# 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.

## 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*(*S*^{1}) ⋉ 𝓛𝓖 reduce to the equivalent problems for Virasoro and affine Kac–Moody algebras (which are central extensions of *Vect*(*S*^{1}) and 𝓛𝓖 respectively). Let *G* be a Lie group and 𝓖 its Lie algebra. The group *Diff*(*S*^{1}) of diffeomorphisms of the circle is included in the group of automorphisms of the Loop group *LG* of smooth maps from *S*^{1} to *G*. For any pairs (*ϕ*, *ψ*) ϵ *Diff*(*S*^{1})^{2} and (*g*, *h*) ϵ *LG*^{2} the composition law of the group *Diff*(*S*^{1}) ⋉ 𝓛𝓖 is

The Lie algebra of *Diff*(*S*^{1}) ⋉ *LG* is the semi-direct product *Vect*(*S*^{1}) ⋉ 𝓛𝓖 of the Lie algebras *Vect*(*S*^{1}) and 𝓛𝓖.

Let 𝓖 be a Lie algebra and 〈., .〉 a non-degenerated invariant bilinear form. *Vect*(*S*^{1}) is the Lie algebra of vector fields on the circle and 𝓛𝓖 the loop algebra (i.e., the Lie algebra of smooth maps from *S*^{1} to 𝓖), *Vect*(*S*^{1})_{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 (λ

*g*+

*μ*

*h*)

_{n}is identified with λ

*g*+

_{n}*μ*

*h*with the relations

_{n}The semi-direct product of *Vect*(*S*^{1}) with 𝓛𝓖 is as a vector space isomorphic to *C*^{∞} (*S*^{1}, ℝ) ⊕ *C*^{∞}(*S*^{1},𝓖) [6]. The Lie bracket of 𝓢𝓤(𝓖) has the form

for any (*u*, *v*) ϵ *C*^{∞}(*S*^{1},ℝ)^{2} and any (*a*, *b*) ϵ *C*^{∞}(*S*^{1},𝓖)^{2}, where prime denote derivative with respect to a coordinate on *S*^{1}. The Lie algebra *Vect*(*S*^{1}) ⋉ 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

We denote by (*u*,*a*, *χ*, *α*) the elements of 𝓢𝓤(𝓖) with *u* ϵ *C*^{∞}(*S*^{1}, ℝ), *a* ϵ *C*^{∞}(*S*^{1}, 𝓖) and (*χ*, *α*) ϵ ℝ^{2}. 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*(

*S*

^{1}) ⋉ 𝓛ℝ has a universal central extension

We denote by (*u*, *a*, *χ*, *α*, *γ*, *δ*) elements of *u* ϵ *C*^{∞}(*S*^{1}, ℝ), *a* ϵ *C*^{∞}(*S*^{1}, 𝓖), and (*χ*, *α*, *γ*) ϵ ℝ^{3}. The Lie bracket of

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 (*u*, *a*, *ξ*, *β*) with *β* ≠ 0. Then introduce two new maps 𝓘(*u*, *a*, *ξ*, *β*) from 𝓢𝓤(𝓖)′ to (*u*, *a*, *ξ*, *β*, *γ*) from 𝓢𝓤(𝓖) to *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*(*S*^{1}) ⋉ *LG* and *Diff*(*S*^{1}) ⋉ _{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 *β**Id*-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

We denote by 𝓢𝓤(𝓖)′ the subset of 𝓢𝓤(𝓖) of elements (*u*, *a*, *ξ*, *β*) with non-vanishing *β*. Let *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 (*β*, let us introduce another new map

## Theorem 2.1

𝓘 *and* 𝓘̃ *are Poisson maps*.

