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It is demonstrated that the standard construction of Lax equations on Lie algebras can be extended to Lie superalgebras, with the even subspace carrying the usual Lax equations. The extended equations inherit the existence of the canonical trace polynomial integrals of motion. An extra set of integrals exists in the odd subspace, with a nontrivial homological structure of the orbit space. This establishes a curious algebraic link between integrable evolution equations, supersymmetry and the deformation theory.
Lie superalgebras, Lax equations, integrals of motion, homological algebra, deformation theory
Introduction
The substantial interest to graded Lie algebras arose
about 60 years ago, in the context of similarity between deformations
of complex-analytic structures on compact manifolds
and deformations of associative algebras and Lie algebras
[1–4], in combination with the relevant cohomological
theories [5, 6]. In these algebras, the interplay of “even”
and “odd” subspaces carrying skew-symmetric and symmetric
multiplication laws plays the crucial role. Later, the new
interest to these structures arose in theoretical physics, in
the context of supergauge symmetries relating particles of
bosonic and fermionic statistics. Although supersymmetry
has not been experimentally discovered, these studies stimulated
an interesting new mathematics [7–10].
At the same time, the advent of the inverse scattering
method gave a boost to the studies of Lie groups and Lie algebras
in mathematical physics, in the context of integrability
of nonlinear evolution equations. In such studies, the nonlinear
dynamics is encoded in the evolution under the “semilinear”
Lax equations possessing trace polynomial integrals
of motion or revealing the isospectrality of the evolving operators
[11–16].
In this work, the standard construction of the Lax equations
on Lie algebras is extended to Lie superalgebras, the
Z2-graded Lie algebras of supersymmetry. The extended
equations possess the canonical trace polynomial integrals of
motion and so can be applied in a similar manner to nonlinear
problems. It is shown that the odd subspace admits extra
polynomial integrals of motion independent of the canonical
integrals. The geometry of the relevant orbit spaces is studied
revealing a nontrivial homological algebra. Thus, an algebraic
link is established between integrable evolution equations,
supersymmetry and the deformation theory. This work
can be regarded as a continuation of the previous work by the
author [17].
It is assumed that the reader is familiar with the basics
of the theory of Lie groups and Lie algebras and their representations
as well as the basics of algebraic geometry and
homological algebra.
1. Lie superalgebras
The algebra of supersymmetry comes from theoretical
physics as an attempt to combine into one unified theory two
statistically different types of particles, bosons and fermions.
According to the method of second quantization, the (complex
finite-dimensional) vector state spaces of these two types are
separated by parity, the one being represented in the even
space l0, the other in the odd space l1. To relate these spaces,
one assumes that the same symmetry (connected) Lie group
G linearly acts on both spaces. The group actions are repre-
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sented by the group homomorphisms
Tk : G → GL(lk), k = 0, 1.
The even action is assumed to be simply the adjoint action of
G, T0 = Ad, i.e. l0 is the Lie algebra of the Lie groupG. The
bilinear skew-symmetric bracket in l0
[, ]0 : l0 × l0 → l0, [x, y]0 = −[y, x]0
is the standard Lie bracket. The differential ad of T0
ad : l0 → End(l0), ad(x)y = [x, y]0, x, y ∈ l0 (1)
represents the adjoint linear action of l0 on itself. It is further
assumed that the odd action T1 is tensorially intertwined with
T0. This means that a symmetric bilinear bracket
[, ]1 : l1 × l1 → l0, [x, y]1 = [y, x]1
is defined on l1 with values in l0 such that
[T1(g)x, T1(g)y]1 = T0(g)[x, y]1,
x, y ∈ l1, g ∈ G. (2)
Using the brackets [, ]k, k = 0, 1, and the differential of
the action T1
ρ : l0 → End(l1), (3)
a bilinear bracket [, ] on the direct sum
l = l0 ⊕ l1
can be defined as
[x, y] =
[x, y]0, x, y ∈ l0,
[x, y]1, x, y ∈ l1,
ρ(x)y, x ∈ l0, y ∈ l1.
