On Symmetry Properties of Frobenius Manifolds and Related Lie-Algebraic Structures

: The aim of this paper is to develop an algebraically feasible approach to solutions of the oriented associativity equations. Our approach was based on a modiﬁcation of the Adler– Kostant–Symes integrability scheme and applied to the co-adjoint orbits of the diffeomorphism loop group of the circle. A new two-parametric hierarchy of commuting to each other Monge type Hamiltonian vector ﬁelds is constructed. This hierarchy, jointly with a specially constructed reciprocal transformation, produces a Frobenius manifold potential function in terms of solutions of these Monge type Hamiltonian systems.

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The Introductory Setting
Let us start with an interesting mathematical structure, suggested in [1][2][3][4][5], on the space of smooth functions: consider a real-valued C ∞ -smooth differentiable Frobenius manifold potential function F ∈ C ∞ (R n ; R) and denote their partial derivatives as for i, j, and k = 1, n, n ∈ N.These partial derivatives are symmetrical, with respect to permutations of their indices.Let us assume additionally that the symmetric matrix η := {η ij (t) := F ij1 (t) : i, j = 1, n} is non-degenerate, and call it an induced metric on the R n .In addition, where, by definition, F ijk (t)η ks (t), ∑ k∈ 1,n η sk (t)η kj (t) = δ s j (1.3) for all i, j, and s ∈ N. Assume now that the set R n represents a local coordinate frame [6,7] of an a finite-dimensional manifold M. Then its tangent space T t (M) at a point t ∈ M is described by means of the local vector field system {∂/∂t i ∈ T t (M) : i = 1, n}, which a priori commute to each other: [∂/∂t i , ∂/∂t j ] = 0 for all i, j = 1, n.Let us now assume that the manifold M is a Frobenius manifold [8][9][10], i.e., its tangent space T t (M) at any point t ∈ M forms an associative Frobenius algebra F M with respect to some multiplication " • " on F M : for any i, j and s = 1, n with the structure constants defined by the expression (1.3).Define now a set of matrices C i (t Then, as it easily follows from (1.4), the structure constants (1.3) should satisfy the following additional constraints: for any t ∈ M and all i, j = 1, n. (1.5) are called the Witten-Dijkgraaf-Verlinde-Verlinde, or oriented associativity WDVV equations.These equations were first investigated in [11][12][13] for problems related with topological and string quantum field theory of elementary particles.A nice introduction into the topic can be found in B. Dubrovin Lecture Notes [2].
A full Frobenius structure on M consists of the data (•, e, η, E).Here • : T(M) ⊗ S T(M) → T(M) is an associative and commutative multiplication on the tangent sheaf, so that T(M) becomes a sheaf of commutative algebras over the ring R{t} of convergent series with identity e ∈ T(M), η is a metric on M (non-degenerate quadratic form T(M) ⊗ S T(M)), and E is a so called Euler vector field.These structures are connected by various constraints and compatibility conditions, and are presented in [2,3] and [32,33].For example, the metric η must be flat and " • "-invariant, i.e., a|b • c η = a • b|c η for the metric •|• η on M and any a, b, and c ∈ T(M).Various weaker versions of the Frobenius structure are interesting in themselves and also appear in [19][20][21] in different contexts.
Let us also mention an additional notion of a unital Frobenius manifold F M , introduced in [10] and further studied in [9].This structure consists of an associative and commutative multiplication " • " on the tangent sheaf as above, satisfying the following properties: 1 0 ) a flat structure T(M) on M subject to a flat connection is compatible with a multiplication " • ", if in a neighborhood of any point there exists a vector field C ∈ Γ(T(M)), such that for arbitrary local flat vector fields X, Y ∈ Γ(T(M)) one has where ) is an identity element, if 1 0 ) holds and moreover, the identity element e := ∂/∂t 1 is flat, that is the corresponding covariant derivative ∇ ω X e = 0 for any X ∈ Γ(T(M)).From (1.6) one easily ensues the relationships (1.5), where for any i, j, and k = 1, n and t ∈ M. As a very interesting example of the above construction can be obtained for the special case n = 3.We can take into account a reduction of the commuting matrices C j ∈ End E 3 , j = 1, 3, presented in [1][2][3].Namely, assume that a smooth Frobenius manifold potential function F ∈ C ∞ (R n ; R) is representable as where a smooth mapping f : R 3 → R satisfies, following from (1.4) in the form for any (t 1 , t 2 , t 3 ) ∈ R 3 .In particular, as it was shown by B. Dubrovin and Y. Manin [2,3,32,33], the Equation (1.9) allows the following system of compatible (for any parameter p ∈ C\{0}) linear differential equations: where ) for t ∈ M, j = 1, 3.An effective Lie-algebraic analysis of the Dubrovin-Manin linear system (1.10) was recently presented in [14,15].In the present work, based on a modification of the Adler-Kostant-Symes integrability scheme, applied to the co-adjoint orbits of the loop diffeomorphism group of circle, a new two-parametric hierarchy of commuting to each other Monge type Hamiltonian vector fields and where the pair of functions (u, v) ∈ C ∞ (M; R 2 ) satisfies the evolution flows (1.13) and (1.14).Then this function F : M → R is a potential function of the Frobenius manifold M, describing the related Frobenius manifold algebraic structures.