## Proof

For any regular function *f*(*u*, *a*, *ξ*, *β*) from *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 (*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)}

and for the bracket

Then the map *π*_{1} from 𝓢𝓤(𝓖) onto *u*, *a*, *ξ*, *β*) onto (*u*′, *ξ*) is a Poisson morphism. The map *π*_{2} from 𝓢𝓤(𝓖) onto *u*, *a*, *ξ*, *β*) to ( *a*, *β*) is a Poisson morphism. For any regular function *f* on *G* on

Indeed, for *i* = 1, 2,

This gives

and

Let *g _{i}*(

*a*,

*β*),

*i*= 1, 2 be two regular functions on the affine Kac–Moody algebra. One notes that

*δ*

*g*

_{1,u}=

*δ*

*g*

_{2,u}= 0. Therefore,

Then,

We have:

The sum of the first two terms is equal to 0. The last term is 𝓙(*δ**f _{u}*〈[

*a*,

*a*],

*δ*

*g*〉), and is equal to zero. One can proceed similarly for 𝓘̃. □

_{a}## 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. Then*H* acts on𝓗̃^{*} by the coadjoint action along coadjoint orbits.

## Proposition 3.1

*The coadjoint actions of the groups**Diff*(*S*^{1}) ⋉ *LG**and**Diff*(*S*^{1}) ⋉ *are given by*

The classification of coadjoint orbits of *Vect*(*S*^{1}) ⋉ 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**to a coadjoint orbits of*

In other words, this means that if *β*_{1} ≠ 0, the elements (*u*_{1}, *a*_{1}, *ξ*_{1}, *β*_{1}) and (*u*_{1},*a*_{1}, *ξ*_{2}, *β*_{2}) are in the same coadjoint orbit if and only if: *ξ*_{1} = *ξ*_{2}, *β*_{1} = *β*_{2}, (*a*_{1}, *β*_{1}) and (*a*_{2}, *β*_{2}) are on the same coadjoint orbit of

## Proof

For any *ϕ* ϵ *Diff*(*S*^{1}), there exists *h* ϵ *LG* such that

By direct computation we check that

This implies Theorem 3.2. □

## Proposition 3.3

*The map* 𝓘̃ *sends the coadjoint orbits of**to coadjoint orbits of*

In other words, this means that if *β*_{1} ≠ 0 the elements (*u*_{1},*a*_{1}, *ξ*_{1}, *β*_{1}, *γ*_{1}) and (*u*_{1},*a*_{1}, *ξ*_{2}, *β*_{2}, *γ*_{2}) are in the same coadjoint orbit if and only if *γ*_{1} = *γ*_{2}, *ξ*_{1} = *ξ*_{2}, *β*_{1} = *β*_{2}, (*a*_{1}, *β*_{1}) and (a_{2}, *β*_{2}) are on the same coadjoint orbit of *β*_{1} = *β*_{2} = 0, then:

## Proposition 3.4

*If the elements* (*u*_{1},*a*_{1}, *ξ*_{1}, *β*_{1}, *γ*_{1}) *and* (*u*_{1},*a*_{1}, *ξ*_{2}, *β*_{2}*γ*_{2}) *are in the same coadjoint orbit then**γ*_{1} = *γ*_{2}, *are in the same coadjoint orbit of the Virasoro Lie algebra*.

## Proof

We have:

Previously, we determined Casemir functions on

## Proposition 3.5

*Let* 𝓒_{Vir}, 𝓒_{K−M} 𝓒_{𝓐}*be Casimir functions for Virasoro*, *affine Kac*–*Moody*, *and the Heisenberg Lie algebras* 𝓐 *correspondingly*. *Let* 𝓢_{P}𝓤(𝓖), *be Poisson submanifolds of* 𝓢𝓤(𝓖) *and**defined by**ξ* = 0. *Then the functions* 𝓒_{Vir}(*u*′, *ξ*), *c*(*u*, *a*, *β*, *ξ*) = 𝓒_{K−M}(*a*, *β*), *and**are Casimir functions on**In particular*, *the functions c*_{𝓐}(*u*, *a*, *β*, *ξ*) = 𝓒_{𝓐}(*a*, *β*), *are Casimir functions on*

## 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**commute pairwise for the Sugawara**and**e*-*braket* {., .}_{e}*with**e* = (1, 0, 0, 2, 0), *and*

The function *∫*_{S1}*u*, and the second one is *∫*_{S1} (*u*^{2} + *γ**u* + ∥ *a* ∥^{2}). A linear combination of these two terms gives the Hamiltonian of equations *H*(*u*, *a*) = *∫*_{S1} (*u*^{2} + ∥ *a* ∥^{2}).

