Classical R-matrices for generalized so(p,q) tops

0.19 MB·English

by  Andrey Tsiganov

Classical R-matrices for generalized so(p, q) tops.
A.V. Tsiganov
Department of Mathematical and Computational Physics, Institute of Physics,
St.Petersburg University, 198 904, St.Petersburg, Russia.
e-mail: [e-mail]
4
0
0
2
An integrable deformation of the known integrable model of two interacting p-dimensional and
n
a q-dimensional spherical tops is considered. After reduction this system gives rise to the generalized
J
Lagrange and the Kowalevski tops. The corresponding Lax matrices and classical r-matrices are
9
1 calculated.

]
I
S 1 Introduction
.
n
i
l The most common examples of classical R-matrices are associated with decompositions of
n
[ Lie algebra into a direct sum of two Lie subalgebras [1]. According to [2] we can consider

various ”perturbations” of the standard decompositions of the loop algebras which may be
1
v associated with integrable deformations of the known integrable systems [3, 4]. In this note
5
we consider integrable deformations of some tops associated with the Lie algebra so(p,q)
2
0 and calculate the corresponding R-matrices.
1 Let g be a self-dual Lie algebra, g ,g ⊂ g its two Lie subalgebras such that g =
0 + −
4 g++˙g− as a linear space. Let P+, P− be the projection operators onto g± parallel to the
0 complimentary subalgebra; the operator
/
n
li R = P+ −P− (1.1)
n
:
v is a classical R-matrix. If a is an intertwining operator
i
X
r a[X,Y] = [aX,Y] = [X,aY] (1.2)
a
for any X,Y ∈ G, then R = R◦a is also a classical R-matrix [1]. So, one decomposition of
a
g determines a family of R-brackets those orbits are related
L 7→ a−1L, (1.3)
if a is invertible. For instance, such change of variables was used in [1] to connect Lax
matrices for the Manakov top on so(4) and for the Clebsch system on e(3).
The most interesting class of examples is provided by loop algebras L
L = g[λ,λ−1] = X(λ) = x λi, x ∈ g . (1.4)
i i
( )
i
X
1
The standard decomposition L = L +˙ L is defined by the natural Z-grading in powers
+ −
of auxiliare variable λ (spectral parameter)
L = g λi, L = g λi (1.5)
+ i − i
i≥0 i<0
M M
TheR-bracketassociatedwithdecomposition(1.5)hasalargecollectionoffinite-dimensional
Poisson subspaces
L = ⊕n g λi.
mn i=−m i
They are ad∗-invariant subspaces
R
ad∗X ·L = (ad∗X ·L) −(ad∗X ·L) , X = P X, L ∈ L , (1.6)
R g + + g − − ± ± mn
which are invariant with respect to the Lax equation
L = [L,M], M = P (dH(L)). (1.7)
t +
Intertwining operators in L are multiplication operators by scalar Laurent polynomials.
The crucial observation in [3, 4] is that to construct new Lax matrices by the rule (1.3) we
can use the matrix Laurent polynomials a.
According to [2], if g is associative algebra, the same space L (1.4) is a linear sum of
L (1.5) and subalgebra Ly
+ −
L = L +˙ Ly, Ly = x λi(1−λ−1y) | x ∈ g, , (1.8)
+ − − i i
( )
i=1
X
defined by some fixed element y ∈ g. This new decomposition of L defines another classical
R-matrix (1.1)
R = Py −Py. (1.9)
y + −
Subalgebra Ly could be regarded as some ”perturbation” of the standard subalgebra L
− −
by element y. In this approach the finite-dimensional Poisson subspaces of the R -bracket
y
(1.9)
n n
g λi(1−λ−1y) or (1−λ−1y)−1 g λi,
i i
i=−m i=−m
M M
are deformations of the known orbits L of R-bracket, which describe integrable deforma-
mn
tions of the known integrable systems.
Our aim here is to show mapping of the Lax matrices
L = λa +l +λ−1s, M = λb (1.10)
associated with Cartan type decomposition of g = so(p,q) into the new Lax matrices asso-
