On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications

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by  A. V. Kiselev

ON THE (NON)REMOVABILITY OF SPECTRAL PARAMETERS
IN Z -GRADED ZERO-CURVATURE REPRESENTATIONS
2
AND ITS APPLICATIONS
A. V. KISELEV† AND A. O. KRUTOV‡
3
Abstract. We generalizeastandardmethodforinspecting the(non)removabilityof
1
0 spectralparametersin zero-curvaturerepresentationsto the case ofZ2-gradedpartial
2 differential equations. In particular, we illustrate a link between deformation tech-
n niques for two types of flat structures over such Z2-graded equations, namely, their
a zero-curvature representations and construction of their parametric families by using
J the Fr¨olicher–Nijenhuis bracket.

0
3

] Introduction. Zero-curvature representations (ZCR) for partial differential equations
G
(PDE) are input data data for a realization of the inverse scattering method [5, 26].
D
Therefore, the most interesting zero-curvature representations are those which contain
.
h a non-removable spectral parameter; in this case the system of PDEs is integrable.
t
a In the papers [21, 20] M. Marvan developed a remarkable method for inspecting the
m
(non)removabiltity of a parameter in a given ZCR. We generalize this method to the
[ case of Z2-graded differential equations (e.g., P. Mathieu’s N = 2 supersymmetric
1 Korteweg–de Vries equations [15]).
v
Another powerful technique for generating parametric families of nonlocal structures
3
4 over PDE is based on the use of Fr¨olicher–Nijenhuis bracket; this approach was de-
1
veloped by I. S. Krasil’schik et al. in [7, 8]. We analyze the link between these two
7
1. methods in the case of Z2-graded PDEs.
This paper is structured as follows. We first fix some notation; then in section 2 we
0
3 generalize Marvan’s approach to proving the (non)removabilty of parameters in ZCRs
1 to the case of Z -gradedpartial differential equations. In section 3 we study the relation
: 2
v between the construction of parametric families of zero-curvature representations and
i
X corresponding families of coverings produced by using the Fr¨olicher–Nijenhuis bracket.
r
a
1. Preliminaries
In this section we recall necessary definitions from supergeometry (we refer to [1, 17]
and [2, 10, 24] for further detail).
Date: January 29, 2012.
2010 Mathematics Subject Classification. 35Q53,37K25, 58J72,58A50.
Key words and phrases. Zero-curvature representation, Gardner’s deformation, Korteweg–de Vries
equation, supersymmetry, Fr¨olicher–Niheuhuis bracket.
†Address: JohannBernoulliInstituteforMathematicsandComputerScience,UniversityofGronin-
gen, P.O.Box 407, 9700AK Groningen, The Netherlands. E-mail: [email protected].
‡Author to whom correspondence should be addressed. Address: Department of Higher Mathe-
matics, Ivanovo State Power University, Rabfakovskaya str. 34, Ivanovo, 153003 Russia. E-mail:
[email protected].
1
2 A.V. KISELEVAND A.O. KRUTOV
1.1. The Z -graded calculus. Let G be a Grassman algebra over the field R with
2 n1
canonicalgeneratorsθ1,··· ,θn1,suchthatθiθj+θjθi = 0. Theparityfunctionp: G →
n1
Z/2Z = Z is defined on G by therule p(a) = ¯0 foreven elements a ∈ G andp(a) = ¯1
2 n1 q
for odd elements a ∈ G .
n1
Let M be a smooth real manifold. Consider the algebra of smooth functions U (M )
0 q 0
on M with values in G . Each function f ∈ U (M ) expands to a power series in {θi}:
0 n1 q 0
f(x,θ) = f (x)θi1 ...θik,
i1,...,ik
Xk≥0i1X,...,ik
here x ∈ M and f ∈ C∞(M ). The functions f are antisymmetric with
0 i1,...,ik 0 i1,...,ik
respect to the indexes i ,...,i .
