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

1 Introduction: The semi-direct product of Virasoro algebra with the Kac–Moody algebra

This paper is a continuation of the paper [1] where we studied bi-Hamiltonian systems associated to the three-cocycle extension of the algebra of diffeomorphisms on a circle. In this note we show that certain natural problems (classification of Verma modules, classification of coadjoint orbits, determination of Casimir functions) [2, 3, 4, 5] for the central extensions of the Lie algebra Vect(S¹) ⋉ 𝓛𝓖 reduce to the equivalent problems for Virasoro and affine Kac–Moody algebras (which are central extensions of Vect(S¹) and 𝓛𝓖 respectively). Let G be a Lie group and 𝓖 its Lie algebra. The group Diff(S¹) of diffeomorphisms 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) ϵ Diff(S¹)² and (g, h) ϵ LG² the composition law of the group Diff(S¹) ⋉ 𝓛𝓖 is

The Lie algebra of Diff(S¹) ⋉ LG is the semi-direct product Vect(S¹) ⋉ 𝓛𝓖 of the Lie algebras Vect(S¹) and 𝓛𝓖.

Let 𝓖 be a Lie algebra and 〈., .〉 a non-degenerated invariant bilinear form. Vect(S¹) is the Lie algebra of vector fields on the circle and 𝓛𝓖 the loop algebra (i.e., the Lie algebra of smooth maps from S¹ to 𝓖), Vect(S¹)_ℂ is the Lie algebra over ℂ generated by the elements L_n, n ϵ ℤ with the relations

We denote by 𝓛𝓖_ℂ the Lie algebra over ℂ generated by the elements g_n, n ϵ ℤ, g ϵ 𝓖 where (λg + μh)_n is identified with λ g_n + μh_n with the relations

The semi-direct product of Vect(S¹) with 𝓛𝓖 is as a vector space isomorphic to C^∞ (S¹, ℝ) ⊕ C^∞(S¹,𝓖) [6]. The Lie bracket of 𝓢𝓤(𝓖) has the form

for any (u, v) ϵ C^∞(S¹,ℝ)² and any (a, b) ϵ C^∞(S¹,𝓖)², where prime denote derivative with respect to a coordinate on S¹. The Lie algebra Vect(S¹) ⋉ 𝓛𝓖 can be extended with a universal central extension 𝓢𝓤(𝓖) by a two-dimensional vector space. Let us denote by 𝓙(u) = ∫_S¹u. Two independent cocycles are given by

We denote by (u,a, χ, α) the elements of 𝓢𝓤(𝓖) with u ϵ C^∞(S¹, ℝ), a ϵ C^∞(S¹, 𝓖) and (χ, α) ϵ ℝ². The algebra 𝓢𝓤(𝓖) can be also represented as the semi-direct product of Virasoro algebra on the affine Kac–Moody algebra. We denote by c_Vir and c_K−M the elements (0, 0, 1, 0) and (0, 0, 0, 1) respectively. If 𝓖 = ℝ, then the Lie algebra Vect(S¹) ⋉ 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 ϵ C^∞(S¹, ℝ), a ϵ C^∞(S¹, 𝓖), and (χ, α, γ) ϵ ℝ³. 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 Affine 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(S1)~⊕LG~)′ the subset of Vect(S1)~⊕LG~ composed of elements (u, a, ξ, β) with β ≠ 0. Then introduce two new maps 𝓘(u, a, ξ, β) from 𝓢𝓤(𝓖)′ to (Vect(S1)~⊕LG~)′, and 𝓘̃(u, a, ξ, β, γ) from 𝓢𝓤(𝓖) to Vect(S1)~⊕LR~. We prove that 𝓘(u, a, ξ, β) and 𝓘̃(u, a, ξ, β, γ) are Poisson maps. In Section 3 we discuss coadjoint orbits and Casemir functions for 𝓢𝓤(𝓖). Let 𝓗̃ be a central extension of a Lie algebra 𝓗 and H be a Lie group with Lie algebra is 𝓗. We find explicit form for the coadjoint actions of the groups Diff(S¹) ⋉ LG and Diff(S¹) ⋉ 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(S1)~⊕LG~ to a coadjoint orbits of Vect(S1)~. We prove that map 𝓘̃ sends the coadjoint orbits of SU(G)~ to coadjoint orbits of Vect(S1)~⊕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_ω𝓗(𝓗). Then we show (Theorem 5.1 that the realizability of the action of 𝓚 in 𝓤_ω𝓗 (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 affine 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.

