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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# 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 Online: 2018-01-31 | DOI: https://doi.org/10.1515/math-2018-0002

## 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.

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) is the Lie algebra over ℂ generated by the elements Ln, n ϵ ℤ with the relations

$[Lm,Ln]=(n−m)Ln+m.$

We denote by 𝓛𝓖 the Lie algebra over ℂ generated by the elements gn, n ϵ ℤ, g ϵ 𝓖 where (λg + μ h)n is identified with λ gn + μ hn 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) = S1 u. 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 cVir and cKM 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 $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}}\left(\mathbb{R}\right)\end{array}$ 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 $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}}\left(\mathbb{R}\right)\end{array}$ with u ϵ C(S1, ℝ), a ϵ C(S1, 𝓖), and (χ, α, γ) ϵ ℝ3. The Lie bracket of $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}}\left(\mathbb{R}\right)\end{array}$ 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 ($\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$)′ the subset of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ composed of elements (u, a, ξ, β) with β ≠ 0. Then introduce two new maps 𝓘(u, a, ξ, β) from 𝓢𝓤(𝓖)′ to ($\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$)′, and 𝓘̃(u, a, ξ, β, γ) from 𝓢𝓤(𝓖) to $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathbb{L}\mathcal{R}}\end{array}$. 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) ⋉ $\begin{array}{}L{\mathbb{R}}_{+}^{\ast }\end{array}$. 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 $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ to a coadjoint orbits of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\end{array}$. We prove that map 𝓘̃ sends the coadjoint orbits of $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathcal{G}\right)}\end{array}$ to coadjoint orbits of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$. Previously, we determined Casemir functions on $\begin{array}{l}\begin{array}{}{\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathcal{G}\right)}}^{\prime }\end{array}\end{array}$ and $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathbb{R}\right)}\end{array}$. We then prove new propositions concerning the exlicit form of Casemir functions on $\begin{array}{l}\begin{array}{}{\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathcal{G}\right)}}^{\prime }\end{array}\end{array}$, and in particular on on $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathbb{R}\right)}\end{array}$′. 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 𝓤ω𝓗(𝓗). Then we show (Theorem 5.1 that the realizability of the action of 𝓚 in 𝓤ω𝓗 (𝓗) leads to the isomorphism $UωK′,ωH′(K⋉H)≃UωK−α(K)⊗UωH(H).$

In subsection 5.2 the case of 𝓢𝓤(𝓖) is considered. In subsection 5.3 we discuss representations of 𝓢𝓤(𝓖). We prove that positive energy representation V of 𝓢𝓤(𝓖) with non-vanishing βId-action of the cocyle cKM delivers a pair of commuting representations of Virasoro and affine Kac–Moody Lie algebras. This proposition determines whether a 𝓢𝓤(𝓖) Verma module is a sub-module of another Verma module of 𝓢𝓤(𝓖). We also prove a proposition regarding a linear form over 𝔥 with non-vanishing λ (cKM). 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 $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ 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 $\begin{array}{}{u}^{\mathrm{\prime }}=u-\frac{\parallel a{\parallel }^{2}}{2\beta }\end{array}$ We denote by ($\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$)′ the subset of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ 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 ($\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$)′. Then for non-vanishing β, let us introduce another new map $\begin{array}{}\stackrel{~}{\mathcal{I}}\left(u,a,\xi ,\beta ,\gamma \right)=\left({u}^{\mathrm{\prime }}-\frac{\gamma }{\beta }{a}^{\prime },a,\xi -\frac{{\gamma }^{2}}{\beta },\beta \right)\end{array}$ from 𝓢𝓤(𝓖) to $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathbb{L}\mathcal{R}}\end{array}$. 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 $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ to ℝ let us define a regular function from 𝓢𝓤(𝓖)′ to ℝ by (u, a, ξ, β) = f(u′, a, ξ, β). For f(u, a, ξ, β) a function on 𝓢𝓤(𝓖) or ($\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$), let us denote by fu the function of the variables a and β that we get when we fix u and ξ. Let us denote fa 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 $\begin{array}{}\left\{.,.{\right\}}^{V\mathcal{L}\mathcal{G}}=\left\{.,.{\right\}}^{\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}}\end{array}$ we have