1 k
Let us extend the parity function p onto the algebra U (M ) as follows: an element
n1 0
f ∈ U (M ) is called even (so that p(f) = ¯0) if p(f(x,θ)) = ¯0 for all x ∈ M , and f is
q 0 0
called odd (hence p(f) = ¯1) if p(f(x,θ)) = ¯0 for all x ∈ M .
0
The derivatives with respect to the respective even and odd variables xi and θj are
Z -graded derivations of U (M ) defined by the formulas
2 q 0
∂ ∂
f(x,θ) = f (x)θi1 ...θik,
∂xi ∂xi i1,...,ik
Xk≥0i1X,...,ik
∂
f(x,θ) = f (x)(δi1θi2···θik −δi2θi1θi3 ···θik +...).
∂θj i1,...,ik j j
Xk≥0i1X,...,ik
where δi = 1 if i = j and δi = 0 if i 6= j is the Kronecker symbol. The derivatives
j j
satisfy the commutation relations
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
f = f, f = f, f = − f. (1)
∂xi ∂xj ∂xj ∂xi ∂xi ∂θj ∂θj ∂xi ∂θi ∂θj ∂θj ∂θi
The Z -graded Leibniz rule is then
2
∂ ∂ ∂ ∂ ∂ ∂
(fh) = f h+f h , (fh) = f h+(−1)p(f)f h .
∂xi (cid:18)∂xi (cid:19) (cid:18)∂xi (cid:19) ∂θi (cid:18)∂θi (cid:19) (cid:18)∂θi (cid:19)
Denote by n the dimension of the smooth real manifold M . With every open subset
0 0
U ⊆ M we associate the algebra U (U) of smooth functions on U with values in
0 n1|n0
the Grassman algebra G . By definition, the manifold M endowed with the sheaf of
n1 0
algebras U(U) is a supermanifold [1]. The manifold M is then called the base manifold.
0
The pair (n |n ) is the dimension of that supermanifold, which we denote by M.
0 1
1.2. The Z -graded differential geometry. We say that a differential form ω ∈
2
Λ⋆(M) on the supermanifold M is the expression
ω = ω dxi1 ...dxikdθj1...dθjl, (2)
i1,...,ik;j1,...,jl
X
where the coefficients ω are antisymmetric w.r.t. the first group of indexes~ı
i1,...,ik;j1,...,jl
and are symmetric w.r.t. the second group ~. The space of differential forms Λ⋆(M)
is the graded R-algebra generated by p anticommuting symbols dx1,··· ,dxp and q
commuting symbols dθ1,··· ,dθq. It isimportant that thegeneratorsdxi anddθj satisfy
the following commutation relations:
dxi ∧dxj = −dxj ∧dxi, dxi ∧dθj = −dθj ∧dxi, dθi ∧dθj = dθj ∧dθi. (3)
ON THE (NON)REMOVABILITY OF SPECTRAL PARAMETERS IN Z2-GRADED ZCR 3
The exterior differential d act on ω ∈ Λ⋆(M) by the rule
∂ ∂
d(ω) = dxi (ω)+ dθj (ω) (4)
∂xi ∂θj
Xi Xj
As it is thecase forclassical manifolds, equality dω = 0 implies that locallyω = dω [16]
1
and equality d(dω) = 0 holds for any ω ∈ Λ⋆(M) [1].
The parity p(ω) of a differential form ω = ω dxi1...dxikdθj1...dθjl is the
i1,...,ik;j1,...,jl
sum of the parity p(ω ) of its coefficient and the parity of p(dxi1) + ··· +
i1,...,ik;j1,...,jl
p(dxik)+p(dθj1)+···+p(dθjl) of generators in it, here p(dxik) = ¯0 and p(dθjl) = ¯1.
The substitution of a Z -graded vector field X into a Z -graded differential form ω is
2 2
defined by the formula i (ω) = (−1)p(X)p(ω)ω(X).