Let {*ϕ _{i}*,

*i*ϵ

*I*} be a set of Casimir functions and

*e*ϵ 𝓖. Define

*x*

_{χ}=

*x*−

*χ*

*e*, for some

*χ*ϵ ℝ.

## Lemma 4.2

*For any* (*i*, *j*)ϵ *I*^{2}*and any* (λ, *μ*)ϵ ℝ^{2}*we have* {*ϕ _{i}*(

*x*

_{λ}),

*ϕ*

_{j}(

*x*

_{μ})} = {

*ϕ*(

_{i}*x*

_{λ}),

*ϕ*

_{j}(

*x*

_{μ})}

_{e}= 0.

## Lemma 4.3

*Suppose**ϕ _{i}*(

*x*

_{λ})

*can be expanded in terms of inverse powers of*λ

*with some extra function*

*f*(λ),

*and modes*

*F*

_{i,k}(

*x*), i.e.,

*then* {*F*_{i,k+1}, *f*}_{e} = {*F*_{i,k}, *f*}_{0}. *We can choose**e**so that the Hamiltonian**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**with respect to the brackets* {., .}_{0}*and* {., .}_{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 *h*_{1}, …, *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

*F*is the vector space generated by the products of at most

_{k}*k*elements of 𝓗. The generalized enveloping algebra is the algebra of the elements of the form ∑

_{k≤n}

*u*where

_{k}*u*

_{k}is an element of weight

*k*of 𝓤𝓗. The product of two such elements is defined by:

where *w _{k}* = ∑

_{iϵℤ}

*u*.

_{i}*v*which is a finite sum. Let

_{ki}*ω*

_{1}, …,

*ω*

_{n}be two-cocycles on the Lie algebra 𝓗, let 𝓗̃ be the central extension associated with and let

*e*

_{1},…,

*e*be the central elements associated with these cocycles.

_{n}The modified generalized enveloping algebra
*e*_{1} – 1,…,*e _{n}* – 1}. We denote again by 1 the neutral element of

*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

*n*

_{0}ϵ ℤ such that for any

*n*>

*n*

_{0}and any

*h*ϵ 𝓗̃

_{n}we have

*h*.

*v*= 0. Such modules are called representations of positive energy, and

*e*acts on

_{i}*V*by λ

_{i}

*Id*. Then

*V*is a module over

*F*_{1} is a Lie sub-algebra isomorphic to the central extension of 𝓗 by the cocycle

*i* be the natural inclusion of 𝓗̃ into

### 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

A two-cocycle *ω*_{𝓚} on 𝓚 defines also a two-cocycle

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

We call the action of 𝓚 on 𝓗 *realizable* in 𝓤_{ω𝓗} (𝓗) when there exists a map *F* : 𝓚 → 𝓤_{ω𝓗} (𝓗) and a two-cocycle *α* on {𝓚} such that for any pair (*g*_{1}, *g*_{2}) in 𝓚^{2}

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_{ω𝓗} (𝓗).

