Россия
Уравнения Лакса dL/dt = [M,L] играют важную роль в теории интегрируемости нелинейных эволюцион- ных уравнений и квантовой динамике. В данной работе предлагаются тензорные расширения уравнений Лакса с M : V → V и L : Tk(V ) → V , k = 1, 2, . . . на ком- плексном векторном пространстве V . Эти расширения от- носятся к обобщенному классу уравнений Лакса (введен- ному ранее Бордеманном) dL/dt = ρk(M)L, где ρk — представление алгебры Ли. Случай k = 1, ρ1 = ad соот- ветствует обычным уравнениям Лакса. Расширенные пары Лакса изучаются с точки зрения изоморфных деформаций полилинейных структур, законов сохранения, внешних ал- гебр и коцепных симметрий.
уравнения Лакса, тензорные расширения, полилинейная алгебра, симметрии
Introduction
The idea of symmetry and conservation laws is fundamental
in natural sciences. Mathematically, it is reduced to the
study of algebraic properties that are invariant under groups
of transformations. From this point of view, linear objects are
much simpler and more symmetric than nonlinear ones. For
instance, nonlinear dynamical systems generally do not admit
conservation laws (integrals of the motion) and manifolds
of their solutions are much harder to describe than those of
linear dynamical systems that always are linear spaces. It is
very tempting then to reduce nonlinear dynamical problems
to linear problems.
The most remarkable success in this direction is the inverse
scattering method of integration of nonlinear evolution
equations. The method is based on including the nonlinear
evolution into a linear operator L that satisfies a linear evolution
equation dL/dt = [M,L] such that the eigenvectors
of L satisfy the linear equation with an operator M, while
the eigenvalues of L do not evolve. The latter property enables
a reconstruction of the nonlinear evolution using a spatial
scattering theory for the operator L. For ordinary differential
equations, the isospectrality ofLis used to find conservation
laws of the nonlinear dynamics. The pairs (M,L) are
called Lax pairs, the equations for the operator L are called
Lax equations [1-7].
In many cases, useful nonlinear relations exist between
solutions to linear dynamical systems. These relations shed
extra light to solutions of the relevant nonlinear problems.
The simplest nonlinear extension of a linear operator is a multilinear
operator. In this work, we realise this idea in replacing
the Lax operator L by a multilinear operator that maps
solutions to the linear problem with the operatorM again to
solutions to the same problem. We call the resulting equations
tensor extensions of the Lax equations.
We show that the extensions thus introduced have a rich
algebraic meaning, closely related to the theory of Lie algebras
and more general multilinear algebraic structures. We
reveal that the extensions we suggest are partial cases of the
generalised Lax equations introduced by Bordemann and related
to representations of Lie algebras other than the adjoint
representation, on which the usual Lax equations are
based [8]. Close connections between the solutions to the extended
Lax equations and Chevalley-Eilenberg cochain complexes
[9, 10] are pointed out. Also, the basic construction
presented in this work is another language for description of
isomorphic deformations of multilinear algebraic structures
on vector spaces with respect to dynamical groups of transformations.
In this sense, this work is a continuation of the
previous work by the author [11].
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1. Basic construction
Let V be a complex vector space and let Vk denote the
vector space of k-linear operators L : V k → V . For any
k = 1, 2, . . ., any linear evolution equation on V
dv/dt = Mv, v ∈ V, M(t) ∈ V1 (1)
generates the linear evolution equation on Vk
dL/dt = ρk(M)L, L∈Vk, ρk(M(t)) ∈ End(Vk) (2)
such that the solution operator L is a k-symmetry of Eq. (1),
i.e., maps k-tuples
(v1(t), . . . , vk(t))
of solutions to Eq. (1) again to solutions to Eq. (1). By multiplication
with respect to t, we can verify that
ρk(M)L(v1, . . . , vk) =
= ML(v1, . . . , vk) − L(Mv1, v2, . . . , vk)−
−L(v1,Mv2, . . . , vk)−. . .−L(v1, v2, . . . ,Mvk). (3)
It is evident that
ρ1(M)L = [M,L], M,L ∈ V1,
where [, ] denotes the commutator, so for k = 1 Eq. (2) is the
usual Lax equation. Using the canonical injection
V k → Tk(V )
of the Cartesian product V k into the k-grade of the tensor
algebra T(V ), due to the universal property of T(V ), any
solution L ∈ Vk to Eq. (2) can be uniquely identified with
a linear operator ¯L : Tk(V ) → V . We call the series of
Eq. (2), k = 2, 3, . . ., tensor extensions of the Lax equation
for k = 1.