X
1.3. The Z -graded infinite jet bundles. Let Σn be an n = (n |n )-dimensional
2 0 1
supermanifold, here1 6 n < ∞and0 6 n < ∞,withevencoordinatesx1,...,xn0 and
0 1
oddcoordinatesθ1,...,θn1 whichgeneratetheGrassmannalgebraG . Letπ: En+m →
n1
Σn be a vector bundle over Σn of fibre dimension m = (m |m ). In particular, we
0 1
let n = (2 | 0) so that the independent variables are x1 = x and x2 = t; we have
that m = (1|0) for the Korteweg–de Vries equation, m = (2|0) for the hierarchy of the
Kaup–Boussinesq equation, and m = (2|2) for the N=2 supersymmetric KdV equation,
see [1, 15, 22].
Consider the jet space J∞(π) of sections of the vector bundle π. The set of local
coordinates on J∞(π) is composed by
• even coordinates xi and odd coordinates θj on Σn,
• even coordinates uk and odd coordinates ξa along the fibres of π; these variables
themselves are elements of the set of
• coordinates uk and ξa along the fibres of the infinite jet bundle π : J∞(π) →
I,J I,J ∞
Σn.
In the above notation we let I be the multi-index that labels partial derivatives of the
unknowns uk and ξa w.r.t. even variables xi, and J the multi-index that labels partial
derivatives of uk and ξa w.r.t. odd variables θj; by convention, uk ≡ uk and ξa ≡ ξa.
∅ ∅
The parity function p: J∞(π) → Z is defined as follows,
2
p(xi) = ¯0, p(θj) = ¯1,
p(uk) = ¯0, p(ξa) = ¯1,
p(uk ) = |J| mod 2 p(ξa ) = |J|+1 mod 2.
I,J I,J
The total derivatives on J∞(π) are
∂ m0 ∂ m1 ∂
D = + uk + ξa ,
xi ∂xi I+1i,J∂uk I+1i,J∂ξa
Xk=1I∈N,JX∈(Z2)n1 I,J Xa=1 I∈N,JX∈(Z2)n1 I,J
∂ m0 ∂ m1 ∂
D = + uk + ξa .
θj ∂θj I,J+1j∂uk I,J+1j∂ξa
Xk=1I∈N,JX∈(Z2)n1 I,J Xa=1I∈N,JX∈(Z2)n1 I,J
These vector fields commute.
4 A.V. KISELEVAND A.O. KRUTOV
Consider a system E of r partial differential equations,
E = Fℓ(xi,θj,uk,...,uk ,ξa,...,ξa ) = 0, ℓ = 1,...,r
I,J I,J
(cid:8) (cid:9)
The system {Fℓ = 0} and all its differential consequences D|I|D|J|Fℓ = 0, which we
xI θJ
assume existing for all |I|+ |J| ≥ 1, generate the infinite prolongation E∞ of E. The
restrictions of D and D onto E∞ determine the Cartan distribution C on the tan-
xi θj
gent space TE∞. Here and in what follows the notation D¯ and D¯ stands for the
xi θj
restrictions of the total derivatives onto E∞.
The exterior differential d¯on Λ(E∞) splits to the sum d¯= d¯ +d¯ of the horizontal
h C
differential d¯ : Λp,q(E∞) → Λp,q+1(E∞) and the vertical differential d¯ : Λp,q(E∞) →
h C
Λp+1,q(E∞). The differential d¯ can be expressed in coordinates by inspection of its
h
actions on elements φ ∈ C∞(E∞) = Λ0,0(E∞), whence
n0 n1
d¯ φ = dxi ∧D¯ φ+ dθj ∧D¯ φ, (5a)
h xi θj
Xi=1 Xj=1
m0 ∂φ m1 ∂φ
d¯ φ = ωk ∧ + ζa ∧ , (5b)
C I,J ∂uk I,J ∂ξa
Xk=1XI,J I,J Xa=1 XI,J I,J
where we put
n0 n1
ωk = duk − uk dxi − uk dθj,
I,J I,J I+1i,J I,J+1j
Xi=1 Xj=1
n0 n1
ζa = dξa − ξa dxi − ξa dθj.
I,J I,J I+1i,J I,J+1j
Xi=1 Xj=1
We have that
i (ωk ) = i (ωk ) = i (ξa ) = i (ξa ) = 0.
