Algebraic proofs for shallow water bi – Hamiltonian systems for three cocycle of the semidirect product of Kac – Moody and Virasoro Lie algebras

This paper is a continuation of the paper [1] wherewe studied bi-Hamiltonian systems associated to the threecocycle extension of the algebra of di eomorphisms on a circle. In this note we show that certain natural problems (classi cation of Verma modules, classi cation of coadjoint orbits, determination of Casimir functions) [2–5] for the central extensions of the Lie algebra Vect(S1)⋉LG reduce to the equivalent problems for Virasoro and a ne Kac–Moody algebras (which are central extensions of Vect(S1) andLG respectively). Let G be a Lie group and G its Lie algebra. The group Di(S1) of di eomorphisms of the circle is included in the group of automorphisms of the Loop group LG of smooth maps from S1 to G. For any pairs (φ,ψ) ∈ Di(S1)2 and (g, h) ∈ LG2 the composition law of the group Di(S1) ⋉LG is (φ, a) ⋅ (ψ, b) = (φ ○ ψ, a.b ○ φ). The Lie algebra ofDi(S1)⋉LG is the semi-direct product Vect(S1)⋉LG of the Lie algebras Vect(S1) andLG. Let G be a Lie algebra and ⟨., .⟩ a non-degenerated invariant bilinear form. Vect(S1) is the Lie algebra of vector elds on the circle and LG the loop algebra (i.e., the Lie algebra of smooth maps from S1 to G), Vect(S)C is the Lie algebra over C generated by the elements Ln , n ∈ Z with the relations [Lm , Ln] = (n −m)Ln+m . We denote by LGC the Lie algebra over C generated by the elements gn , n ∈ Z, g ∈ G where (λg + μh)n is identi ed with λgn + μhn with the relations [gn , hm] = [g, h]n+m .

The semi-direct product of Vect(S ) with LG is as a vector space isomorphic to C ∞ (S , R) ⊕ C ∞ (S , G) [6]. The Lie bracket of SU(G) has the form for any (u, v) ∈ C ∞ (S , R) and any (a, b) ∈ C ∞ (S , G) , where prime denote derivative with respect to a coordinate on S . The Lie algebra Vect(S ) ⋉ LG can be extended with a universal central extension SU(G) by a two-dimensional vector space. Let us denote by J (u) = ∫ S u. Two independent cocycles are given by We denote by (u, a, χ, α) the elements of SU(G) with u ∈ C ∞ (S , R), a ∈ C ∞ (S , G) and (χ, α) ∈ R . The algebra SU(G) can be also represented as the semi-direct product of Virasoro algebra on the a ne Kac-Moody algebra. We denote by c Vir and c K−M the elements ( , , , ) and ( , , , ) respectively. If G = R, then the Lie algebra Vect(S ) ⋉ LR has a universal central extension SU(R) by a three-dimensional vector space. The third independent cocycle is given by We denote by (u, a, χ, α, γ, δ) elements of SU(R) with u ∈ C ∞ (S , R), a ∈ C ∞ (S , G), and (χ, α, γ) ∈ R . The Lie bracket of SU(R) is given by In this paper we discuss a few questions. Let us mention the main results. First, in Section 2 we consider Kirillov-Kostant Poisson brackets [7] of the regular dual of the semi-direct product of Virasoro Lie algebra with the A ne Kac-Moody Lie algebra. Let us denote by SU(G) ′ the subset of SU(G) of elements (u, a, ξ, β) with non-vanishing β. We denote by ( Vect(S ) ⊕ LG) ′ the subset of Vect(S ) ⊕ LG composed of elements (u, a, ξ, β) with β ≠ . Then introduce two new maps I(u, a, ξ, β) from SU(G) ′ to ( Vect(S ) ⊕ LG) ′ , and I(u, a, ξ, β, γ) from SU(G) to Vect(S ) ⊕ LR. We prove that I(u, a, ξ, β) andĨ(u, a, ξ, β, γ) are Poisson maps. In Section 3 we discuss coadjoint orbits and Casemir functions for SU(G). LetH be a central extension of a Lie algebra H and H be a Lie group with Lie algebra is H. We nd explicit form for the the coadjoint actions of the groups Di (S ) ⋉ LG and Di (S ) ⋉ LR * + . As a result we obtain the following new theorem. We prove that a coadjoint orbit of SU(G) is mapped by I to a coadjoint orbit of Vect(S ) ⊗ LG to a coadjoint orbits of Vect(S ). We prove that the mapĨ sends the coadjoint orbits of SU(G) to coadjoint orbits of Vect(S ) ⊗ LG. Previously, we determined Casemir functions on SU(G) ′ and SU(R). We then prove new propositions concerning the exlicit form of Casemir functions on SU(G) ′ , and in particular on on SU(R) ′ . This paper was partially inspired by the construction of bi-Hamiltonian systems as natural generalization of the classical Korteweg-de Vries equation. [1,[8][9][10][11]. It has been showed in [1], that the dispersive water waves system equation [9,10,12] is a bi-Hamiltonian system related to the semi-direct product of a Kac-Moody and Virasoro Lie algebras, and the hierarchy for this system was found. In Section 4 some results of [1] are obtained from another point of view. We prove new proposition for pairwise commuting functions under certain brackets. In section 5 we discuss properties of the universal enveloping algebra of SU(G). In subsection 5.1 we consider a decomposition of the enveloping algebra of a semi-direct product. We introduce the notion of realizability of the action of K on H in U ω H (H). Then we show (Theorem 5.1) that the realizability of the action of K in U ω H (H) leads to the isomorphism In subsection 5.2 the case of SU C (G) is considered. In subsection 5.3 we discuss representations of SU(G). We prove that positive energy representation V of SU C (G) with non-vanishing βId-action of the cocyle c K−M delivers a pair of commuting representations of Virasoro and a ne Kac-Moody Lie algebras. This proposition determines whether a SU C (G) Verma module is a sub-module of another Verma module of SU C (G). We also prove a proposition regarding a linear form over h with non-vanishing λ(c K−M ). In this paper we present proofs for corresponding theorems and lemmas.