2. Isomorphic deformations and conservation
laws
Eq. (3) enables the solutions to Eqs. (2) for any k and any
initial operator L(0) ∈ Vk to be written in the form
L(t)(v1, . . . , vk) = Φ(t)L(0)(Φ−1(t)v1, . . . ,Φ−1(t)vk)
(4)
where Φ(t) ∈ V1 is the operator that maps any vector v ∈ V
to the solution ¯v(t) = Φ(t)v to Eq. (1) with the initial value
¯v(0) = v,
dΦ/dt = MΦ, Φ(0) = 1 ∈ V1. (5)
Due to Eqs. (4), (5), solutions L(t) of the extended Lax equations
(2) are k-multiplicative algebraic structures on V that
are isomorphic to their initial values L(0) under the evolution
of Eq. (1). Eqs. (2) describe then isomorphic deformations
of k-multiplicative algebraic structures on V . In fact, Eq. (4)
is equivalent to
L(t)(Φ(t)v1, . . . , Φ(t)vk) = Φ(t)L(0)(v1, . . . , vk).
(6)
For finite values of time, the fundamental operator Φ(t) of
Eq. (1) is an isomorphism between L(0) and L(t). By the action
(6), the group generated by the operators Φ(t)
G = gen{Φ(t), t ∈ R} ⊂ GL(V ) (7)
maps any structure L(0) to structures isomorphic to L(0).
Stationary solutions L(t) = L(0) to Eqs. (2) that do not
explicitly depend on time describe k-multiplicative structures
that are automorphic with respect to the operators Φ(t) for
all values of t,
L(0)(Φ(t)v1, . . . , Φ(t)vk) = Φ(t)L(0)(v1, . . . , vk).
The group G of Eq. (7) is then a subgroup of the automorphisms
group of L(0),
G ⊂ Aut(L(0)).
For k > 1, evolutions under Eq. (1) on the vector space
V generate symmetries of the stationary solutions to Eq. (2)
as multiplicative k-linear algebraic structures on V . On the
other hand, by definition, solutions to Eq. (2) are k-symmetries
of Eq. (1) as they map k-tuples of solutions to Eq. (1) again
to solutions to Eq. (1) . We can say that Eqs. (1), (2) describe
mutual symmetries of the extended Lax pair (M,L).
The operator ρk(M) : Vk → Vk as a linear function of
M defined by Eq. (3) has the property
[ρk(M), ρk(N)] = ρk([M,N]) ∀M,N ∈ V1.
Hence, the linear map
ρk : V1 → End(Vk)
is a representation of the general Lie algebra V1 = gl(V ) on
the vector space Vk, i.e., a Lie algebra homomorphism
ρk : gl(V ) → gl(Vk). (8)
Thus, each Eq. (2) is a partial case of the generalised Lax
equation suggested by Bordemann [8]. For the usual Lax
equation k = 1, we have ρ1 = ad is the adjoint representation.
By exponentiation, the representation ρk of the Lie algebra
gl(V ) generates the linear action (representation) ¯ρk of
the general Lie group GL(V ) on the same space Vk. Then
any scalar function f : Vk → C invariant under this action,
f(¯ρk(m)L) = f(L), ∀ m ∈ GL(V ), L ∈ Vk, (9)
is a conservation law for Eq. (2), i.e., the values f(L(t))
are time-independent and do not change along the solutions
L(t). It is directly seen by differentiation of Eq. (9) by m at
the identity element e of the group GL(V ) and the fact that
M belongs to the tangent space TeGL(V ). For k = 1 and
a finite-dimensional vector space V , we have
ρ1 = ad, ¯ρ1(m)L = mLm−1
and the trace polynomial functions
fn(L) = Tr (Ln), n = 1, 2, . . . ,
are conservation laws for the usual Lax equation.