### Theorem 5.1

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

### Proof

Let 𝓤_{g} = {*ĝ* | *g* ϵ 𝓚} with be the unitary subalgebra of
*ĝ* = *g* – *F*(*g*), and 𝓤_{j} = {*j*(*h*), *h* ϵ 𝓗̃} be the unitary subalgebra of
*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_{ω𝓚–α}(𝓚). Let us check that the generators *ĝ*|*g* ϵ 𝓚} of this algebra satisfy the relations of the generators of 𝓤_{ω𝓚–α}(𝓚):

Since *F*(*g*_{1}) is an element of 𝓤_{j} and since the algebras 𝓤_{g} and 𝓤_{j} commute [*F*(*g*_{1}),*g*_{2}] = [*F*(*g*_{1}), *F*(*g*_{2})] and [*g*_{1}, *F*(*g*_{2})] = [*F*(*g*_{1}), *F*(*g*_{2})]. Therefore:

and finally

The subalgebra 𝓤_{j} is obviously isomorphic to 𝓤_{ω𝓗} (𝓗). The generalized modified enveloping algebra
_{ω𝓚 –α} (𝓚) with 𝓤_{ω𝓗}(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 {*K*_{1},…,*K _{n}*} a basis of 𝓖, and the dual basis

*Vect*(

*S*

^{1}), 𝓗 = 𝓛𝓖,

*ω*

_{𝓚}=

*ξ*

*ω*

_{Vir}, and

*ω*

_{𝓗}=

*β*

*ω*

_{K–M}. In this case,

*η*=

*β*+

*C*≠ 0, the Sugawara construction, delivers a map

_{φ}*F*:

*Vect*(

*S*

^{1})

_{ℂ}→ 𝓤

_{ω𝓖}(𝓛𝓖

_{ℂ}) defined by

where

(here dots denote the normal ordering), i.e., the action of *Vect*(*S*^{1}) is realizable in 𝓤_{βωK–M} (𝓛𝓖), with *α* = *β**ω*_{Vir}/12*η*. Thus we obtain

### Proposition 5.2

*If**η* ≠ 0, *then* 𝓤_{ξωVir}, *β**ω*_{K–M} (𝓢𝓤_{ℂ}𝓖) ≃ 𝓤_{βωK–M} (*Vect*(*S*^{1})_{ℂ}}_{(ξ – α)}) ⊗ 𝓤(𝓛𝓖).

The Lie algebra *Vect*_{ℂ}(*S*^{1}) acts on the Heisenberg algebra by

In this case, on has
*F* : *Vect*(*S*^{1})_{ℂ} → 𝓢𝓤_{ℂ}(ℂ) defined by

for a cocycle *α^* = (*α* + *γ*^{2}*β*_{-1})*ω*_{Vir}. For 𝓢𝓤_{ℂ}(𝓒) we obtain

### Proposition 5.3

*For**β* ≠ 0, *we have*

*with* θ = *ξ* – *γ*^{2}/*β* – 1/12.

### 5.3 Representations of 𝓢𝓤(𝓖)

### Proposition 5.4

*A positive energy representation**V**of* 𝓢𝓤_{ℂ}(𝓖) *with non-vanishing**β**Id*-*action of the cocyle**c*_{K–M}*brings 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 {*h*_{1}, …, h_{k}}. The Lie subalgebra 𝔨 of 𝓢𝓤_{ℂ}(𝓖) is generated by the elements {*c _{Vir}*,

*c*

_{K–M},

*u*

_{0}, (

*h*

_{1})

_{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
*V*, *μ* over the spaces generated by *c _{Vir}* and

*u*

_{0},

*c*

_{K–M}and {(

*h*

_{1})

_{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

*v*satisfies

*v*(

*u*

_{0})

*e*= λ(

*u*

_{0}–

*F*(

*u*

_{0}))

*e*, i.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

*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*

*where**μ* (*e _{i}*) = λ (

*e*),

_{i}*i*= 1, …,

*n*,

*defines*

*μ*,

*μ*(

*c*

_{K–M}) = λ (

*c*

_{K−M}),

*and*

*v*(

*c*) = λ(

_{Vir}*c*) –

_{Vir}*defines*

*v*,

*v*(

*u*

_{0}) = λ(

*u*

_{0}) –

### References

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**Received:**2016-11-30

**Accepted:**2018-01-04

**Published Online:**2018-01-31

© 2018 Zuevsky, published by De Gruyter

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