ciated with R -bracket
y
L = (1−λ−1y)−1(λa +l′ +λ−1s), M = M(1−λ−1y), (1.11)
y y
which describe new integrable deformations of so(p,q) tops.
2
2 Interacting spherical tops on so(p,q) algebra
Let G = SO(p,q) is the group of pseudo-orthogonal matrices with signature (p,q), p ≥ q.
The Lie algebra so(p,q) consists of all the (p+q)×(p+q) matrices satisfying
XT = −J XJ,
where J = diag(1,...,1;−1,...,−1), trJ = p−q and T means a matrix transposition.
In the natural (p,q)-block notation an element X ∈ so(p,q) has the form
ℓ s
X = , (2.1)
sT m
(cid:18) (cid:19)
where ℓT = −ℓ, mT = −m are p×p and q ×q matrices, s is an arbitrary p×q matrix .
The Cartan involution is given by σX = −XT and g = f+p is the corresponding Cartan
decomposition. The maximal compact subalgebra f = so(p)⊕so(q) consists of matrices X
with s = 0
ℓ 0
f = so(p)⊕so(q) =
0 m
(cid:26)(cid:18) (cid:19)(cid:27)
The subspace p consists of matrices X with ℓ = 0, m = 0
0 s
p = .
sT 0
(cid:26)(cid:18) (cid:19)(cid:27)
The pairing between so(p,q) and so(p,q)∗ is given by invariant inner product (X,Y) =
−1 trXY which is positive definite on f.
2
We extend the involution σ to the loopalgebraL(g,σ) by setting (σX)(λ) = σ(X(−λ)).
By definition, the twisted loop algebra L(so(p,q),σ) consists of matrices X(λ) such that
X(λ) = −XT(−λ). (2.2)
The pairing between L(g,σ) and L(g,σ) is given by < X,Y >= Res λ−1(X,Y).
2.1 Quadratic Hamiltonian
Let k ∈ SO(p), r ∈ SO(q) and ℓ ∈ so(p), m ∈ so(q) be the configuration and momentum
variables on the phase space T∗SO(p)×T∗SO(q). All the non-zero Lie-Poisson brackets are
equal to
{ℓ ,ℓ } = δ ℓ +δ ℓ −δ ℓ −δ ℓ ,
ij mn in jm jm in im jn jn im
{ℓ ,k } = δ k −δ k ,
ij nm jm ni im nj
(2.3)
{m ,m } = δ m +δ m −δ m −δ m
ij kn in jk jk in ik jn jn ik
{m ,r } = δ r −δ r .
ij nk jk ni ik nj
ThePoissonmappingfromT∗SO(p)×T∗SO(q)intoL(so(p,q),σ)isdefinedbythefollowing
theorem [1].
3
Theorem 1 (Reyman, Semenov-Tian-Shansky)
The spectral invariants of the Lax matrix
L = λa +l +λ−1s
(2.4)
0 kArT kℓkT 0 0 F
= λ− +λ−1 ,
rAT kT 0 0 rmrT FT 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
are in involution with respect to the canonical Poisson brackets (2.3) on T∗SO(p)×T∗SO(q).
Here A and F are constant p×q matrices.
We refer the reader to [1] for a complete proof of this theorem which uses that L(λ) (2.4) is
an orbit of classical R-matrix (1.1) associated with a standard decomposition of the twisted
loop algebra L(so(p,q),σ).
Here we present an elementary proof using a tensor form of the R-bracket which is more
familiar in the inverse scattering method [1]. According to [6] functions φ (L) = trLk, k ≥ 2
k
are in the involution if and only if matrix L(λ) satisfies relation
1 2 1 2
L(λ),L(µ) = r (λ,µ),L(λ) − r (λ,µ),L(µ) . (2.5)
12 21
(cid:26) (cid:27) (cid:20) (cid:21) (cid:20) (cid:21)
Here we used the standard notations
1 2
L(λ) = L(λ)⊗1, L(µ) = 1⊗L(µ),
and matrices r , r are kernels of the operators R and R∗ such that
12 21
r (λ,µ) = Πr (µ,λ)Π,
21 12
where Π is a permutation operator ΠX ⊗Y = Y ⊗XΠ.
One checks without difficulty that L(λ) (2.4) satisfies (2.5) with the following kernel of
the R-matrix
λ2 λµ
r (λ,µ) = − P + P , (2.6)
12 λ2 −µ2 f λ2 −µ2 p
where P and P are kernels of the standard Casimir operators acting in the orthogonal
f p
subspaces f and p respectively. They are equal to
p(p−1)+q(q−1)
pq
2
P = Z ⊗Z , P = S ⊗S ,
f α α p β β
α=1 β=1
X X
where antisymmetric Z and symmetric S matrices form two orthonormal basises in f and
α β
p. For instance we can use
0 1 0 ··· 0 0 1 ···
−1 0 0 ··· 0 0 0 0 ··· 0
   