D¯xi I,J D¯θl I,J D¯xi I,J D¯θl I,J
These equalities mean that the Cartan distribution can be equivalently described in
terms of Cartan forms ωk and ξa.
The restriction of Cartan distribution on J∞(π) onto E∞ is horizontal with respect
to the projection π∞ E∞: E∞ → M. This determines the connection CE∞: D(M) →
D(E∞), where D(M)(cid:12)and D(E∞) are the modules of vector fields on M and E∞ over
(cid:12)
the ring of smooth functions C∞(M) and C∞(E∞), respectively. The connection form
UE∞ ∈ D(Λ1(E∞)) of CE∞ is called the structural element of theequation E∞; we denote
by D(Λ1(E∞)) the C∞(E∞)-module of derivations C∞(E∞) → Λ1(E∞) taking values in
the C∞(E∞)-module of one-forms on E∞.
2. (Non)removability of parameters in Z -graded zero-curvature
2
representations
Consider the tensor product (over the ring of G -valued smooth function on E∞) of
n1
the exterior algebra Λ¯(E∞) = ∗Λi,0(E∞) with a finite-dimensional matrix complex Lie
superalgebra g. The product iVs endowed with the bracket
[A⊗µ,B ⊗ν] = (−1)p(B)p(µ)[A,B]⊗µ∧ν
ON THE (NON)REMOVABILITY OF SPECTRAL PARAMETERS IN Z2-GRADED ZCR 5
for µ,ν ∈ Λ¯(E∞) and A,B ∈ g. We now define the operator d¯ that acts on elements
h
of Λ¯(E∞)⊗g by the rule
¯ ¯
d (A⊗µ) = A⊗d µ,
h h
¯
where the horizontal differential d in the right-hand side is (5a). The tensor product
h
Λ¯(E∞)⊗g is a differential graded associative algebra with respect to the multiplication
(A⊗µ)·(B⊗ν) = (−1)p(B)p(µ)(A·B)⊗µ∧ν inducedbytheordinarymatrixmultiplication
so that
[ρ,σ] = ρ·σ −(−1)rs(−1)p(ρ)p(σ)σ ·ρ,
d¯ (ρ·σ) = d¯ ρ·σ +(−1)rρ·d¯ σ
h h h
for ρ ∈ Λ¯r(E∞) ⊗ g and σ ∈ Λ¯s(E∞) ⊗ g. Elements of C∞(E∞) ⊗ g are called g-
matrices [20].
Definition 1 ([19, 20, 21]). A horizontal 1-form α ∈ Λ¯1(E∞)⊗g is called a g-valued
zero-curvature representation for the equation E if the Maurer–Cartan condition
1
¯
d α = [α,α]. (6)
h
2
holds by virtue of E and its differential consequences.
Let G be the Lie supergroup of the Lie superalgebra g. Given equation E, for any
zero-curvature representation α there exists the zero-curvature representation αS such
that
αS = d¯ S ·S−1 +S ·α·S−1, S ∈ C∞(E∞)⊗G. (7)
h
The zero-curvature representation αS is called gauge-equivalent to α, and S is the gauge
transformation. Elements of C∞(E∞)⊗G are called G-matrices.
Definition 2. Let α be a family of zero-curvature representations depending on a
λ
complex parameter λ ∈ I ⊆ C. The parameter λ is removable if the forms α are
λ
gauge-equivalent at different values of λ ∈ I.
The following proposition and its proof are proper Z -generalizations of M. Marvan’s
2
result for classical, non-graded systems of partial differential equations [20, 21].
Proposition 1. The parameter λ in a family of zero-curvature representations α for
λ
the system E is removable if and only if for each λ ∈ I there is a g-matrix Q , depending
λ
smoothly on λ, such that p(Q ) = ¯0 and
λ
∂
¯
α = d Q −[α ,Q ].