2 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 Affine 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 ϵ 𝓚 if there exists an element ∇ f (x) such that

for any a ϵ 𝓚. For two regular functions f,g : 𝓚 ⟶ ℝ, we define the Kirillov-Kostant structure as a Poisson structure on 𝓚 with

Then for any e ϵ 𝓖, the second Poisson structure {f, g}_e(x) compatible with the Kirillov-Kostant Poisson structure is defined by

A non-degenerated bilinear form on 𝓢𝓤(𝓖) and Vect(S1)~⊕LG~ is defined by

We denote by 𝓢𝓤(𝓖)′ the subset of 𝓢𝓤(𝓖) of elements (u, a, ξ, β) with non-vanishing β. Let u′=u−∥a∥22β We denote by (Vect(S1)~⊕LG~)′ the subset of Vect(S1)~⊕LG~ composed of elements (u, a, ξ, β) with β ≠ 0. Let us introduce a new map 𝓘(u, a, ξ, β) = (u′, a, ξ, β) from 𝓢𝓤(𝓖)′ to (Vect(S1)~⊕LG~)′. Then for non-vanishing β, let us introduce another new map I~(u,a,ξ,β,γ)=(u′−γβa′,a,ξ−γ2β,β) from 𝓢𝓤(𝓖) to Vect(S1)~⊕LR~. Here we give a proof for the following new theorem:

3 Coadjoint orbits Casimir functions and for 𝓢𝓤(𝓖)

Let 𝓗̃ be a central extension of a Lie algebra 𝓗, and H be a Lie group with Lie algebra is 𝓗. ThenH acts on𝓗̃^* by the coadjoint action along coadjoint orbits.

The classification of coadjoint orbits of Vect(S¹) ⋉ 𝓛𝓖 can be known from the classification of coadjoint orbits of the Virasoro and affine Kac-moody algebra. Here we obtain the following new

In other words, this means that if β₁ ≠ 0, the elements (u₁, a₁, ξ₁, β₁) and (u₁,a₁, ξ₂, β₂) are in the same coadjoint orbit if and only if: ξ₁ = ξ₂, β₁ = β₂, (a₁, β₁) and (a₂, β₂) are on the same coadjoint orbit of LG~, (u1−∥a1∥2β1),ξ1)and(u2−∥a2∥2β2),ξ2) are elements of the same coadjoint orbit of Vect(S1)~.

In other words, this means that if β₁ ≠ 0 the elements (u₁,a₁, ξ₁, β₁, γ₁) and (u₁,a₁, ξ₂, β₂, γ₂) are in the same coadjoint orbit if and only if γ₁ = γ₂, ξ₁ = ξ₂, β₁ = β₂, (a₁, β₁) and (a₂, β₂) are on the same coadjoint orbit of LG~,(u1−a122β,ξ1−γ12β1)and(u2−a222β,ξ2−γ22β2) are elements of the same coadjoint orbit of Vect(S1)~. In a particular case, if β₁ = β₂ = 0, then:

Previously, we determined Casemir functions on SU(G)~′ and SU(R)~. We gave the following proposition:

4 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↦ϕ1(A(u+Bdadx+C)) has an asymptotic development. The coefficients of this development form a hierarchy. The first term of this development is ∫_S¹u, and the second one is ∫_S¹ (u² + γu + ∥ a ∥²). A linear combination of these two terms gives the Hamiltonian of equations H(u, a) = ∫_S¹ (u² + ∥ a ∥²).

Let {ϕ_i, i ϵ I} be a set of Casimir functions and e ϵ 𝓖. Define x_χ = x − χe, for some χ ϵ ℝ.

5 The universal enveloping algebra of 𝓢𝓤(𝓖)

When 𝓗 = ∑_kϵℤ 𝓗_k has a structure of graded algebra, its universal enveloping algebra 𝓤𝓗 is also naturally endowed with a structure of a graded Lie algebra. Indeed, the weight of a product h₁, …, h_n ϵ 𝓤𝓗 of homogeneous elements is defined to be the sum of the weights of the elements h_i, i = 1, …, n. The universal enveloping algebra 𝓤𝓗 admits a filtration UH=∪i=0∞Fk 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≤nu_k where u_k is an element of weight k of 𝓤𝓗. The product of two such elements is defined by:

where w_k = ∑_iϵℤu_i.v_ki which is a finite sum. Let ω₁, …, ω_n be two-cocycles on the Lie algebra 𝓗, let 𝓗̃ be the central extension associated with and let e₁,…,e_n be the central elements associated with these cocycles.

The modified generalized enveloping algebra Uω1,…,ωnH is defined to be the quotient of the generalized enveloping algebra of 𝓗̃ by the ideal generated by the elements {e₁ – 1,…,e_n – 1}. We denote again by 1 the neutral element of

UλG. The algebra Uω1,…,ωnH is by construction a graded algebra and a filtered algebra. We denote by F_n, n ϵ ℕ its filtration. Let us recall shortly the main properties of the modified generalized enveloping algebra. Let V is be a module over 𝓗̃ such that for any v ϵ V, there exists n₀ ϵ ℤ such that for any n > n₀ and any h ϵ 𝓗̃_n we have h.v = 0. Such modules are called representations of positive energy, and e_i acts on V by λ_iId. Then V is a module over

Uω1,…,ωnH Such modules are named modules of positive energy. The anticommutator provides a structure of Lie algebra on Uω1,…,ωnH For this bracket F₁ is a Lie sub-algebra isomorphic to the central extension of 𝓗 by the cocycle

ω=∑i=1nωi. We denote by i be the natural inclusion of 𝓗̃ into UωG given by this identification.