Z = 0 0 0 ··· , Z = −1 0 0 ··· ,... ,
1 2
 ... ... ... ...   ... ... ... ... 
   
 0 0   0 0 
   
   
4
and
0 1 0 ··· 0 0 1 ···
.. .. .. .. .. ..
. . . . . .
S =  , S =  ,....
1 1 0 ··· 2 0 0 ···
 ... ... ... 0   1 ... ... 0 
   
   
Corollary 1 The equations of motion generated by the spectral invariants of the matrix L(λ)
(2.4) are Hamiltonian equations with respect to canonical brackets (2.3), which give rise to
Lax equation (1.7).
Let A be the truncated diagonal matrix A = a δ . The spectral invariants φ (L) of L(λ)
i ij k
(2.4) may be combined in the following Hamiltonian
a b −a b a b −a b
H = i i j j (ℓ2 +m2 )−2 i j j i ℓ m +
a2 −a2 ij ij a2 −a2 ij ij
i<j6q i j i<j6q i j
X X
p q p
b
+ j ℓ2 +γ ℓ2 −2 b k F r , (2.7)
a ij ij i ij js is
j
i=q+1 j=1 i,j=q+1 i,j,s
X X X X
which formally describes an interaction of p-dimensional and q-dimensional spherical tops.
Here a ,b , i = 1...q and γ are arbitrary parameters [1]. In this case the second Lax matrix
i i
M in (1.7) is equal to
0 kBrT
M(λ) = P (dH(L)) = − λ, (2.8)
+ rBT kT 0
(cid:18) (cid:19)
where diagonal matrix B = b δ includes parameters b .
i ij i
2.2 Integrable deformations
Let us introduce matrix
0 F
y = c , c ∈ C, (2.9)
−FT 0
(cid:18) (cid:19)
which is a generic solution of the following equations
0 F
yXs = σ(yXs), ∀X ∈ g, s = ∈ p. (2.10)
FT 0
(cid:18) (cid:19)
This solution is parametrized by an arbitrary numerical parameter c.
The matrix y (2.9), y ∈ so(p+q) = f+ip, does not belong to the algebra so(p,q) (2.1).
So, to consider deformations Ly (1.8) of L we have to embed initial algebra g and the
− −
element y into some algebra g and only then to discuss mapping of the known orbit (1.10)
of R-bracket (1.5) into the orbit (1.11) of R -bracket (1.9) defined in L(g)∗.
y
Therefore, for brevity, we shall consider kernels r(λ,µ) and ry(λ,µ) instead of the cor-
e
responding operators R (1.1,1.5) and Ry (1.9,1.8). e
5
Proposition 1 The spectral invariants of the Lax matrix
L = (1−λ−1y)−1(λa +l′ +λ−1s), (2.11)
y
where
1
l′ = l − ya +ay , (2.12)
2
(cid:16) (cid:17)
are in involution with respect to the canonical Poisson brackets (2.3) on T∗SO(p)×T∗SO(q).
To proof this proposition we have to check equation (2.5) with the kernel of the operator R
y
ry (λ,µ) = 1⊗(1−µ−1y)−1 r (λ,µ) (1−λ−1y)⊗1 . (2.13)
12 12
h i h i
by using properties of the element y (2.10) and definition of matrices r (λ,µ) (2.6).
ij
Corollary 2 The equations of motion generated by the spectral invariants of the matrix L
y
(2.11) are Hamiltonian equations with respect to canonical brackets (2.3). They give rise to
Lax equation (1.7), where
M = Py(dH(L)) = M(1−λ−1y). (2.14)
y +
The Lax equation (1.7) for L (2.11) may be rewritten as a Lax triad
y
d
L = L M L −L M L , i = 1,2 (2.15)
i i 2 2 i
dt