The Kirillov-Kostant structure of SU (G)
Now we consider Kirillov-Kostant Poisson brackets of the regular dual of the semi-direct product of Virasoro Lie algebra with the A ne Kac-Moody Lie algebra. Let K be a Lie algebra with a non-degenerated bilinear form ⟨., .⟩. A function f ∶ K → R is called regular at x ∈ K if there exists an element ∇f (x) such that for any a ∈ K. For two regular functions f ,g ∶ K → R, we de ne the Kirillov-Kostant structure as a Poisson structure on K with Then for any e ∈ G, the second Poisson structure {f , g} e (x) compatible with the Kirillov-Kostant Poisson structure is de ned by A non-degenerated bilinear form on SU(G) and Vect(S ) ⊕ LG is de ned by Here we give a proof for the following new theorem: Proof. For any regular function f (u, a, ξ, β) from Vect(S ) ⊕ LG to R let us de ne a regular functionf from us denote by f u the function of the variables a and β that we get when we x u and ξ. Let us denote f a the function of the variables u and ξ that we get when we x a and β. With the previous notations, one has for β ≠ for the bracket {., and for the bracket {., For any regular function f on Vect(S ) and any regular function g on LG we have Indeed, for i = , , δ a − a β δ u f i (û, ξ) = . We have: This gives Let g i (a, β), i = , be two regular functions on the a ne Kac-Moody algebra. One notes that δg ,u = δg ,u = . Therefore, Then, We have: The sum of the rst two terms is equal to . The last term is J (δf u ⟨[a, a], δg a ⟩), and is equal to zero. One can proceed similarly forĨ.

Coadjoint orbits Casimir functions and for SU (G)
LetH be a central extension of a Lie algebra H, and H be a Lie group with Lie algebra is H. Then H acts oñ H ⋆ by the coadjoint action along coadjoint orbits.

Proposition 3.1. The coadjoint actions of the groups Di (S ) ⋉
LG and Di (S ) ⋉ LR * + are given by The classi cation of coadjoint orbits of Vect(S )⋉LG can be known from the classi cation of coadjoint orbits of the Virasoro and a ne Kac-moody algebra. Here we obtain the following new Proof. For any φ ∈ Di (S ), there exists h ∈ LG such that By direct computation we check that I(Ad * φ, g (u, a, ξ, β) = (Ad * φ, g.h I(u, a, ξ, β).
Proof. We have: Ad(φ, g) Previously, we determined Casemir functions on SU(G) ′ and SU(R). We gave the following proposition:

Bi-hamiltonian dispersive water waves systems associated to SU (G)
It has been showed in [1], that the dispersive water waves system equation [9,10,12] is a bi-Hamiltonian system related to the semi-direct product of a Kac-Moody and Virasoro Lie algebras, and the hierarchy for this system was found. In this section some results of [1] are obtained from another point of view. We obtain new The function λ ↦ φ A(u + B da dx + C) has an asymptotic development. The coe cients of this development form a hierarchy. The rst term of this development is ∫ S u, and the second one is ∫ S (u + γu+ ∥ a ∥ ). A linear combination of these two terms gives the Hamiltonian of equations H(u, a) = ∫ S (u + ∥ a ∥ ).
Let {φ i , i ∈ I} be a set of Casimir functions and e ∈ G. De ne x χ = x − χe, for some χ ∈ R.

