = u u , g ij,α =4 = u
00 0 0 0 u . Since the first metric is non-degenerate one may easily solve the Gauss-Manin equationsto find its flat coordinates. Applying this transformation to the full set of structures gives g ij,α =1 = g ij,α =3 = − v v v , g ij,α =4 = − ( v − v ) 0 v
00 0 0 0 v . This example may be easily generalized.5.
Monodromy and multi-dimensional Hamiltonian structures
Throughout this section we assume that one of the metrics, say g ij,α =1 , is non-degenerate, and that the associated Novikov algebra A has a right identity. The flatcoordinates are found by solving the Gauss-Manin equations g α =1 ∇ d v = 0 , where g α =1 ∇ is the Levi-Civita connection corresponding to the metric g α =1 . This isan over-determined system, but the compatibility conditions are precisely the flatnessof the metric and hence are automatically satisfied.Solutions of the Gauss-Manin equations will exhibit branching around the discrimi-nant Σ = { u | ∆( u ) = 0 } where ∆( u ) = det (cid:0) g ij ( u ) (cid:1) , and this is encapsulated in the associated monodromy group W ( M ) = µ ( π ( M \ Σ))(see, for example, [6]). Explicitly, the continuation of a solution under a closed path γ on M \ Σ yields a transformation˜ v a ( u ) = A ab ( γ ) v b ( u ) + B a ( γ )with A orthogonal with respect to the metric g ( u ) , and these generate a subgroup of O ( N ) . Thus to every Novikov algebra A there is an associated monodromy group whichwe denote W ( A ) . Example 11.
For the algebra A defined by the first metric in Example 9 the monodomygroup is W ( A ) ∼ = Z [1 , , , which acts on the v -variables via v ε v ,v ε v ,v ε v where ε = 1 . This monodromy group also controls the structure of the remaining metrics g ijα for α ≥ . On applying the transformation u = u ( v ) to the entire pencil of metrics g ij ( u ) = D X α =1 k α g ijα ( u ) , k α ∈ C one knows that this transforms to the pencil g ij ( v ) = η ij + D X α =2 k α d ijαk v k for some constants d ijαk since the linearity property must be preserved under this changeof variables. On applying the monodromy transformation to this pencil gives the followsinvariance properties for the component parts of the pencil: A ia η ab A jb = η ij and A ia d abαk A jb = d abαk and hence (assuming that the monodromy group acts such that A ia = diag( ε d , . . . , ε d M )where ε is the ( d + d M ) th root of unity) the constraints ε d a + d b η ab = η ab ,ε d a + d b − d c d abαc = d abαc and this constrains where the non-zero terms can appear: if ε d a + d b = 0 then η ab = 0 ,ε d a + d b − d c = 0 then d abαc = 0 . This, coupled to the triangular structure of the transformation u = u ( v ) which impliesthe the metrics g ijα ( v ) for α ≥ Example 12.
In the above Example 11, the monodromy group W ( A ) ∼ = Z [1 , , andthe only non-zero entries are d , d and hence the pencil of metric takes the skeletalform g ij = c c c + d v d v d v . This methods only fixes the skeletal forms of the pencil: it does note fix the values ofthe constants. These have to be found by ensuring the resulting structures satisfy the B -algebra conditions. CONSTRUCTION OF MULTIDIMENSIONAL DUBROVIN-NOVIKOV BRACKETS 11 Conclusion
The above construction could be generalized: underlying the construction is a singlecommutative, associative algebra. One could start with pencils of compatible com-mutative, associative algebras and construct their derivations. One could then deriveconditions for the resulting structures to satisfy the conditions in the definition og the B -algebra. Another direction of possible research is to connect this construction withthe theory of Segre forms, as used in [9]. Conversely, normal forms has played no rolein the theory of Novikov algebras, which has been more algebraic than geometric: itwould be of interest to reconcile these two approaches.Finally, it should be stressed that the construction presented here is far from ex-haustive: it does not, in any way, claim to be part of a classification scheme - just amethod to construct examples. But in an area where few examples are known, it doesprovide a method for the construction of multi-component, multi-dimensional bracketsof Dubrovin-Novikov type. Acknowledgements
I would also like to thank Blazej Szablikowski and Dafeng Zuo for various conversa-tions concerning Novikov algebras and their role in the theory of Hamiltonian structures.
References [1] Bai, C. and Meng, D.,
The classification of Novikov algebras in low dimensions , J. Phys. A (2001) 1581–1594[2] Bai, C. and Meng, D., Addendum: invariant bilinear forms , J. Phys. A (2001) 8193–8197[3] Balinskii, A.A. and Novikov, S.P., Poisson brackets of hydrodynamic type, Frobenius algebras andLie algebras , Soviet Math. Dokl. (1985) 228–231.[4] Burde, D. and de Graaf, W., Classification of Novikov algebras , Appl. Algebra Engrg. Comm.Comput. (2013) 1–15[5] Casati, M. On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type , Com-mun. Math. Phys. (2015), 851-894.[6] Dubrovin, B.A.,
Geometry of 2D topological field theories , in:
Integrable Systems and QuantumGroups , (Montecatini, Terme, 1993). Lecture Notes Maths. 1620, 120-348.[7] Dubrovin, B.A. and Novikov, S.P.,
Hamiltonian formalism of one-dimensional systems of thehydrodynamic type and the Bogolyubov-Whitham averaging method , Soviet Math. Dokl. (1984)651–654.[8] Dubrovin, B.A. and Novikov, S.P., On Poisson brackets of hydrodynamic type , Soviet Math. Dokl. (1984) 651–654.[9] Ferapontov, E.V., Lorenzoni, P. and Savoldi, A., Hamiltonian Operators of Dubrovin-NovikovType in 2D , Lett. Math. Phys. (2015) 341-377.[10] Filipov, V. T.
A class of simple nonassociative algebras , Mat. Zametki (1989) 101-105.[11] Gelfand I.M. and Dorfman, I.Y., Hamiltonian operators and algebraic structures related to them ,Funct. Anal. Appl. 13 (1979), no. 4, pp. 248-262.[12] Mokhov, O.I.,
Dubrovin-Novikov type Poisson brackets (DN-brackets) , Funct. Anal. Appl. 22(1988), no. 4 , pp. 336-338.[13] Mokhov, O.I.
Classification of nonsingular multidimensional Dubrovin-Novikov brackets , Funct.Anal. Appl. 42 (2008), no. 1, pp39-52, 95-96.[14] Savoldi, A.,
Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D , J. Phys.A (2015), Article Number: 265202.[15] Strachan, I.A.B.,
Darboux coordinates for Hamiltonian structures defined by Novikov algebras ,arxiv 1804.07073.[16] Xu, X.,
Classification of simple Novikov algebras and their irreducible modules of characteristic0 , J. Algebra (1997) 253-279.
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