on the pair of matrices
1
L = L− (ya +ay) and L = 1−λ−1y,
1 2
2
entering in the definition of L (2.11).
y
For future reference we present two equivalent forms Hy of the deformed Hamiltonian
1,2
Hy using additional canonical transformations of the phase space
c
l → l ± (sa −as). (2.16)
2
Compositions of these canonical transformations with noncanonical map (2.12)
1
l → l = l − (y ∓cs)a +a(y ±cs)
1,2
2
(cid:16) (cid:17)
act either in the subalgebra so(p)
ℓ → ℓ = ℓ−c kT F rAT −ArTFTk , (2.17)
1
(cid:16) (cid:17)
either in the subalgebra so(q)
m → m = m−c rATkTF −FT kArT . (2.18)
1
(cid:16) (cid:17)
6
The corresponding perturbations of the initial Hamiltonian H (2.7) depend either on ℓ
variables
c2
H = H −c·tr(kℓBrT FT)+ tr(BTArFTF rT) (2.19)
1
2
either on m variables
c2
H = H +c·tr(rmBT kT F)+ tr(BAT kTF FTk). (2.20)
2
2
The corresponding Lax matrices are equal to
FrATkT −kArTFT 0
L(1) = (1−λ−1y)−1 L(λ)+c (2.21)
y 0 0
(cid:20) (cid:18) (cid:19)(cid:21)
or
0 0
L(2) = (1−λ−1y)−1 L(λ)−c . (2.22)
y 0 FrATkT −kArTFT
(cid:20) (cid:18) (cid:19)(cid:21)
We have discussed so far Lax matrices for the tops in the stationary reference frame,
i.e. used Euler-Lagrange description of motion. According to [1], we can go over to the Lax
matrices in the frame moving with the body which amounts to the gauge transformation
L(λ) = gTLg = λa˜ +˜l +λ−1s˜ =
e 0 A ℓ 0 0 kT F r
= λ− +λ−1 ,
AT 0 0 m rT FT k 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(2.23)
0 B d
M(λ) = gTM(λ)g+w = − λ+gT g.
BT 0 dt
(cid:18) (cid:19)
f
k 0
Here g = and w = gTg˙ is the block-matrix of angular velocities, related to ℓ and
0 r
(cid:18) (cid:19)
m by a linear transformations.
Proposition 2 The Lax matrix L(λ) (2.23) associated with the Euler-Poison description of
the motion satisfies equation (2.5) with the following matrices r (λ,µ)
ij
e
r (λ,µ) = −r (λ−1,µ−1), r (λ,µ) = −r (λ−1,µ−1), (2.24)
12 12 21 2e1
which are obtained from r (λ,µ) (2.6) by change of the spectral parameters.
e 12 e
Using the similar gauge transformation to the matrices L (λ) (2.11) and M (λ) (2.14)
y y
one gets
1
L (λ) = gTL g = (1−λ−1y˜)−1 L(λ)− (y˜a˜ +a˜ y˜) ,
y y
2
(cid:18) (cid:19)
(2.25)
e e
d
M (λ) = gTM (λ)g+gT g = M(1−λ−1y).
y y
dt
f f e
7
Here matrices w = gTg˙ are calculated with respect to cubic Hamiltonian Hy and, therefore,
y
it depends on all the dynamical variables. As above (2.15), a pair of matrices
1
L = L+ (y˜a +ay˜) and L = 1−λ−1y˜,
1 2
2
entering in the definition Ley (2.25)e, satiesfy to the Lax triad
d
L (λe) = L (λ)M(λ)+M T(−λ)L (λ), i = 1,2. (2.26)
i i i
dt
These equations (2.26) have a morefcomplicafted structure with respect to Lax triad in the
rest frame (2.15)
In the body frame the matrix
0 kT F r
y˜= gT yg = c
−rT FT k 0
(cid:18) (cid:19)