With this bracket, the graded vector space l becomes a
(complex) Lie superalgebra, i.e., a Z2-graded algebra whose
bracket satisfies the conditions
[x, y] ⊆ lξ+η, [x, y] = −(−1)ξη[y, x],
(−1)ξν[x, [y, z]] + (−1)ξη[y, [z, x]]+
+(−1)ην[z, [x, y]] = 0
∀ x ∈ lξ, y ∈ lη, z ∈ lν, ξ, η, ν = 0, 1.
(4)
The skew-symmetry between l0 and l1 and the graded Jacobi
identity are externally imposed on x, y, z ∈ l1 (to naturally
extend the representation theory) while the rest of the conditions
follow the intrinsic properties of the construction above.
The combined action T = (T0, T1) of the Lie group G
on the Lie superalgebra l = (l0, l1) generates the structural
group of automorphisms of l,
[T(g)x, T(g)y] = T(g)[x, y], x, y ∈ l, g ∈ G. (5)
The differential (ad, ρ) of this action generates a representation
of the Lie algebra l0 on l.
2. Representations
Representations of Lie superalgebras are Lie superalgebra
homomorphisms
ϕ : l → L, ϕ([x, y]) = [ϕ(x), ϕ(y)]L (6)
into operator Lie superalgebras L. The latter are constructed
as follows. For a Z2-graded (complex finite-dimensional)
vector space
V = V 0 ⊕ V 1,
let L0,L1 be the spaces of linear operators V → V of homogeneous
degrees 0,1. This means that operators from L0
act on the grades while those from L1 permute the grades,
L0V k ⊆ V k, k = 0, 1, L1V 0,1 ⊆ V 1,0.
On the Z2-graded vector space
L = L0 ⊕ L1
define a bracket [, ]L by the rule
[X, Y ]L = XY − (−1)ξηY X,
X ∈ Lξ, Y ∈ Lη, ξ, η = 0, 1.
(7)
With this bracket, L is a Lie superalgebra (the graded Jacobi
identity follows from Eq. (7)). Representations of l are homomorphisms
of Eqs. (6), (7) such that
ϕ(lk) ⊆ Lk, k = 0, 1. (8)
In particular, the restrictions
ϕ0 = ϕ|
l0 (9)
to the even subspace are representations of the Lie algebra
l0.
Nontrivial representations of Lie superalgebras always
exist. For example, the homomorphism
l → Ider(l), x → ∂x ≡ [x, ·] (10)
to the space of inner derivations of l satisfies the requirement.
This representation generalizes the adjoint representation of
a Lie algebra. The existence of faithful representations (in a
more general context of graded Lie algebras over commutative
rings) has been proved in Ref. [3]. Each faithful representation
of l0 (guaranteed by Ado’s theorem) can be extended to
a faithful representation of l.
3. Invariants and Lax equations
The group action T on l admits a set of canonical invariants,
the (complex) trace polynomial functions on l
Is[ϕ](x) = Tr ([ϕ(x)]s) , x ∈ l (11)
taken for any power s ⩾ 0 and any representation ϕ of l.
In fact, according to Eq. (7), for X ∈ L0 and Y ∈ L, the
bracket [X, Y ]L is the commutator of operators. Hence T
acts on operators of algebra representations by conjugation
and so preserves the traces of their powers. The restrictions
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of Eq. (9) generate the set of canonical invariants in the even
subspace,
I0
s [ϕ0](x) = Is[ϕ0](x), x ∈ l0. (12)
These are the standard trace polynomial invariants generated
by the Lie bracket in l0.
The intertwining of Eq. (2) enables extra invariants to be
built for the group action T1 on l1. Precisely, for any invariant
f of the action T0, the function
I1[f](x) = f([x, x]), x ∈ l1, (13)
is an invariant of the action T1. In fact, the map
w : l1 → l0, w(x) = [x, x] (14)
gives a (nonlinear) intertwining of T1 with T0. Its composition
fw with any invariant f of T0 is an invariant of T1. The
invariants given by Eq. (13) are called derived invariants.
For each k = 0, 1, the canonical invariants Is[ϕ] of
Eq. (11) are integrals of motion (conservation laws) of evolution
equations of the form
dl/dt = [m, l], m ∈ l0, l ∈ lk (15)
where m = m(t) is any time-independent or (continuously)
time-dependent magnitude. In fact, for any m, including the
case where m depends on l, the trajectories of the solutions
to Eq. (15) in the subspaces lk belong to orbits of the group actions
Tk determined by the initial values. Eqs. (15) are called
Lax equations on the Lie superalgebra l.
