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

A. Zuevsky

doi:10.1515/math-2018-0002

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

A. Zuevsky

Published/Copyright: January 31, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

We prove new theorems related to the construction of the shallow water bi-Hamiltonian systems associated to the semi-direct product of Virasoro and affine Kac–Moody Lie algebras. We discuss associated Verma modules, coadjoint orbits, Casimir functions, and bi-Hamiltonian systems.

Keywords: Affine Kac–Moody Lie algebras; Bi-Hamiltonian systems; Verma modules; Coadjoint orbits

MSC 2010: 17B69; 17B08; 70G60; 82C23

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

(ϕ,a)⋅(ψ,b)=(ϕ∘ψ,a.b∘ϕ−1).

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

[Lm,Ln]=(n−m)Ln+m.

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

[gn,hm]=[g,h]n+m.

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

[(u,a),(v,b)]=([.,∂t.].u⊗v,va′−ub′+[a,b]),

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

ωVir((u,a),(v,b))=J(u‴v),ωK−M((u,a),(v,b))=J(〈a′,b〉).

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

ωsp((u,a),(v,b))=J(ub″−va″).

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

[(u,a,ϕ,α,γ),(v,b,ξ,β,δ)]=(uv′−u′v,[a,b]−ub′+va′,J(u‴v),J(〈a′,b〉),J(ub″−va″)).

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_{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

UωK′,ωH′(K⋉H)≃UωK−α(K)⊗UωH(H).

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

f(x+ϵa)=f(x)+ϵ〈∇f(x),a〉+o(ϵ),

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

{f,g}(x)=〈x,[∇f(x),∇g(x)]〉.

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

{f,g}e(x)=〈e,[∇f(x),∇g(x)]〉.

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

〈(u1,a1,β1,ξ1),(u2,a2,β2,ξ2)〉=∫S1u1u2+∫S1〈a1,a2〉+ξ1ξ2+β1β2.

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:

Theorem 2.1

𝓘 and 𝓘̃ are Poisson maps.

Proof

For any regular function f(u, a, ξ, β) from Vect(S1)~⊕LG~ to ℝ let us define a regular function f̂ from 𝓢𝓤(𝓖)′ to ℝ by f̂ (u, a, ξ, β) = f(u′, a, ξ, β). For f(u, a, ξ, β) a function on 𝓢𝓤(𝓖) or (Vect(S1)~⊕LG~), let us denote by f_u the function of the variables a and β that we get when we fix u and ξ. Let us denote f_a the function of the variables u and ξ that we get when we fix a and β. With the previous notations, one has for β ≠ 0 for the bracket {., .}^U = {., .}^{𝓢𝓤(𝓖)}

{f,g^}U(u,a,ξ,β)=[{fu,gu}U+{fu,ga}U+{fa,gu}U+{fa,ga}U](u,a,ξ,β),

and for the bracket {.,.}VLG={.,.}Vect(S1)~⊕LG~ we have

{f,g}VLG(u,a,ξ,β)={fu,gu}VLG+{fa,ga}VLG.

Then the map π₁ from 𝓢𝓤(𝓖) onto Vect(S1)~ which sends (u, a, ξ, β) onto (u′, ξ) is a Poisson morphism. The map π₂ from 𝓢𝓤(𝓖) onto LG~ which sends (u, a, ξ, β) to ( a, β) is a Poisson morphism. For any regular function f on Vect(S1)~ and any regular function G on LG~ we have

{π1∗f,π2∗g}U=0.

Indeed, for i = 1, 2, (δa−aβδu)fi(u^,ξ)=0. We have:

{f1(u^,ξ),f2(u^,ξ)}ξ,βU(u,a,ξ,β)=J([ξ(δf1,u(u^),ξ)xxxδf2,u(u^,ξ)+2(δf1,u(u^,ξ))xδf2,u(u^,ξ)u+δf1,u(u^,ξ)uxδf2,u(u^,ξ)−β−1(δf1,u(u^,ξ)x)∥a∥2δf2,u(u^,ξ)−〈(δf1,u(u^)a/β,ξ)x,δf2,u(u^,ξ)〉+β−1〈(δf1,u(u^)a,ξ)x,δf2,u(u^,ξ)a〉]).