depends on the dynamical variables and, therefore, the kernel (2.13) of the corresponding
operator R depends on these variables too.
y
However inthemovingframewecanconstructanotherLaxmatrixsuchthatthekernelof
the corresponding R-matrix does not depend on dynamical variables. Namely, using matrix
0 A
z˜= c (2.27)
−AT 0
(cid:18) (cid:19)
such that
0 A
z˜Xa = σ(z˜Xa), ∀X ∈ g, a = ∈ p, (2.28)
AT 0
(cid:18) (cid:19)
we can consider perturbation of the Lax matrix L(λ) (2.23) by z˜
1
L (λ) = (1−λz˜)−1 L(λe)− (z˜s˜ +s˜z˜) . (2.29)
z
2
(cid:18) (cid:19)
e e
Proposition 3 The Lax matrix L (λ) (2.29) satisfies equation (2.5) with the following nu-
z
merical r-matrix
e
rz (λ,µ) = 1⊗(1−λz˜)−1 r (λ,µ) (1−µz˜)⊗1 . (2.30)
12 12
h i h i
The proof is straightforward.
e e
Applying inverse gauge transformation to L (λ) one gets another Lax matrix L (λ) in
z z
the stationary frame
e
1
L (λ) = gL gT = (1−λz)−1 L(λ)− (zs +sz) , z = gz˜gT. (2.31)
z z
2
(cid:18) (cid:19)
The corresponding r-matrix is dynamical.
So, for initial Hamiltonian H (2.7) we know two Lax matrices L(λ) (2.4) and L(λ)
(2.23) associated with the physically different coordinate systems. For these Lax matrix we
constructed by two perturbations L (2.11), L (2.31) in the rest frame and L (2.25)e, L
y z y z
(2.29) in the body frame.
e e
8
3 Further examples
According to [1], we shall try to exclude the nonphysical degrees of freedom using symmetry
of the Hamiltonian H (2.7) and its perturbations H (2.19,2.20). Assume that A = E is
1,2
the truncated identity matrix E = δ .
ij ij
Below all the Hamiltonians will be expressed through kinetic momentum ℓ ∈ so(p) and
ij
the entries of the Poisson vectors x = kT f , where f are the column vectors of the matrix
i i i
F. These variables are canonical coordinates on the algebra e(p,q)∗ = so(p)⋉(Rp⊗Rq) [1].
3.1 Algebra so(p,1), Lagrange top and spherical pendulum.
If q = 1 the phase space is T∗SO(p). We can put
m = 0, r = 1, (3.1)
without loss of generality. In this particular case the deformations (2.19) and (2.20) of the
initial Hamiltonian are quadratic polynomials instead of cubic ones.
Following to [1], we can rewrite Hamiltonian (2.7) in the form
p p
1 γ
H = ℓ2 + ℓ2 −(e ,x), x = kT f. (3.2)
2 ij 2 ij 1
i,j=1 i,j=2
X X
Here e is a first vector of the standard basis in Rp.
1
The Hamiltonian (3.2) describes rotation of a rigid body around a fixed point in a
homogeneous gravity field. The vector x is the vector along the gravity field, with respect
to the body frame and e is the vector pointing from the fixed point to the center of mass
1
of the body.
It is Lagrange case because the body is rotationally symmetric and the fixed point lies
on the symmetry axis e . The perturbations (2.19) and (2.20) of the Hamiltonian (3.2)
1
c2 c2
H = H + (e , kT f)2 = H + (e ,x)2