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

Then the map π1 from 𝓢𝓤(𝓖) onto $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\end{array}$ which sends (u, a, ξ, β) onto (u′, ξ) is a Poisson morphism. The map π2 from 𝓢𝓤(𝓖) onto $\begin{array}{}\stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ which sends (u, a, ξ, β) to ( a, β) is a Poisson morphism. For any regular function f on $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\end{array}$ and any regular function G on $\begin{array}{}\stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ we have

${π1∗f,π2∗g}U=0.$

Indeed, for i = 1, 2, $\begin{array}{}\left({\delta }_{a}-\frac{a}{\beta }{\delta }_{u}\right){f}_{i}\left(\stackrel{^}{u},\xi \right)=0.\end{array}$ 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 gi(a, β), i = 1, 2 be two regular functions on the affine Kac–Moody algebra. One notes that δ g1,u = δ g2,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(δfu〈[a, a], δga〉), 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 groups Diff(S1) ⋉ LG and Diff(S1) ⋉ $\begin{array}{}L{\mathbb{R}}_{+}^{\ast }\end{array}$ 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 of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$ to a coadjoint orbits of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\end{array}$.

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 $\begin{array}{}\stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$, $\begin{array}{}\left({u}_{1}-\frac{\parallel {a}_{1}\parallel }{2{\beta }_{1}}\right),{\xi }_{1}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left({u}_{2}-\frac{\parallel {a}_{2}\parallel }{2{\beta }_{2}}\right),{\xi }_{2}\right)\end{array}$ are elements of the same coadjoint orbit of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\end{array}$.

#### 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 of $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathcal{G}\right)}\end{array}$ to coadjoint orbits of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\oplus \stackrel{~}{\mathcal{L}\mathcal{G}}\end{array}$.

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 $\begin{array}{}\stackrel{~}{\mathcal{L}\mathcal{G}},\left({u}_{1}-\frac{{a}_{1}^{2}}{2\beta },{\xi }_{1}-\frac{{\gamma }_{1}^{2}}{{\beta }_{1}}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left({u}_{2}-\frac{{a}_{2}^{2}}{2\beta },{\xi }_{2}-\frac{{\gamma }_{2}^{2}}{{\beta }_{2}}\right)\end{array}$ are elements of the same coadjoint orbit of $\begin{array}{}\stackrel{~}{Vect\left({S}^{1}\right)}\end{array}$. 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 then γ1 = γ2, $\begin{array}{}\left({a}_{1}^{2}+{\gamma }_{1}{a}_{1}^{\prime },{\gamma }_{1}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left({a}_{2}^{2}+{\gamma }_{2}{a}_{2}^{\prime },{\gamma }_{2}\right)\end{array}$ are in the same coadjoint orbit of the Virasoro Lie algebra.

#### Proof

We have: $\begin{array}{}Ad\left(\varphi ,g\right)\left({a}_{1}^{2}+{\gamma }_{1}{a}_{1}^{\prime }\right)=\left({a}_{1}^{2}+{\gamma }_{1}{a}_{1}^{\prime }\right)\circ \varphi +{\gamma }_{1}S\left(\varphi \right).\end{array}$ □

Previously, we determined Casemir functions on $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathcal{G}\right)}\end{array}$′ and $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathbb{R}\right)}\end{array}$. We gave the following proposition:

#### Proposition 3.5

Let 𝓒Vir, 𝓒KM 𝓒𝓐 be Casimir functions for Virasoro, affine KacMoody, and the Heisenberg Lie algebras 𝓐 correspondingly. Let 𝓢P𝓤(𝓖), $\begin{array}{}\stackrel{~}{{\mathcal{S}}_{P}\mathcal{U}\left(\mathbb{R}\right)}\end{array}$ be Poisson submanifolds of 𝓢𝓤(𝓖) and $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathbb{R}\right)}\end{array}$ defined by ξ = 0. Then the functions 𝓒Vir(u′, ξ), c(u, a, β, ξ) = 𝓒KM(a, β), and $\begin{array}{}{\int }_{{S}^{1}}|{u}^{\mathrm{\prime }}{|}^{1/2},\end{array}$ are Casimir functions on $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathcal{G}\right)}\end{array}$′. In particular, the functions c𝓐(u, a, β, ξ) = 𝓒𝓐(a, β), $\begin{array}{}{\mathcal{C}}_{Vir}\left({u}^{\mathrm{\prime }}-\frac{{\gamma }^{2}}{\beta }{a}^{\prime },\xi \right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\int }_{{S}^{1}}|{u}^{\mathrm{\prime }}-\frac{{\gamma }^{2}}{\beta }{a}^{\prime }{|}^{1/2},\end{array}$ are Casimir functions on $\begin{array}{}\stackrel{~}{\mathcal{S}\mathcal{U}\left(\mathbb{R}\right)}\end{array}$′.

## 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 $\begin{array}{}\left\{{\varphi }_{1}\left(A\left(u+B\frac{da}{dx}+C\right)\right)|\lambda ϵ\mathbb{R}\right\}\end{array}$ commute pairwise for the Sugawara $\begin{array}{}\left\{.,.{\right\}}_{Sug}^{\prime }\end{array}$ and e-braket {., .}e with e = (1, 0, 0, 2, 0), and $\begin{array}{}A=\left(\xi -\frac{\gamma }{\beta -2\lambda }{\right)}^{-2},B=-\frac{\gamma }{\beta -2\lambda },C=-\frac{\parallel a{\parallel }^{2}}{2\beta -4\lambda }-\lambda .\end{array}$ □

The function $\begin{array}{}\lambda ↦{\varphi }_{1}\left(A\left(u+B\frac{da}{dx}+C\right)\right)\end{array}$ has an asymptotic development. The coefficients of this development form a hierarchy. The first term of this development is S1 u, and the second one is S1 (u2 + γ u + ∥ a2). A linear combination of these two terms gives the Hamiltonian of equations H(u, a) = S1 (u2 + ∥ a2).

Let {ϕi, i ϵ I} be a set of Casimir functions and e ϵ 𝓖. Define xχ = xχ e, for some χ ϵ ℝ.

#### Lemma 4.2

For any (i, jI2 and any (λ, μ)ϵ ℝ2 we have {ϕi(xλ), ϕj(xμ)} = {ϕi(xλ), ϕj(xμ)}e = 0.

#### Lemma 4.3

Suppose ϕi(xλ) can be expanded in terms of inverse powers of λ with some extra function f(λ), and modes Fi,k(x), i.e.,

$ϕi(xλ)=f(λ)∑k∈Rλ−kFi,k(x),$

then {Fi,k+1, f}e = {Fi,k, f}0. We can choose e so that the Hamiltonian $\begin{array}{}H\left(x\right)=\frac{1}{2}〈x,x〉\end{array}$ commute with these functions.

#### Lemma 4.4

If an element e ϵ 𝓖 satisfies two conditions: (i) ad*(e)e = 0; (ii) for any u ϵ 𝓖, ad*(u) e belongs to the tangent space to the coadjoint orbit of u (i.e., for any u ϵ 𝓖 there exists v ϵ 𝓖 such that ad*(u)e = ad*(v)u). then the functions ϕ(a − λ e) commute with the Hamiltonian of the geodesics $\begin{array}{}H\left(a\right)=\frac{1}{2}\parallel a{\parallel }^{2}\end{array}$ with respect to the brackets {., .}0 and {., .}e.