This gives

{f1(u^,ξ),f2(u^,ξ)}ξ,βU(u,a)=J(ξ[(δf1,u(u^,ξ))xxxδf2,u(u^,ξ)+2(δf1,u(u^,ξ))xδf2,u(u^,ξ)(u^,ξ)+δf1,u(u^,ξ)(u^)xδf2,u(u^,ξ)]),

and

{f1(u^,ξ),f2(u^,ξ)}ξ,βU(u,a)={f1,f2}ξVir(u^,ξ).

Let g_i(a, β), i = 1, 2 be two regular functions on the affine Kac–Moody algebra. One notes that δg_1,u = δg_2,u = 0. Therefore,

{g1,g2}ξ,βU(u,a)=βJ(〈dx(δg1,a(a,β)),δg2,a(a,β)〉+〈[a,δg1,a(a,β)],δg2,a(a,β)〉).

Then,

{g1,g2}U(u,a,ξ,β)={f,g}LG~(a,β).

We have:

{f(u^,ξ),g(a,β)}U=J(〈(δfu(u^)xa,ξ),δg,a(a,β)〉−βdx(δfu(u^,ξ)a),δga(a,β)〉+[a,δfua],δga〉).

The sum of the first two terms is equal to 0. The last term is 𝓙(δf_u〈[a, a], δg_a〉), and is equal to zero. One can proceed similarly for 𝓘̃. □

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.

Proposition 3.1

The coadjoint actions of the groupsDiff(S¹) ⋉ LGandDiff(S¹) ⋉ LR+∗are given by

Ad∗(ϕ,g)−1(u,a,ξ,β)=((u∘ϕ)ϕ′2+ξS(ϕ)+〈g−1g′,a〉ϕ′2+12β∥g−1g′∥2,ϕ′Ad(g−1)a∘ϕ+βg−1g′,ξ,β),((u∘ϕ)ϕ′2+ξS(ϕ)+〈g′g−1,a〉ϕ′2+12β(g′g−1)2+γg″g−1,ϕ′Ad(g−1)a∘ϕ+βg−1g′−γg″g−1,ξ,β,γ).

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

Theorem 3.2

A coadjoint orbit of 𝓢𝓤 (𝓖) is mapped by 𝓘 to a coadjoint orbit ofVect(S1)~⊕LG~to a coadjoint orbits ofVect(S1)~.

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)~.

Proof

For any ϕ ϵ Diff(S¹), there exists h ϵ LG such that

hah−1+β∂h(x)∂x.h−1=a∘ϕ.ϕ′.

By direct computation we check that

I(Ad∗(ϕ,g)(u,a,ξ,β)=(Ad∗(ϕ,g.h)I(u,a,ξ,β).

This implies Theorem 3.2. □

Proposition 3.3

The map 𝓘̃ sends the coadjoint orbits ofSU(G)~to coadjoint orbits ofVect(S1)~⊕LG~.

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:

Proposition 3.4

If the elements (u₁,a₁, ξ₁, β₁, γ₁) and (u₁,a₁, ξ₂, β₂γ₂) are in the same coadjoint orbit thenγ₁ = γ₂, (a12+γ1a1′,γ1)and(a22+γ2a2′,γ2)are in the same coadjoint orbit of the Virasoro Lie algebra.