1 1 1
2 2
H = H +c(ℓe , kT f) = H +c(ℓe ,x).
2 1 1
are quadratic polynomials and, therefore, the corresponding equations of motion have the
form of the Kirchhoff equations.
Let x = kTf and π are canonical coordinates in R2p, {π ,x } = δ . Substituting ℓ =
i j ij i,j
x π −π x in L(λ) (2.4) one gets a Lax matrix for the spherical pendulum in appropriate
i j i j
physical coordinates [1]. The corresponding Hamiltonian
1
H = π2 − A x .
2 i i i
X X
describes a motion on the sphere Sp−1, (x,x) = 1, (x,π) = 0, in a homogeneous gravity field.
Its deformations (2.20) and (2.19) look like
c2 2
H = H +c A π , H = H + A x .
1 i i 2 i i
2
X (cid:16)X (cid:17)
9
3.2 The Kowalevski top.
According [1, 5], the generalized p-dimensional Kowalevski top is the reduced system (2.7)
with respect to the action of the subgroup SO(q). The reduction amounts to imposing the
constraints
m+P ℓP = 0, r = 1. (3.3)
Here P means the orthogonal projection from Rp onto subspace of Rq spanned by first q
vectors ofthestandardbasis. Thereduced phase spaceisT∗SO(p)withitscanonicalPoisson
structure.
Inserting the constraints (3.3) into (2.4) one gets the Lax matrix for the generalized
Kowalevski top
0 kE kℓkT 0 0 F
LKow(λ) = λ− +λ−1 . (3.4)
ET kT 0 0 −ET ℓE FT 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Proposition 4 The spectral invariants of the Lax matrices LKow(λ) (3.4) and L (λ) (2.11)
y
by (3.3) are in the involution with respect to Lie-Poisson brackets on T∗SO(p).
We present a direct proof of the involutivity of invariants without recourse to Hamiltonian
reduction.
In the rest frame it can easily be shown that the reduced Lax matrices satisfy equation
(2.5) with the same kernels r (λ,µ) (2.6) and ry (λ,µ) (2.13). These kernels do not change
12 12
by the Poisson reduction (3.3) and the corresponding operators R (1.1) and R (1.9) remain
y
differences of the same projectors.
It should be pointed out that constraints (3.3) agree with the Euler-Lagrange description
of the top, but disagree with its Euler-Poisson description [1]. In the body frame the Lax
matrix is given by
0 E ℓ 0 0 kTF
LKow(λ) = λ− +λ−1 . (3.5)
ET 0 0 −ET ℓE FTk 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
e
Proposition 5 The spectral invariants of the Lax matrices LKow(λ) (3.5) and L (λ) (2.29)
z
by (3.3) are in the involution with respect to Lie-Poisson brackets on T∗SO(p).
e e
In the body frame reduction (3.3) does not Poisson mapping which changes R-brackets.
Thus, after reduction the Lax matrix L˜(λ) (2.23) satisfies (2.5) with the reduced kernel
p(p−1)+q(q−1)
2
rKow(λ,µ) = r (λ,µ)− Z ⊗PZ P . (3.6)
12 12 α α
α=1
X
e e b b
Here
Z , 0 < α ≤ q(q−1);
α+p(p−1) 2
2
0 E 
P = ET 0 , PZαP =  0, q(q2−1) < α ≤ p(p2−1);
(cid:18) (cid:19) 

b b b Z , p(p−1) < α.
 α 2




10