Frobenius Manifolds, the Related Compatible Co-Adjoint Loop Lie Algebra and Integrability
Consider now the functional Lie algebra G ≃ (C ∞ (T * (S 1 ); R); {•, •}), generated by special Hamiltonian vector fields on the cotangent space T * (S 1 ) to the circle S 1 and endowed with the canonical Lie commutator {a, b}(x; p) for any a, b ∈ G at point (x, p) ∈ T * (S 1 ).This algebra possesses the following symmetric and non-degenerate bi-linear form: (a|b and where, by definition, ∇h ± (l) := ∇h (k) (l)| G ± , are commuting to each other subject to the corresponding evolution parameters t k ∈ R, k ∈ Z + , for arbitrary infinite hierarchy of smooth functionally independent Casimir functionals h (k) ∈ I(G * ), k ∈ Z + .The latter is, evidently, equivalent to the following Lax-Sato type vector field representations: for all k, m ∈ Z + , where, by definition, any element a ∈ G via the expression ã(x; p) := ))) generates a canonical Hamiltonian vector field on T * (S 1 ) at point (x; p) ∈ T * (S 1 ).
Take now an analytic at the momentum p ∈ R element l ∈ G * ≃ G in the following asymptoptic as p → ∞ form: where the element p ∈ G * is considered here as an infinitesimal Lie algebra G character, satisfying the conditions [G ± , p] ∈ G ± , that can be easily checked by direct computations.The flows (2.6) are equivalent to the following co-adjoint action on G * ≃ G with respect to the evolution parameters t k ∈ R for all k ∈ Z + .
It is worthy to observe now that in the case of the Casimir functionals h (k) := where ψ(•; z) ∈ C ∞ (T * (S 1 ); C) is the eigenfunction corresponding to an eigenvalue z ∈ C, which is a priori invariant with respect to all vector fields (2.10).The latter naturally allows to apply to (2.12) the modified inverse scattering transform technique developed in [40] and describe many classes of symbols l ∈ G, generating important dispersion-less heavenly type [41] dynamical systems, important for applications in modern mathematical physics.
As the point variables (x; p) ∈ T * (S 1 ) are constant parameters for the evolution flows (2.10) on analytic at p = ∞ element l ∈ G * , one can put, by definition, l(x; p) = z ∈ C and resolve the functional equation l(x; p) = z with respect to the symbol parameter p ∈ R, obtaining the following expression: with coefficients ξ j ∈ C ∞ (S 1 ; R), j ∈ N, characterized by the following lemma.

.14)
where the elements H k (x; z) := l k + (x; ξ(x; z)), k ∈ N, are determined, using the following simple algebraic expressions: which hold jointly with compatibility relationships for all k, s ∈ N.
Proof.Making use of the Equation (2.10), one can easily calculate for any k ∈ N the evolution equations giving rise to the following expressions which hold for all k ∈ N and all z ∈ R. The compatibility relationships are obvious, following from the commuting to each other flows (2.14).
Consider now the functional identity which is satisfied as z → ∞, owing to the following residuum calculation: which holds for any k ∈ N. Consider now Hamiltonian functions H k : T * (S 1 ) → R, k ∈ N, and consider the related canonical Hamiltonian vector fields on the cotangent space T * (R) : with respect to a point (x, p) ∈ T * (S 1 ) subject to the evolution parameter t k ∈ R, k ∈ N. Taking into account the evolution flows (2.20) and the fact that ∂/∂t 1 = ∂/∂x, the identity (2.18) can be rewritten as from which and the relationships (2.16) one ensues the functional representation for some smooth function F : M → R. with respect to the evolution parameters x = t 1 ∈ R, t 2 , t 3 ∈ R, etc., where, for instance, and so on.The above commutator expressions with respect to the evolution parameters t 1 , t 2 and t 3 ∈ R reduce to the next commuting to each other non-linear Monge type evolution systems and being also compatible dispersion-less Hamiltonian flows on the corresponding functional phase.Moreover, the evolution systems (2.27) and (2.28) are equivalent to the Lax-Sato vector field commutator representation (2.7), where ∇h ∇h The vector fields (2.29), being considered as elements of the Lie algebra G ≃ di f f (S 1 × C) of holomorphic with respect to the variable p ∈ C vector fields on S 1 × C, naturally splits into the direct sum of two sub-algebras G = G+ ⊕ G− , holomorphic in the parameter p ∈ C inside D 1 + (0) of the unit circle D 1 + (0) ⊂ C and outside D 1 − (0) of this disk, respectively, appear to be generated by the corresponding Casimir functionals on the adjoint space G * ≃ Ω 1 (S 1 × C) at some root element l ∈ G * subject to the following canonical non- degenerate bi-linear form on G * × G : where we put, by definition, l := l|dx , ã := a|∂/∂x ,x:= (p; where the pair of functions (u, v) ∈ C ∞ (M; R 2 ) satisfies the evolution flows (2.27) and (2.28).Then it is a potential function of the Frobenius manifold M, describing the related Frobenius manifold algebraic structures.
This result makes it possible to describe a wide variety of Frobenius manifold potential functions in terns of solutions to these Monge type Hamiltonian systems (2.27) and (2.28).
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