6
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In fact, functions f satisfying Eq. (9) are conservation laws
for Eq. (2) with any operator M. For k > 1, the explicit description
and even existence of such functions is a nontrivial
problem even if V is a finite-dimensional vector space. The
“isospectrality” of Eq. (2) is closely related to symmetries of
the operator M and manifests itself in the following observations.
Let V have a finite dimension N and a basis v1, . . . , vN.
Any initial operator L(0) ∈ Vk is defined by its values on the
basic vectors of the tensor k-grade Tk(V ),
L(0)(vi1 , . . . , vik ) =
XN
s=1
λ(s)
i1...ik
vs, (10)
where the indices i1, . . . , ik independently take all values
from the set {1, . . . ,N} and λ(s)
i1...ik
are complex coefficients,
the “structure constants” of the multiplicative algebraic
structure L(0). The solution L(t) to Eq. (2) with the
initial value L(0) has the property
L(t)(¯vi1(t), . . . , ¯vik (t)) =
XN
s=1
λ(s)
i1...ik
¯vs(t), (11)
where ¯vj(t) are the solutions to Eq. (1) with the initial values
vj and the coefficients λ(s)
i1...ik
remain time-independent.
This directly follows from Eq. (6) for any k. This does not mean
(even for k = 1) that the structure constants of the initial operator
L(0) are conservation laws for the solution L(t). In
fact, according to Eq. (4),
L(t)(vi1 , . . . , vik ) =
XN
s=1
¯λ
(s)
i1...ik
(t)vs =
= Φ(t)L(0)(Φ−1(t)vi1 , . . . ,Φ−1(t)vik ).
The expansion of the initial value by Eq. (10) generates the
evolution
λ(s)
i1...ik
→ ¯λ(s)
i1...ik
(t) (12)
of the structure constants of L(0) to those of L(t). This evolution
is another characteristic of the isomorphism between
L(0) and L(t).
The special case where Eq. (2) is explicitly solved is
where the operator M is time-independent and the basis
v1, . . . , vN is composed of eigenvectors ofM with eigenvalues
m1, . . . ,mN. In this case, Φ(t) = etM and the group
G defined in Eq. (7) is a 1-parameter subgroup of GL(V ):
Φ(t + s) = Φ(t)Φ(s). According to Eq. (4), the evolution
(12) takes the simple form
¯λ
(s)
i1...ik
(t) = etϕ(s)
i1...ik λ(s)
i1...ik
,
ϕ(s)
i1...ik
= ms −
Xk
p=1
mip.
(13)
It follows from Eq. (13) that the structural constants of the initial
operator L(0) that satisfy the condition
ϕ(s)
i1...ik
λ(s)
i1...ik
= 0
do not change under the evolution L(t), i.e., are conservation
laws of Eq. (2). In particular, the zero structural constants
are always conserved. A nonzero structural constant λ(s)
i1...ik
is conserved if the “resonance” ϕ(s)
i1...ik
= 0 takes place between
the eigenvalues m1, . . . ,mN of the operatorM.
The stationary solutions L(0) to Eq. (2) that are automorphic
with respect to the groupG are defined then by the condition
ϕ(s)
i1...ik
λ(s)
i1...ik
= 0 ∀s, i1, . . . , ik.
It follows, for instance, that if all the eigenvalues are “nonresonant”
ϕ(s)
i1...ik
̸= 0 ∀s, i1, . . . , ik
then all stationary solutions to Eq. (2) are trivial L(0) = 0.
Note that the case k = 2 with skew-symmetric bilinear
operators L corresponds to Lie algebraic structures if additionally
the Jacobi identity is satisfied. The finite limit transitions
¯λ
(s)
i1i2(t) → ˜λ(s)
i1i2 , t → ±∞,
are closely related to Inönü-Wigner contractions and lead to
stationary solutions to Eq. (2), automorphic with respect to
the “dynamical” group G. This situation has been considered
in more detail in the previous work by the author [11].