The universal enveloping algebra of SU (G)
When H = ∑ k∈Z H k has a structure of graded algebra, its universal enveloping algebra U H is also naturally endowed with a structure of a graded Lie algebra. Indeed, the weight of a product h , . . . , h n ∈ UH of homogeneous elements is de ned to be the sum of the weights of the elements h i , i = , . . . , n. The universal enveloping algebra U H admits a ltration U H = ∪ ∞ i= F k where F k is the vector space generated by the products of at most k elements of H. The generalized enveloping algebra is the algebra of the elements of the form ∑ k≤n u k where u k is an element of weight k of U H. The product of two such elements is de ned by: where w k = ∑ i∈Z u i .v k i which is a nite sum. Let ω , . . . , ω n be two-cocycles on the Lie algebra H, letH be the central extension associated with and let e , . . . , e n be the central elements associated with these cocycles.
The modi ed generalized enveloping algebra U H ω ,...,ω n is de ned to be the quotient of the generalized enveloping algebra ofH by the ideal generated by the elements {e − , . . . , e n − }. We denote again by the neutral element of U G λ . The algebra U H ω ,...,ω n is by construction a graded algebra and a ltered algebra. We denote by F n , n ∈ N its ltration. Let us recall shortly the main properties of the modi ed generalized enveloping algebra. Let V is be a module overH such that for any v ∈ V, there exists n ∈ Z such that for any n > n and any h ∈H n we have h.v = . Such modules are called representations of positive energy, and e i acts on V by λ i Id. Then V is a module over U H ω ,...,ω n . Such modules are named modules of positive energy. The anticommutator provides a structure of Lie algebra on U H ω ,...,ω n . For this bracket F is a Lie sub-algebra isomorphic to the central extension of H by the cocycle ω = ∑ n i= ω i . We denote by i be the natural inclusion ofH into U G ω given by this identi cation.

. Decomposition of the enveloping algebra of a semi-direct product
In some very particular cases, the modi ed generalized enveloping algebra of a semi-direct product K ⋉ H of two Lie algebras is isomorphic to the tensor product of some modi ed generalized enveloping algebras of K and of H. LetH be the central extension of H with the two-cocycle ω H . Denote by ⋅ the action of the Lie algebra K on the Lie algebraH. Let us introduce the semi-direct product K ⋉H which is a central extension of K ⋉ H by a two-cocycle ω ′ H with A two-cocycle ω K on K de nes also a two-cocycle ω ′ K by of K ⋉ H. Let I be the natural inclusion ofH into U ω H (H) and J be the natural inclusion ofH into We call the action of K on H realizable in U ω H (H) when there exists a map F ∶ K → U ω H (H) and a two-cocycle α on K such that for any pair (g , g ) in K Proof. Let U g = {ĝ g ∈ K} with be the unitary subalgebra of U ω ′ K ,ω ′ H (K ⋉ H) generated by the elementŝ g = g − F(g), and U j = {j(h), h ∈H} be the unitary subalgebra of U ω ′ K ,ω ′ H (K ⋉ H). For any (g, h) this implies that the generators of U g and U j commute, i.e., [ĝ, j(h)] = . The subalgebras U g and U j therefore commute. The subalgebra U g is isomorphic to U ω K −α (K). Let us check that the generators {ĝ g ∈ K} of this algebra satisfy the relations of the generators of U ω K −α (K): Since F(g ) is an element of U j and since the algebras U g and U j commute [F(g ), g ] = [F(g ), F(g )] and [g , F(g )] = [F(g ), F(g )]. Therefore: and nally The subalgebra U j is obviously isomorphic to U ω H (H). The generalized modi ed enveloping algebra

. The case of SU C (G)
Let G be a simple complex Lie algebra and C ϕ its dual Coxeter number. Introduce the {K , . . . , K n } a basis of G, and the dual basis K * , . . . , K * n with respect to the Killing form ⟨., .⟩. We apply Theorem 5.1 for K = Vect(S ), H = LG, ω K = ξω Vir , and ω H = βω K−M . In this case, In this case, on has ω ′ H = βω H + γω sp . The map F ∶ Vect(S ) C → SU C (C) de ned by βF(L n ) = i∈Z ∶ a i a n−i ∶ +γa n , for a cocycleα = (α + γ β − )ω Vir . For SU C (C) we obtain , are associated to linear forms ν, µ over the spaces generated by c Vir and u , c K−M and {(h ) , . . . , (h k ) } correspondingly. For any λ ∈ k * , the Verma module V λ (SU C (G)) is a positive energy representation. Thus, V λ (SU C (G)) is Virasoro and a ne Kac-Moody algebra module. The generator e of V λ (SU C (G)) brings about a Verma module V Vir ν for Virasoro algebra. It generates also a Verma module V Vir ν for the a ne Kac-Moody algebra. The linear form ν satis es ν(u )e = λ(u − F(u ))e, i.e., (u − (β + η) − K ⋅ K * e = ν(u )e.
Suppose the action of a Casimir element of G is given by acts by D(λ)Id for D(λ) ∈ C. We then have