λ h λ λ λ
∂λ
Proof. Suppose that λ is removable. This means that for any fixed λ there exists a
0
G-matrix S such that αSλ = α and S = 1 ∈ G. The matrix S˙ = ∂/∂λ| S
λ λ0 λ λ0 λ0 λ=λ0 λ
belongs to the tangent space at unit of G, i.e., to the Lie superalgebra g. Note that
6 A.V. KISELEVAND A.O. KRUTOV
p(S˙ ) = ¯0. We have that
λ0
0 = ∂ α = ∂ αSλ−1 = ∂ d¯ (S−1)S +S−1α S
∂λ(cid:12) λ0 ∂λ(cid:12) λ ∂λ(cid:12) h λ λ λ λ λ
(cid:12)λ=λ0 (cid:12)λ=λ0 (cid:12)λ=λ0(cid:0) (cid:1)
(cid:12) (cid:12) (cid:12)
= ∂ (cid:12) −S−1d¯ S +S(cid:12) −1α S = − ∂(cid:12) S−1d¯ (S )−S−1dS˙ −S−1S˙ S−1α S
∂λ(cid:12) λ h λ λ λ λ ∂λ λ0 h λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0
(cid:12)λ(cid:0)= λ0 (cid:1)
(cid:12)
(cid:12) +S−1α˙ S +S−1α S˙ = −dS˙ −S˙ α +α S˙ +α˙ .
λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0
This implies that α˙ = d¯ S˙ −[α ,S˙ ], where S˙ ∈ g⊗Λ¯0(E∞).
λ0 h λ0 λ0 λ0
Conversely, suppose now that α˙ = d¯ Q −[α ,Q ] for some Q ∈ g⊗C∞(E∞). Let
λ h λ λ λ λ
S ∈ G be a solution of the matrix equation ∂S/∂λ = Q S with initial data S = 1.
λ λ λ λ0
Note that solutions S exist only for even g-matrices Q . Consider the expression
λ λ
Z = dS +S α −α S = (αSλ −α )S . We have that
λ λ λ λ0 λ λ λ0 λ λ
∂ ∂
¯
Z = (d S +S α −α S )
∂λ λ ∂λ h λ λ λ0 λ λ
= d¯ (S˙ )+S˙ α −α˙ S −α S˙
h λ λ λ0 λ λ λ λ
¯
= d (Q S )+Q S α −α˙ S −α Q S =
h λ λ λ λ λ0 λ λ λ λ λ
¯ ¯
= d Q S +Q d S +Q S α −α˙ S −α Q S +(Q α S −Q α S )
h λ λ λ h λ λ λ λ0 λ λ λ λ λ λ λ λ λ λ λ
¯ ¯
= (d Q −α Q +Q α −α˙ )S +Q (d S +S α −α S )
h λ λ λ λ λ λ λ λ h λ λ λ0 λ λ
α˙λ
= Q| Z . {z }
λ λ
It is obvious that Z = 0, whence αSλ − α = 0. Therefore, the parameter λ is
λ0 λ0 λ
removable. (cid:3)
Example 1. Let us consider the four-component generalization of the KdV equation,
namely, the N=2 supersymmetric Korteweg–de Vries equation (SKdV) [15]:
a−1 ∂ d
u = −u +3 uD D u + D D u2 +3au2u , D = +θ · , (8)
t xxx 1 2 x 2 1 2 x x i ∂θ i dx
(cid:0) (cid:1) (cid:0) (cid:1) i
where
u(x,t;θ ,θ ) = u (x,t)+θ ·u (x,t)+θ ·u (x,t)+θ θ ·u (x,t) (9)
1 2 0 1 1 2 2 1 2 12
is the complex bosonic super-field, θ ,θ are Grassmann variables such that θ2 = θ2 =
1 2 1 2
θ θ +θ θ = 0, u , u are bosonic fields (p(u ) = p(u ) = ¯0), and u , u are fermionic
1 2 2 1 0 12 0 12 1 2
fields (p(u ) = p(u ) = ¯1). Expansion (9) converts (8) to the four-component system1
1 2
u = −u + au3 −(a+2)u u +(a−1)u u , (10a)
0;t 0;xxx 0 0 12 1 2 x
u = −u +(cid:0) (a+2)u u +(a−1)u u −(cid:1)3u u +3au2u , (10b)
1;t 1;xxx 0 2;x 0;x 2 1 12 0 1 x
u = −u +(cid:0)−(a+2)u u −(a−1)u u −3u u +3au2u (cid:1) , (10c)
2;t 2;xxx 0 1;x 0;x 1 2 12 0 2 x
u = −u −(cid:0)6u u +3au u +(a+2)u u (cid:1)
12;t 12;xxx 12 12;x 0;x 0;xx 0 0;xxx
+3u u +3u u +3a u2u −2u u u . (10d)
1 1;xx 2 2;xx 0 12 0 1 2 x
(cid:0) (cid:1)
1The Korteweg–de Vries equation upon u12, see (16) below, is underlined in (10d).