In the subspace l1, Eq. (15) is rewritten as
dl/dt = ρ(m)l, l ∈ l1 (16)
where ρ is the representation of l0 on l1 given by the differential
of the group action T1 (see Eq. (3)). This is a generalization
of the standard Lax equation on the Lie algebra l0 to
another representation subspace l1. Similar generalizations
(outside the Lie superalgebras theory) have been considered,
for example, in Ref. [16]. The derived invariants I1[f] given
by Eq. (13) are integrals of motion of Eq. (16) additional to the
canonical invariants.
The property of the Lax equations to have the “m-universal”
conservation laws is very useful. It enables one to integrate
nonlinear evolution equations (15) generated by any
(continuous) dependences of m on l and t.
4. Geometry of orbits
Since the invariants are integrals of motion, the trajectories
of evolution under Eq. (15) belong to the intersections of
integral surfaces, on which the invariants take constant values
determined by the initial states. Each such intersection
is filled with orbits of the group action T on l. The form of
the canonical invariants Is[ϕ] suggests their strong dependence
on the representations ϕ. The representations (on the
same vector space) are subdivided into equivalence classes
with respect to the canonical invariants,
∃g ∈ G : ϕ′ = ϕT(g) −→ Is[ϕ′] = Is[ϕ].
Besides this, it is hard to formulate anything general about
the integral surfaces created by the invariants Is[ϕ].
The derived invariants I1[f] on the odd subspace l1 are
different. They are written as compositions of any T0-invariant
with the map w of Eq. (14) that is independent of representations
of l. By Eq. (5), we have
[T1(g)x, T1(g)x] = T0(g)[x, x],
∀ g ∈ G, x ∈ l1. (17)
Hence, if the vector v = [x0, x0] ∈ l0 is fixed under the
group action T0,
T0(g)v = v ∀g ∈ G, (18)
then the trajectory of the solution l(t) to Eq. (16) starting from
x0 is completely contained in the set
Sv = {x ∈ l1 : [x, x] = v}. (19)
In fact, in this case, any constant function f is suitable for the
derived invariant I1[f]. The space of the vectors v defined by
Eq. (18) is the zeroth cohomology group h0(G, l0) of the group
G with coefficients in l0. This space also forms the centre of
the Lie algebra l0.
For any v, the relation that defines the set Sv is quadratically
polynomial with respect to the coordinates in l1, so the
set Sv is an (affine) algebraic variety. By Hilbert’s Nullstellensatz,
it is defined by the zero locus of a proper ideal in the
polynomial ring C[l1] containing these quadratic polynomials.
There is an obvious link of Eq. (19) to the classical problem
of intersections of quadrics. The variety Sv is symmetric
under the reflection with respect to the origin x → −x.
It is non-compact in general: the homotheties v → λv,
x →
√
λ x (λ ̸= 0) make the varieties Sv and Sλv isoomorphic.
In the case v = 0, removing the trivial orbit x = 0, Sv
becomes compact as a projective variety.
The special property of the variety Sv of Eq. (19) is that
it lies in the intersection of integral surfaces of all canonical
polynomial invariants passing through the point x0. In fact,
for any representation ϕ of l and any x ∈ l1, in accordance
with Eqs. (6), (7), (8),
ϕ([x, x]) = 2[ϕ(x)]2
and any odd power of the operator ϕ(x) permutes the even
and odd subspaces and so has a zero trace. Hence, we obtain
for any integer s > 0 and any representation ϕ
I2s[ϕ](x) = 2−sTr ([ϕ([x, x])]s) ,
I2s−1[ϕ](x) = 0, x ∈ l1.
(20)
According to Eqs. (17), (18), the set Sv is filled with orbits
of the group action T1 on l1. This generates the orbit space
Sv/G that classifies points of Sv. Two points belong to the
same equivalence class if they belong to the same orbit. In
the case v ̸= 0, we will assume that the group action T1 is
irreducible on l1.
The classification problem Sv/G can be approached as
follows. For any x ∈ Sv, let ∂x be the inner derivation defined
by the homomorphism of Eq. (10). In other words,
∂xy = [x, y], x ∈ Sv, y ∈ l.