Proof

We have: Ad(ϕ,g)(a12+γ1a1′)=(a12+γ1a1′)∘ϕ+γ1S(ϕ). □

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

Proposition 3.5

Let 𝓒_Vir, 𝓒_K−M 𝓒_𝓐be Casimir functions for Virasoro, affine Kac–Moody, and the Heisenberg Lie algebras 𝓐 correspondingly. Let 𝓢_P𝓤(𝓖), SPU(R)~be Poisson submanifolds of 𝓢𝓤(𝓖) andSU(R)~defined byξ = 0. Then the functions 𝓒_Vir(u′, ξ), c(u, a, β, ξ) = 𝓒_K−M(a, β), and∫S1|u′|1/2,are Casimir functions onSU(G)~′. In particular, the functions c_𝓐(u, a, β, ξ) = 𝓒_𝓐(a, β), CVir(u′−γ2βa′,ξ),and∫S1|u′−γ2βa′|1/2,are Casimir functions onSU(R)~′.

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

Proposition 4.1

The functions{ϕ1(A(u+Bdadx+C))|λϵR}commute pairwise for the Sugawara{.,.}Sug′ande-braket {., .}_ewithe = (1, 0, 0, 2, 0), andA=(ξ−γβ−2λ)−2,B=−γβ−2λ,C=−∥a∥22β−4λ−λ. □

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.

Lemma 4.2

For any (i, j)ϵ I²and any (λ, μ)ϵ ℝ²we have {ϕ_i(x_λ), ϕ_j(x_μ)} = {ϕ_i(x_λ), ϕ_j(x_μ)}_e = 0.

Lemma 4.3

Supposeϕ_i(x_λ) can be expanded in terms of inverse powers of λ with some extra functionf(λ), and modesF_i,k(x), i.e.,

ϕi(xλ)=f(λ)∑k∈Rλ−kFi,k(x),

then {F_i,k+1, f}_e = {F_i,k, f}₀. We can chooseeso that the HamiltonianH(x)=12〈x,x〉commute with these functions.

Lemma 4.4

If an elemente ϵ 𝓖 satisfies two conditions: (i) ad^*(e)e = 0; (ii) for anyu ϵ 𝓖, ad^*(u) ebelongs to the tangent space to the coadjoint orbit ofu (i.e., for anyu ϵ 𝓖 there exists v ϵ 𝓖 such thatad^*(u)e = ad^*(v)u). then the functionsϕ(a − λ e) commute with the Hamiltonian of the geodesicsH(a)=12∥a∥2with respect to the brackets {., .}₀and {., .}_e.

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:

∑k=−≤nuk.∑k≤mvk=∑k∈Zwk,

where w_k = ∑_iϵℤu_i.v_{k_i} 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.

5.1 Decomposition of the enveloping algebra of a semi-direct product

In some very particular cases, the modified generalized enveloping algebra of a semi-direct product 𝓚 ⋉ 𝓗 of two Lie algebras is isomorphic to the tensor product of some modified generalized enveloping algebras of 𝓚 and of 𝓗. Let 𝓗̃ be the central extension of 𝓗 with the two-cocycle ω_𝓗. Denote by · the action of the Lie algebra 𝓚 on the Lie algebra 𝓗̃. Let us introduce the semi-direct product 𝓚 ⋉ 𝓗̃ which is a central extension of 𝓚 ⋉ 𝓗 by a two-cocycle ωH′ with

ωH′((0,h1),(0,h2))=ωH(h1,h2).

A two-cocycle ω_𝓚 on 𝓚 defines also a two-cocycle ωK′ by

ωK′((g1,h1),(g2,h2))=ωK(g1,g2),

of 𝓚 ⋉ 𝓗. Let I be the natural inclusion of 𝓗̃ 𝓤_{ω_𝓗} (𝓗) and J be the natural inclusion of 𝓗̃ into UωK′,ωH′(K⋉H).

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 (g₁, g₂) in 𝓚²

F([g1,g2])=[F(g1),F(g2)]+α(g1,g2)1,

and the map F satisfies the compatibility condition, i.e., for any g ϵ 𝓚 and h ϵ 𝓗̃ with the anti–commutator [F(g), I(h)] = I(g · h), of the algebra 𝓤_{ω_𝓗} (𝓗).