ON THE (NON)REMOVABILITY OF SPECTRAL PARAMETERS IN Z2-GRADED ZCR 7
The SKdV equation is most interesting (in particular, bi-Hamiltonian, whence com-
pletely integrable) if a ∈ {−2,1,4}, see [6, 11, 15]. Let us consider the bosonic limit
u = u = 0 of system (10): by setting a = −2 we obtain the triangular system which
1 2
consists of the modified KdV equation upon u and the equation of KdV-type; in the
0
case a = 1 we obtain the Krasil’shchik–Kersten system; for a = 4, we obtain the third
equation in the Kaup–Boussinesq hierarchy. In what follows we consider the case of
a = 4.
The N=2 supersymmetric a=4-KdV equation (10) admits [4, 12] the sl(2 | 1)-valued
zero-curvature representation αN=2 = Adx+Bdt, where
−iu ε−1(u2 +u )−ε−2u i −ε−1(u +iu )
0 0 12 0 2 1
A =  −ε −iu −ε−1 0 
0
0 iu −u −2iu −ε−1
1 2 0
 
and the sl(2 | 1)-matrix B, which is relatively large, is contained in Appendix 3 on
p. 15.
We claim that there is no sl(2 | 1)-matrix Q satisfying the equalities
∂ ∂
A = D¯ (Q)−[A,Q], B = D¯ (Q)−[B,Q].
x t
∂ε ∂ε
Consequently, the parameter ε in αN=2 is non-removable under gauge transformations.
We notethat the zero-curvature representation αN=2 can beused for theconstruction
of a solution, which is an alternative to the first solution reported in [6], of Gardner’s
deformation problem [15, 22] for the N = 2, a = 4 SKdV equation (we refer to [12] for
detail). The parameter ε which we use here is the parameter in the classical Gardner
deformation of the KdV equation [23], therefore we denote this parameter by ε instead
of λ.
Example 2. Consider another sl(2 | 2)-value zero-curvature representation β = Adx+
Bdt for the N=2, a=4-SKdV equation: we let
λ−iu −λ2 −(u2 +u ) −iu −u
0 0 12 1 2
A =  1 −λ−iu 0 
0
0 u −iu −2iu
2 1 0
 
and again, the sl(2 | 1)-matrix B is written in Appendix B on p. 15.
The sl(2 | 1)-matrix
0 1 0
Q = 0 0 0
0 0 0
 
satisfies the equations
∂ ∂
A = D¯ (Q)−[A,Q], B = D¯ (Q)−[B,Q].
x t
∂λ ∂λ
Solving the Cauchy problem
∂
S = QS, S| = 1,
λ=0
∂λ
8 A.V. KISELEVAND A.O. KRUTOV
we obtain the SL(2 | 1)-matrix
1 λ 0
S = 0 1 0.
0 0 1
 
This matrix S defines the gauge transformation that removes the parameter λ from the
zero-curvature representation β, i.e., (β)S−1 = β| . Consequently, the parameter λ in
λ=0
β is removable.
3. Families of coverings and the Fro¨licher–Nijenhuis bracket
From now on we assume that the base M of the superbundle π is just the base man-
ifold M , i.e., we let n = 0 and consider systems E of partial differential equations
0 1
upon m +m unknowns that may depend only on the even independent variables xi,
0 1
i = 1,...,n .