## 5 The universal enveloping algebra of 𝓢𝓤(𝓖)

When 𝓗 = ∑kϵℤ 𝓗k 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, …, hn ϵ 𝓤𝓗 of homogeneous elements is defined to be the sum of the weights of the elements hi, i = 1, …, n. The universal enveloping algebra 𝓤𝓗 admits a filtration $\begin{array}{}\mathcal{U}\mathcal{H}={\cup }_{i=0}^{\mathrm{\infty }}{F}_{k}\end{array}$ where Fk 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 ∑kn uk where uk is an element of weight k of 𝓤𝓗. The product of two such elements is defined by:

$∑k=−≤nuk.∑k≤mvk=∑k∈Zwk,$

where wk = ∑iϵℤ ui.vki which is a finite sum. Let ω1, …, ωn be two-cocycles on the Lie algebra 𝓗, let 𝓗̃ be the central extension associated with and let e1,…,en be the central elements associated with these cocycles.

The modified generalized enveloping algebra $\begin{array}{}{\mathcal{U}}_{{\omega }_{1},\dots ,{\omega }_{n}}^{\mathcal{H}}\end{array}$ is defined to be the quotient of the generalized enveloping algebra of 𝓗̃ by the ideal generated by the elements {e1 – 1,…,en – 1}. We denote again by 1 the neutral element of

$\begin{array}{}{\mathcal{U}}_{\lambda }^{\mathcal{G}}.\end{array}$ The algebra $\begin{array}{}{\mathcal{U}}_{{\omega }_{1},\dots ,{\omega }_{n}}^{\mathcal{H}}\end{array}$ is by construction a graded algebra and a filtered algebra. We denote by Fn, 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 𝓗̃n we have h.v = 0. Such modules are called representations of positive energy, and ei acts on V by λi Id. Then V is a module over

$\begin{array}{}{\mathcal{U}}_{{\omega }_{1},\dots ,{\omega }_{n}}^{\mathcal{H}}\end{array}$ Such modules are named modules of positive energy. The anticommutator provides a structure of Lie algebra on $\begin{array}{}{\mathcal{U}}_{{\omega }_{1},\dots ,{\omega }_{n}}^{\mathcal{H}}\end{array}$ For this bracket F1 is a Lie sub-algebra isomorphic to the central extension of 𝓗 by the cocycle

$\begin{array}{}\omega =\sum _{i=1}^{n}{\omega }_{i}.\end{array}$ We denote by i be the natural inclusion of 𝓗̃ into $\begin{array}{}{\mathcal{U}}_{\omega }^{\mathcal{G}}\end{array}$ 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 ω𝓗. 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 $\begin{array}{}{\omega }_{\mathcal{H}}^{\prime }\end{array}$ with

$ωH′((0,h1),(0,h2))=ωH(h1,h2).$

A two-cocycle ω𝓚 on 𝓚 defines also a two-cocycle $\begin{array}{}{\omega }_{\mathcal{K}}^{\prime }\end{array}$ by

$ωK′((g1,h1),(g2,h2))=ωK(g1,g2),$

of 𝓚 ⋉ 𝓗. Let I be the natural inclusion of 𝓗̃ 𝓤ω𝓗 (𝓗) and J be the natural inclusion of 𝓗̃ into $\begin{array}{}{\mathcal{U}}_{{\omega }_{\mathcal{K}}^{\prime },{\omega }_{\mathcal{H}}^{\prime }}\left(\mathcal{K}⋉\mathcal{H}\right).\end{array}$

We call the action of 𝓚 on 𝓗 realizable in 𝓤ω𝓗 (𝓗) when there exists a map F : 𝓚 → 𝓤ω𝓗 (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 𝓤ω𝓗 (𝓗).