Theorem 5.1

If the action of 𝓚 is realizable in 𝓤_{ω_𝓗} (𝓗) then

UωK′,ωH′(K⋉H)≃UωK−α(K)⊗UωH(H).

Proof

Let 𝓤_g = {ĝ | g ϵ 𝓚} with be the unitary subalgebra of UωK′,ωH′(K⋉H) generated by the elements ĝ = g – F(g), and 𝓤_j = {j(h), h ϵ 𝓗̃} be the unitary subalgebra of UωK′,ωH′(K⋉H). For any (g, h) this implies that the generators of 𝓤_g and 𝓤_j commute, i.e., [ĝ, j(h)] = 0. 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 ϵ 𝓚} of this algebra satisfy the relations of the generators of 𝓤_{ω_𝓚–α}(𝓚):

[g^1,g^2]=[g1,g2]+ωK(g1,g2)1+[F(g1),F(g2)]−[F(g1),g2]−[g1,F(g2)].

Since F(g₁) is an element of 𝓤_j and since the algebras 𝓤_g and 𝓤_j commute [F(g₁),g₂] = [F(g₁), F(g₂)] and [g₁, F(g₂)] = [F(g₁), F(g₂)]. Therefore:

[g^1,g^2]=[g1,g2]+ωK(g1,g2)1−[F(g1),F(g2)],

and finally

[g^1,g^2]=[g1,g2]−F([g1,g2])+(ωK(g1,g2)−α(g1,g2))1.

The subalgebra 𝓤_j is obviously isomorphic to 𝓤_{ω_𝓗} (𝓗). The generalized modified enveloping algebra UωK′+ωH′(K⋉H) is therefore isomorphic to the tensor product over ℂ of 𝓤_{ω_{𝓚 –α}} (𝓚) with 𝓤_{ω_𝓗}(𝓗)

□

5.2 The case of 𝓢𝓤_ℂ(𝓖)

Let 𝓖 be a simple complex Lie algebra and C_φ its dual Coxeter number. Introduce the {K₁,…,K_n} a basis of 𝓖, and the dual basis {K1∗,…,Kn∗} with respect to the Killing form 〈., .〉. We apply Theorem 5.1 for 𝓚 = Vect(S¹), 𝓗 = 𝓛𝓖, ω_𝓚 = ξω_Vir, and ω_𝓗 = βω_K–M. In this case, ωH′=βωK−M. For η = β + C_φ ≠ 0, the Sugawara construction, delivers a map F : Vect(S¹)_ℂ → 𝓤_{ω_𝓖}(𝓛𝓖_ℂ) defined by

(β+η)F(Ln)=K⋅K∗,

where

K⋅K∗=∑i∈Zj=1,…,n:(Kj)i(Kj∗)n−i:,

(here dots denote the normal ordering), i.e., the action of Vect(S¹) is realizable in 𝓤_{βω_K–M} (𝓛𝓖), with α = βω_Vir/12η. Thus we obtain

Proposition 5.2

Ifη ≠ 0, then 𝓤_{ξω_Vir}, βω_K–M (𝓢𝓤_ℂ𝓖) ≃ 𝓤_{βω_K–M} (Vect(S¹)_ℂ}_{(ξ – α)}) ⊗ 𝓤(𝓛𝓖).

The Lie algebra Vect_ℂ(S¹) acts on the Heisenberg algebra by

Ln.am=man+m+δn,−mm2cK−M.

In this case, on has ωH′=βωH+γωsp. The map F : Vect(S¹)_ℂ → 𝓢𝓤_ℂ(ℂ) defined by

βF(Ln)=12∑i∈Z:aian−i:+γan,

for a cocycle α^ = (α + γ²β_-1)ω_Vir. For 𝓢𝓤_ℂ(𝓒) we obtain

Proposition 5.3

Forβ ≠ 0, we have

UξωVir,βωK−M,γωsp(SUC(C)~)≃UθωVir(Vect(S1)C)⊗UωK−M(LG),

with θ = ξ – γ²/β – 1/12.