For k = 1 (regardless of whetherM is time-independent
or not), eigenvectors of the operator L(t) ∈ V1 that evolves
under the usual Lax equation are solutions to Eq. (1) and the
relevant eigenvalues are time-independent (being eigenvalues
of the initial operator L(0)). This underlies the inverse
scattering method of integration of nonlinear evolution equations
[1-7].
3. Exterior algebras and cochain symmetries
It can be verified that, for any k = 1, 2, . . ., if L′ ∈ V1
and L ∈ Vk are solutions to Eq. (2) then the operator composition
L′L ∈ Vk is also a solution to Eq. (2). In this sense, the
left multiplication by the solutions to the usual Lax equation
is a symmetry of the extended Lax equations (2).
For anyM and any k, the operator ρk(M) : Vk → Vk is
invariant under the action of the symmetric group Sk on Vk.
For any permutation σ ∈ Sk of the indices 1, . . . , k,
ρk(M)σ(L) = σ(ρk(M)L),
σ(L)(v1, . . . , vk) ≡ L(vσ(1), . . . , vσ(k)).
(14)
Thus, Sk is a symmetry group for Eq. (2). For any solution
L(t) and any permutation σ ∈ Sk, the “braided” operator
σ(L(t)) is also a solution.
This symmetry and the idea of considering only k-tuples
of linearly independent solutions to Eq. (1) leads to the restriction
from the infinite-dimensional tensor algebra T(V ) to the
finite-dimensional exterior (Grassmann) algebra
V
(V ) that
is a quotient of the tensor algebra with respect to the leftright
ideal generated by the tensors of the form v ⊗ v. In
terms of Eq. (2), it means that only alternating k-linear operators
L are to be considered, i.e., those with
σ(L) = sgn(σ)L, σ ∈ Sk.
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The vector spaces Vk will denote now the vector spaces of
alternating operators L : V k → V . Each such operator can
be identified with a linear operator from the k-grade of the
exterior algebra, ¯L :
Vk(V ) → V . We assume that the
vector space V is finite-dimensional, dim V = N.
The construction related to the representation (8) can be
extended to a representation of any Lie algebra. In fact, let
a be a Lie algebra and let
ρ : a → gl(V ) (15)
be its representation on V . Then the composition
πk = ρkρ : a → gl(Vk)
is a representation of a on Vk. The extended Lax equations
(2) are written then as
dL/dt = πk(a)L, a ∈ a.
Let now the underlying vector space of the Lie algebra
a be V and the representation ρ in Eq. (15) be the adjoint
representation. This enables the Chevalley-Eilenberg cochain
complex to be built,
V −→δ V1
−→δ V2
−→δ . . . −→δ VN
where δ : Vk−1 → Vk, δ2 = 0, is the exterior derivative
(δL)(v1, . . . , vk) =
=
Xk
s=1
(−1)s+1ρ(vs)L(v1, . . . , ˆvs, . . . , vk)+
+
X
s<s′
(−1)s+s
′
L([vs, vs′ ], v1, . . . , ˆvs, . . . , ˆvs′ , . . . vk),
k > 1, (δL)v = ρ(v)L, L ∈ V.
Here [, ] is the Lie bracket in a and the hat means that the relevant
variable should be omitted [9,10]. The solutions L ∈ Vk
to Eq. (2) are then naturally identified with (time-dependent)
k-cochains of this complex.
It can be verified that the exterior derivative δ is a symmetry
of the set of the extended Lax equations (2). In fact,
if L ∈ Vk is a solution in the k-grade then δL ∈ Vk+1 is
a solution in the next (k + 1)-grade. We call this symmetry
cochain symmetry. In the case of the exterior algebra,
according to Eqs. (4), (5), for k = 1,N the extended Lax
equations (2) are solved explicitly as
L(t) = Φ(t)L(0)Φ−1(t), k = 1,
L(t) =
Φ(t)
det Φ(t)
L(0), k = N.
4. Conclusion
It has been demonstrated that the classical Lax equations,
important in the integrability theory and quantum dynamics,
can be extended in a manner closely related to symmetries
of multilinear algebraic structures and representations of Lie
algebras other than the adjoint.
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