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It follows from the Jacobi identity (see Eq. (4)) that
2∂x∂xy = [v, y]. (21)
Eq. (18) is equivalent to the condition
[m, v] = 0 ∀m ∈ l0.
By Eq. (21), this implies
∂x∂xl0 = 0. (22)
Also, we have
[v, y] = [[y, y], y] = 0 ∀y ∈ Sv
where we again used the Jacobi identity. Hence, Sv is a subset
of the centralizer of v in l1. This centralizer isG-invariant
because v is G-fixed. For v ̸= 0, we assumed that the T1-
action is irreducible, so the whole l1 centralizes v (otherwise,
there would exist a smaller invariant subspace of T1),
[l1, v] = 0. (23)
By Eqs. (21), (23), we conclude then that
∂x∂xl1 = 0. (24)
Combination of Eqs. (22), (24) gives
∂x∂xl = 0.
Considering the restrictions on the even and odd subspaces
∂k
x = ∂x|
lk , k = 0, 1,
we have on l
∂1
x∂0
x = ∂0
x∂1
x = 0.
This enables the Lie superalgebra l to be represented as the
“loop” chain complex
l0
∂0x
−↽−−−−⇀−
∂1x
l1
with respect to the differential ∂x. Introducing the kernels
and images (the cycles and boundaries)
Zk
x = ker ∂k
x, B0,1
x = im ∂1,0
x ,
we assign to each point x ∈ Sv the even and odd homology
groups as the quotients
Hk
x = Zk
x/Bk
x, k = 0, 1. (25)
The groups Hk
x , Hk
x′ are isomorphic if x, x′ belong to the
same G-orbit.
Introducing the vector spaces
Zx = Z0
x
⊕ Z1
x, Bx = B0x
⊕ B1x, Hx = H0
x
⊕ H1
x,
we see that Zx is a Lie superalgebra that is a subalgebra of
l, Bx is an ideal in Zx and so Hx = Zx/Bx also becomes a
Lie superalgebra.
The subspaces Z1
x,B1x
are respectively the tangent space
to Sv and the tangent space to the orbit of x at the point x.
If the odd homology group is trivial, H1
x = 0, then the orbit
of x covers a whole neighbourhood of the point x in Sv.
All small deformations of x within Sv areG-orbit equivalent.
Such points x are called rigid. For Z1
x = 0 (for v ̸= 0), the
set Sv consists of one point x (which in this case is fixed under
the group action, B1x
= 0). If H1
x
̸= 0 then the orbit of x
tends to lie strictly inside Sv.
The subspace Z0
x is the Lie subalgebra of l0 that centralizes
x: [Z0
x, x] = 0. The subspace B0x
is the image of x
under the odd inner derivations: B0x
= [l1, x]. By the Jacobi
identity and Eqs. (19), (23), it is a Lie subalgebra (actually an
ideal) of Z0
x. If the even homology group is trivial, H0
x = 0,
then x is a simple point of Sv. In fact, let H0
x = 0 and let
x + u ∈ Sv be a deformation of the point x in Sv. Then u
satisfies the deformation equation
2∂1
xu + [u, u] = 0. (26)
We can write the solution to Eq. (26) as a formal power series
u = zu1 + z2u2 + . . . (27)
in some (complex) scalar parameter z. The first term
2∂1
xu1 = 0 −→ u1 ∈ Z1
x
can be chosen arbitrarily. To find the higher terms, the following
induction can be applied. Let the first q terms be known.
Then they satisfy the equations
2∂1
xur + Jr = 0,
Jr =
Σr−1
p=1
[up, ur−p], r = 1, . . . , q.
(28)
To find the (q + 1)th term, the following equation should be
solved
2∂1
xuq+1 + Jq+1 = 0. (29)
Let
u(q) =
Σq
r=1
zrur
be the qth partial sum. Using the Jacobi identity, we have
[x + u(q), [x + u(q), x + u(q)]] = 0.
Taking the (q+1)th power of z, with the use of Eqs. (23), (24),
we obtain
[x, Jq+1] +
Σq
r=1
[uq+1−r, 2[x, ur] + Jr] = 0.
By Eqs. (28), this gives
∂0
xJq+1 = 0.