#### Theorem 5.1

If the action of 𝓚 is realizable in 𝓤ω𝓗 (𝓗) then

$UωK′,ωH′(K⋉H)≃UωK−α(K)⊗UωH(H).$

#### Proof

Let 𝓤g = {ĝ | g ϵ 𝓚} with be the unitary subalgebra of $\begin{array}{}{\mathcal{U}}_{{\omega }_{\mathcal{K}}^{\prime },{\omega }_{\mathcal{H}}^{\prime }}\left(\mathcal{K}⋉\mathcal{H}\right)\end{array}$ generated by the elements ĝ = gF(g), and 𝓤j = {j(h), h ϵ 𝓗̃} be the unitary subalgebra of $\begin{array}{}{\mathcal{U}}_{{\omega }_{\mathcal{K}}^{\prime },{\omega }_{\mathcal{H}}^{\prime }}\left(\mathcal{K}⋉\mathcal{H}\right).\end{array}$ For any (g, h) this implies that the generators of 𝓤g and 𝓤j commute, i.e., [ĝ, j(h)] = 0. The subalgebras 𝓤g and 𝓤j therefore commute. The subalgebra 𝓤g is isomorphic to 𝓤ω𝓚–α(𝓚). Let us check that the generators ĝ|g ϵ 𝓚} of this algebra satisfy the relations of the generators of 𝓤ω𝓚–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 𝓤j and since the algebras 𝓤g and 𝓤j 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 𝓤j is obviously isomorphic to 𝓤ω𝓗 (𝓗). The generalized modified enveloping algebra $\begin{array}{}{\mathcal{U}}_{{\omega }_{\mathcal{K}}^{\prime }+{\omega }_{\mathcal{H}}^{\prime }}\left(\mathcal{K}⋉\mathcal{H}\right)\end{array}$ is therefore isomorphic to the tensor product over ℂ of 𝓤ω𝓚 –α (𝓚) with 𝓤ω𝓗(𝓗)

□

## 5.2 The case of 𝓢𝓤ℂ(𝓖)

Let 𝓖 be a simple complex Lie algebra and Cφ its dual Coxeter number. Introduce the {K1,…,Kn} a basis of 𝓖, and the dual basis $\begin{array}{}\left\{{K}_{1}^{\ast },\dots ,{K}_{n}^{\ast }\right\}\end{array}$ with respect to the Killing form 〈., .〉. We apply Theorem 5.1 for 𝓚 = Vect(S1), 𝓗 = 𝓛𝓖, ω𝓚 = ξωVir, and ω𝓗 = βωKM. In this case, $\begin{array}{}{\omega }_{\mathcal{H}}^{\prime }=\beta {\omega }_{K-M}.\end{array}$ For η = β + Cφ ≠ 0, the Sugawara construction, delivers a map F : Vect(S1) → 𝓤ω𝓖(𝓛𝓖) 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 𝓤βωKM (𝓛𝓖), with α = βωVir/12η. Thus we obtain

#### Proposition 5.2

If η ≠ 0, then 𝓤ξωVir, βωKM (𝓢𝓤𝓖) ≃ 𝓤βωKM (Vect(S1)}(ξα)) ⊗ 𝓤(𝓛𝓖).

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

$Ln.am=man+m+δn,−mm2cK−M.$

In this case, on has $\begin{array}{}{\omega }_{\mathcal{H}}^{\prime }=\beta {\omega }_{\mathcal{H}}+\gamma {\omega }_{sp}.\end{array}$ The map F : Vect(S1) → 𝓢𝓤(ℂ) defined by

$βF(Ln)=12∑i∈Z:aian−i:+γan,$

for a cocycle α^ = (α + γ2β-1)ωVir. For 𝓢𝓤(𝓒) we obtain

#### Proposition 5.3

For β ≠ 0, we have

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

with θ = ξγ2/β – 1/12.

## 5.3 Representations of 𝓢𝓤(𝓖)

#### Proposition 5.4

A positive energy representation V of 𝓢𝓤(𝓖) with non-vanishing βId-action of the cocyle cKM brings about a pair of commuting representations of Virasoro and affine KacMoody Lie algebras.

This proposition determines whether a 𝓢𝓤(𝓖) Verma module is a sub-module of another Verma module of 𝓢𝓤(𝓖). Let 𝔥 be a Cartan algebra of 𝓖 with a basis {h1, …, hk}. The Lie subalgebra 𝔨 of 𝓢𝓤(𝓖) is generated by the elements {cVir, cKM, u0, (h1)0,…,(hk)0}. A Verma module Vλ(𝓢𝓤(𝓖)) of 𝓢𝓤(𝓖) is associated to any linear form λ ϵ 𝔥*.