5.3 Representations of 𝓢𝓤(𝓖)

Proposition 5.4

A positive energy representationVof 𝓢𝓤_ℂ(𝓖) with non-vanishingβId-action of the cocylec_K–Mbrings about 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). Let 𝔥 be a Cartan algebra of 𝓖 with a basis {h₁, …, h_k}. The Lie subalgebra 𝔨 of 𝓢𝓤_ℂ(𝓖) is generated by the elements {c_Vir, c_K–M, u₀, (h₁)₀,…,(h_k)₀}. A Verma module V_λ(𝓢𝓤_ℂ(𝓖)) of 𝓢𝓤_ℂ(𝓖) is associated to any linear form λ ϵ 𝔥^*.

Verma modules VνVir,VμK−M, are associated to linear forms V, μ over the spaces generated by c_Vir and u₀, c_K–M and {(h₁)₀,…,(h_k)₀} correspondingly. For any λ ϵ 𝔨^*, the Verma module V_λ(𝓢𝓤_ℂ(𝓖)) is a positive energy representation. Thus, V_λ(𝓢𝓤_ℂ(𝓖)) is Virasoro and affine Kac–Moody algebra module. The generator e of V_λ(𝓢𝓤_ℂ(𝓖)) brings about a Verma module VνVir for Virasoro algebra. It generates also a Verma module VνVir for the affine Kac–Moody algebra. The linear formv satisfies v(u₀)e = λ(u₀ – F(u₀))e, i.e.,

(u0−(β+η)−1K⋅K∗e=ν(u0)e.

Suppose the action of a Casimir element of 𝓖 is given by acts by D(λ)Id for D(λ) ϵ ℂ. We then have

(u0−(β+η)−1K⋅K∗.e=(u0−(β+η)−1∑j=1…n:(Kj)0(Kj∗)0:).e,

(λ(u0)−D(λ)2η)e. This implies ν(u0)=λ(u0)−D(λ)2η. The other values of μ and v can be computed by the same method.

Proposition 5.5

Let λ be a linear form over {𝔥} uwith non-vanishing λ (c_K–M). Then

Vλ(SUC(G))≃VνVir⊗VCK−Mμ,

whereμ (e_i) = λ (e_i), i = 1, …, n, definesμ, μ (c_K–M) = λ (c_K−M), andv(c_Vir) = λ(c_Vir) – β12ηdefinesv, v(u₀) = λ(u₀) – D(λ)2η.

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Received: 2016-11-30

Accepted: 2018-01-04

Published Online: 2018-01-31

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2018-0002

Keywords for this article

Affine Kac–Moody Lie algebras; Bi-Hamiltonian systems; Verma modules; Coadjoint orbits

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

Article

Abstract

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

2 The Kirillov-Kostant structure of 𝓢𝓤(𝓖)

Theorem 2.1

Proof

3 Coadjoint orbits Casimir functions and for 𝓢𝓤(𝓖)

Proposition 3.1

Theorem 3.2

Proof

Proposition 3.3

Proposition 3.4

Proof

Proposition 3.5

4 Bi-hamiltonian dispersive water waves systems associated to 𝓢𝓤(𝓖)

Proposition 4.1

Lemma 4.2

Lemma 4.3

Lemma 4.4

5 The universal enveloping algebra of 𝓢𝓤(𝓖)

5.1 Decomposition of the enveloping algebra of a semi-direct product

Theorem 5.1

Proof

5.2 The case of 𝓢𝓤ℂ(𝓖)

Proposition 5.2

Proposition 5.3

5.3 Representations of 𝓢𝓤(𝓖)

Proposition 5.4

Proposition 5.5

References

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5.2 The case of 𝓢𝓤_ℂ(𝓖)