0
Consider a(k |k )-dimensional covering τ: E˜= W×E∞ → E∞ witheven nonlocalco-
0 1
ordinates w1,...,wk0 and odd nonlocal coordinates f1,...,fk1 on a (k |k )-dimensional
0 1
auxiliary supermanifold W. The prolongations D˜ of the total derivatives D¯ to the
xi xi
covering equation E˜ are given by the formula [2, 14]
∂ ∂
D˜ = D¯ +wp +fq .
xi xi xi∂wp xi∂fq
These total derivatives D˜ determine the Cartan distribution C(E˜) on the covering
xi
equation E˜. In turn, the Cartan distribution C(E˜) yields the connection C : D(M) →
E˜
D(E˜); the corresponding connection form U ∈ D(Λ1(E˜))) is called the characteristic
E˜
element of the covering τ. Expressing U in coordinates, we obtain
E˜
∂ ∂ ∂ ∂
U = d¯ (uk ) +d¯ (ξa ) +(dwp −wp dxi) +(dfq −fq dxi) .
E˜ C I,J ∂uk C I,J ∂ξa xi ∂w xi ∂f
I,J I,J
Next, let us recall that the Fr¨olicher–Nijenhuis bracket [·,·]FN on D(Λ⋆(E˜)) is defined
by the formula [14]
[Ω,Θ]FN(g) = L (Θ(g))−(−1)rs+p(Ω)p(Θ)L (Ω(g)),
Ω Θ
where Ω ∈ D(Λr(E˜)), Θ ∈ D(Λs(E˜)), and f ∈ C∞E˜; here L = i ◦d+d◦i is the Lie
Ω Ω Ω
derivative.
Let τ : E˜ = W × E∞ → E∞ be a smooth family of coverings over E∞ depending
λ λ λ
on a parameter λ ∈ C and U be the corresponding characteristic element of τ . The
λ λ
evolution of U with respect to λ is described by the equation [8, 7]
λ
d
U = [X,U ]FN, (11)
λ λ
dλ
where X ∈ D(E˜) is some vector field on E˜ .
λ
Let g ⊆ gl(k + 1|k ) be a finite-dimensional Lie superalgebra with basis e , here
0 1 i
k ,k > 0. Let us consider two representation of g:
0 1
(1) ρ: g → Mat(k +1,k ) — a matrix representation;
0 1
ON THE (NON)REMOVABILITY OF SPECTRAL PARAMETERS IN Z2-GRADED ZCR 9
(2) ̺: g → Vect(C;poly) — the representation in the space of vector fields with
polynomial coefficients on the (k |k )-dimensional supermanifold W with local
0 1
coordinates w1,...,wk0, f1,...,fk1.
Wenowrecallanimportantexampleofsuchrepresentation̺; thisstandardconstruction
will be essential in what follows.
Example 3 (The projective substitution and nonlinear representations of Lie algebras
in the spaces of vector fields [25]). Suppose g is a finite-dimensional Lie superalgebra.
We shall use the projective substitution [25] to construct a representation of g in the
space of vector fields.
Let N be an (k +1|k )-dimensional supermanifold with local coordinates
0 1
v = (v1,v1,...,vk0+1,f1,...,fk1) ∈ N, and put∂v = (∂w1,∂w2,...,∂wk0+1,∂f1,...,∂fk1)t.
For any g ∈ g ⊆ gl(k +1|k ), its image V under the representation of g in the space
0 1 g
of vector fields on N is given by the formula
Vg = vg∂v.
We note that V is linear in vi and fj. By construction, the representation preserves
g
the commutation relations in the initial Lie algebra g:
[Vg,Vh] = [vg∂v,vh∂v] = v[g,h]∂v = V[g,h], h,g ∈ g.
Locally, at all points of N where v 6= 0 we consider the projection
1
p: vi 7→ wi−1 = µvi/v1, p: fj 7→ fj = µfj/v1, µ ∈ R, (12)
and its differential dp: ∂v → ∂w. The transformation p yields new coordinates on the
open subset of N where v1 6= 0 and on the corresponding subset of TN:
w = (µ,w1,...,wk0,f1,...,fk1),
1 k0 k1
∂w = (−µ( wi∂wi + fj∂fj),∂w2,...,∂wk0,∂f1,...,∂fk1)t.