Verma modules $\begin{array}{}{V}_{\nu }^{Vir},{V}_{\mu }^{K-M}\end{array}$, are associated to linear forms V, μ over the spaces generated by cVir and u0, cKM and {(h1)0,…,(hk)0} correspondingly. For any λ ϵ 𝔨*, the Verma module Vλ(𝓢𝓤(𝓖)) is a positive energy representation. Thus, Vλ(𝓢𝓤(𝓖)) is Virasoro and affine Kac–Moody algebra module. The generator e of Vλ(𝓢𝓤(𝓖)) brings about a Verma module $\begin{array}{}{V}_{\nu }^{Vir}\end{array}$ for Virasoro algebra. It generates also a Verma module $\begin{array}{}{V}_{\nu }^{Vir}\end{array}$ for the affine Kac–Moody algebra. The linear formv satisfies v(u0)e = λ(u0F(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

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

$\begin{array}{}\left(\lambda \left({u}_{0}\right)-\frac{D\left(\lambda \right)}{2\eta }\right)e.\end{array}$ This implies $\begin{array}{}\nu \left({u}_{0}\right)=\lambda \left({u}_{0}\right)-\frac{D\left(\lambda \right)}{2\eta }.\end{array}$ 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 (cKM). Then

$Vλ(SUC(G))≃VνVir⊗VCK−Mμ,$

where μ (ei) = λ (ei), i = 1, …, n, defines μ, μ (cKM) = λ (cKM), and v(cVir) = λ(cVir) – $\begin{array}{}\frac{\beta }{12\eta }\end{array}$ defines v, v(u0) = λ(u0) – $\begin{array}{}\frac{D\left(\lambda \right)}{2\eta }.\end{array}$

## References

• [1]

Zuevsky A., Hamiltonian structures on coadjoint orbits of semidirect product G = Diff+(S1) ⋉ C(S1, ℝ). Czechoslovak J. Phys., 2004, 54, no. 11, 1399-1406

• [2]

Arnold V.I., Mathematical methods of classical mechanics, 1978, Springer Google Scholar

• [3]

Kirillov A., Infinite dimensional Lie groups; their orbits, invariants and representations. The geometry of moments, Lecture Notes in Math., 1982, Vol. 970, 101-123

• [4]

Segal G., The geometry of the KdV equation, Int. J. of Modern Phys., 1991, Vol. 6, No. 16, 2859-2869

• [5]

Witten E., Coadjoint orbits of the Virasoro Group, Comm. in Math. Phys., 1998, v. 114, 1-53 Google Scholar

• [6]

Ovsienko V., Roger C., Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor densities on S1, 1998, Indag. Math. (N.S.) 9, no. 2, 277-288

• [7]

Enriquez B., Khoroshkin S., Radul A., Rosly A., Rubtzov V. Poisson-Lie aspects of classical W-algebras. The interplay between differential geometry and differential equations, 1995, Amer. Math. Soc. Transl. Ser. 2, 167, Adv. Math. Sci., 24, Amer. Math. Soc., Providence, RI, 37-59 Google Scholar

• [8]

Das A., Integrable models, 1989, World Scientific Publishing Google Scholar

• [9]

Harnad J., Kupershmidt B.A., Sympletic geometries on T*, Hamiltonian group actions and integrable systems, 1995, J. Geom. Phys., 16, no. 2, 168-206 Google Scholar

• [10]

Kupershmidt B.A., Mathematics of dispersive water waves, Commun. Math. Phys., 1985, v. 99, No.1, 51-73

• [11]

Reiman A. G., Semenov-Tyan-Shanskii M. A., Hamiltonian structure of equations of Kadomtsev-Petviashvili type. (Russian. English summary) Differential geometry, Lie groups and mechanics, VI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 1984, 133, 212-227 Google Scholar

• [12]

Vishik S.M., Dolzhansky F.V., Analogues of the Euler-Poisson equations and magnetic hydrodynamics connected to Lie groups. Reports of the Academy of Science of the USSR, 1978, vol. 238, No. 5 (in Russian). Google Scholar

## About the article

Accepted: 2018-01-04

Published Online: 2018-01-31

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1–8, ISSN (Online) 2391-5455,

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