This means that Jq+1 ∈ Z0
x and so Jq+1 ∈ B0x
because we
assumed H0
x = 0. Then Eq. (29) can be resolved for uq+1,
uniquely if we take the zero projection to Z1
x. Hence, all the
terms of the power series of Eq. (27) can be uniquely found.
This series converges for any u1 as long as |z| is sufficiently
small. We obtain that the point x ∈ Sv can be analytically
deformed within Sv in any direction given by the space Z1
x of
tangent vectors to Sv. Thus, for H0
x = 0, the point x is simple.
A structure of a complex manifold on Sv can be defined
in a neighbourhood of x. The situation is very similar to that
described in Ref. [1].
8
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The consideration of dimensions gives the following relations
dimHk
x = dim lk −
Σ
p=0,1
dimBp
x,
dimZk
x = dim lk − dimB(k+1)
x , k = 0, 1.
(30)
Here in addition to Eq. (25) we used the isomorphisms
B0,1
x
≃ l1,0/Z1,0
x .
Also, since v ∈ B0x
⊆ Z0
x, we obtain
v ̸= 0 −→ dimZ0
x ⩾ dimB0x
⩾ 1 (31)
(in particular, if Sv ̸= ∅ for v ̸= 0 then the group action T1
on l1 cannot be free). It immediately follows from Eq. (30) that
if the subspaces l0,1 are not isomorphic, dim l1 ̸= dim l0
(i.e., the representation spaces of ad and ρ are not isomorphic
as vector spaces), then the groups H0,1
x are not simultaneously
trivial and are not isomorphic, dimH1
x
̸= dimH0
x.
This is valid for each point x ∈ Sv. This means, for instance,
that neither point x ∈ Sv can be simultaneously
a simple point of the variety Sv and have its orbit covering
the whole neighbourhood of x in Mv. In particular, for
dim l1 ̸= dim l0, the variety Sv cannot be a (nontrivial
smooth) homogeneous space of the G-action.
Eqs. (30), (31) enable an estimation of possible orbit
classes in the space Sv/G to be made. The existence of functions
on l1 that separate orbits in Sv and their links to the
homology on Sv are interesting open questions. In addition to
Eq. (20), note that, for the “adjoint representation” of Eq. (10),
the canonical integrals take the zero values on Sv,
ϕ(x) = ∂x −→ Is[ϕ](x) = 0, x ∈ Sv, s > 0.
In fact, for x ∈ Sv, we have ∂2
x = 0, so the operator ∂x is
nilpotent and its all positive powers have a zero trace.
5. Conclusion
We have shown that the well-known construction of the
Lax equations on Lie algebras can be extended to Lie superalgebras,
important in mathematics and theoretical physics
in their relation to the deformation theory and supersymmetry.
Like the usual Lax equations, the extended ones admit the
canonical trace polynomial integrals of motion which can be
used in the integrability theory for nonlinear evolution equations.
Besides the canonical integrals, the extra set of derived
integrals occurs in the odd subspace, as a result of tensorial
intertwining with the even subspace. This new feature is due
to the symmetric character of multiplication within the odd
subspace. The orbit spaces generated by constant values of
the derived integrals [x, x] = v, where v belongs to the 0th
cohomology group of the underlying Lie group action, possess
the natural (co)homological structure with respect to the inner
derivations ∂x. This structure is generically nontrivial,
giving obstacles for the integral surfaces to be locally homogeneous
spaces. These results algebraically relate the integrability
theory of evolution equations with supersymmetry
and the deformation theory.
The future work can be focused on possible connections
of the orbit space Sv/G with the “intrinsic properties” of the
algebraic variety Sv independent of its embedding into the
odd subspace l1 (say, in the spirit of the Zariski and Mumford
theories). An extension of the described algebraic structures
to the general graded Lie algebras should be possible
in terms of their natural grading into the even and odd subspaces.
From the point of view of physical applications, it can
be interesting to relate the above constructions to integrable
nonlinear dynamics and supersymmetry (for instance, to connect
Eqs. (19), (26) to symplectic geometry and Hamiltonian
dynamics as well as to the extended supersymmetry theory,
say, for the Poincaré algebra). Possible relations of the deformation
Eq. (26) to the Maurer-Cartan formalism and the
gauge theories can also be interesting.
The author declares no conflict of interest.
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