Xi=1 Xj=1
Consider the vector field X = dp(V ). In coordinates, we have
g g
Xg = wg∂w. (13)
We note that, generally, X is nonlinear with respect to wi and fj. The commutation
g
relations between the vector fields of such type are also inherited from the relations in
the Lie algebra g:
[X ,X ] = [dp(V ),dp(V )] = dp([g,f]) = dp(V ) = X .
g f g f [g,f] [g,f]
We now take X for the representation ̺(g) of elements g of the Lie superalgebra g.
g
For the sake of definition we now set n = 2, n = 0, x1 = x, x2 = t, k = 1, k = 0,
0 1 0 1
w1 = w.
Using the representation ̺ we construct the prolongations of total derivatives
∂ ∂
D˜ = D¯ +w D˜ = D¯ +w
x x x t t t
∂w ∂w
10 A.V. KISELEVAND A.O. KRUTOV
by inspecting the way in which they must act on the nonlocal variable w ∈ W:
w = −a ̺(e ) dw, (14a)
x i i
w = −bj̺(e ) dw. (14b)
t j
We thus obtain a one-dimensional covering τ: E˜ = W × E∞ → E∞ with nonlocal
variable w.
Example 4. Let us consider the N=2, a=4 SKdV equation (10) and a family of
coverings over it derived from the zero-curvature representations which we considered
for this super-system in Example 1. We now solve equation (11) in three steps.
We start from the covering derived from the Gardner’s deformation [23],
w = 1(w −u )−εw2, (15a)
x ε 12
w = 1(u +2u2 )+ 1 u + 1 u + −2u − 2u − 1 w + 2εu + 1 w2,
t ε 12;xx 12 ε2 12;x ε3 12 12;x ε 12 ε3 12 ε
(cid:0) (cid:1) (cid:0) (cid:1)(15b)
of the Korteweg–de Vries equation
u = −u −6u u . (16)
12;t 12;xxx 12 12;x
For a vector field
∂ ∂ ∂ ∂
X = a +b +ω +ϕ ,
σ
∂x ∂t ∂u ∂w
12;σ
containing undetermined coefficients a, b, ω and ϕ equation (11) for (15) splits into a
σ
system
d ∂w ∂w ∂w ∂w ∂b
˜ x x x x ˜
− w = D ϕ−ϕ −ω +b u + w −D w −w
x x σ 12;σt t x t t
dλ ∂w ∂u (cid:18)∂u ∂w (cid:19) ∂x
12;σ 12;σ
∂w ∂w ∂a
˜ x x
+a −D w + u + w −w ,
x x 12;σx x x
(cid:18) ∂u ∂w (cid:19) ∂x
12;σ
d ∂w ∂w ∂w ∂w ∂b
˜ t t t t ˜
− w = D ϕ−ϕ −ω +b u + w −D w −w
t t σ 12;σt t t t t
dλ ∂w ∂u (cid:18)∂u ∂w (cid:19) ∂t
12;σ 12;σ
∂w ∂w ∂a
˜ t t
+a −D w + u + w −w ,
t x 12;σx x x
(cid:18) ∂u ∂w (cid:19) ∂t
12;σ
∂b ∂a ∂b ∂a
ω = D˜ ω −u −u , ω = D˜ ω −u −u .
σx x σ 12;σt 12;σx σt t σ 12;σt 12;σx
∂x ∂x ∂t ∂t
By introducing the vertical vector field X′ = X U = X −aD˜ −bD˜ = ϕ′∂/∂x+
E˜ x t
ω′∂/∂u as in [14], we simplify this system:
σ 12;σ
d ∂w ∂w
− w = D˜ ϕ′ −ϕ′ x −ω′ x ,
dλ x x ∂w σ∂u
12;σ
d ∂w ∂w
− w = D˜ ϕ′ −ϕ′ t −ω′ t ,
dλ t t ∂w σ∂u
12;σ
ω′ = D˜ ω′,
σx x σ
ω′ = D˜ ω′.
σt t σ