Automorphic algebras
of dynamical systems and generalised
In¨on¨u-Wigner contractions
Автоморфные алгебры
динамических систем и обобщенные
контракции Иненю-Вигнера
A. Karabanov А. Карабанов
Cryogenic Ltd,
London, W3 7QE, UK
karabanov@hotmail.co.uk
ООО «Криогеника»,
г. Лондон, W3 7QE, Великобритания
karabanov@hotmail.co.uk
Abstract
Lie algebras a with a complex underlying vector space V are
studied that are automorphic with respect to a given linear
dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂
Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a
Lie algebraic structure to the vector space of trajectories of
the group Gt. The basics of the general structure of automorphic
algebras a are described in terms of the eigenspace
decomposition of the operatorM ∈ der(a) that determines
the dynamics. Symmetries encoded by the presence of nonabelian
automorphic algebras are pointed out connected to
conservation laws, spectral relations and root systems. It is
shown that, for a given dynamics Gt, automorphic algebras
can be found via a limit transition in the space of Lie algebras
on V along the trajectories of the group Gt itself. This procedure
generalises the well-known Inönü-Wigner contraction
and links adjoint representations of automorphic algebras to
isospectral Lax representations on gl(V ). These results can
be applied to physically important symmetry groups and their
representations, including classical and relativistic mechanics,
open quantum dynamics and nonlinear evolution equations.
Simple examples are given.
Аннотация
Изучаются алгебры Ли a с комплексным базовым вектор-
ным пространством V , автоморфные относительно задан-
ной линейной динамической системы на V , т. е. 1-пара-
метрической подгруппы Gt ⊂ Aut(a) ⊂ GL(V ). Каж-
дая автоморфная алгебра сообщает Ли-алгебраическую
структуру векторному пространству траекторий группы
Gt. Основы общей структуры автоморфных алгебр a опи-
саны в терминах разложения по собственным подпро-
странствам оператора M ∈ der(a), определяющего ди-
намику. Указаны симметрии, кодируемые наличием неабе-
левых автоморфных алгебр, связанные с законами сохра-
нения, спектральными соотношениями и системами кор-
ней. Показано, что при заданной динамике Gt автоморф-
ные алгебры могут быть найдены посредством предельно-
го перехода в пространстве алгебр Ли на V вдоль траекто-
рий самой группы Gt. Эта процедура обобщает известную
контракцию Иненю-Вигнера и связывает присоединенные
представления автоморфных алгебр с изоспектральными
представлениями Лакса на gl(V ). Полученные результа-
ты можно применить к физически важным группам сим-
метрии и их представлениям, включая классическую и ре-
лятивистскую механику, открытую квантовую динамику
и нелинейные эволюционные уравнения. Приведены про-
стые примеры.
Keywords:
automorphic algebras, dynamical systems, generalised
In¨on¨u-Wigner contractions
Ключевые слова:
автоморфные алгебры, динамические системы, обобщен-
ные контракции Иненю-Вигнера
Introduction
Lie groups and Lie algebras are a powerful mathematical
tool that has a variety of physical applications. The local properties
of a Lie group are described in terms of its Lie algebra.
Lie algebras have also applications, fully separate from Lie
groups. This makes the theory of Lie algebras independently
useful. Finite-dimensional complex and real semisimple Lie
algebras and their representations are fully classified [1–3].
The modern theory of Lie algebras mostly concerns infinite-
dimensional generalisations (with links to modern problems
of theoretical physics [4]) and geometric extensions
(with links to algebraic groups and algebraic topology [5,
6]). Automorphisms of Lie algebras (and adjacent algebraic
structures) describe the algebra symmetries and so play an
important role in the theory. Normally, the direct problem
is tackled, i.e., the problem of finding the group of automorphisms
of a given Lie algebra. We address here the inverse
problem — the problem of description of Lie algebras that
have a given group of automorphisms.
The inverse problem is quite useful and intricate as well,
even in the finite-dimensional case. For instance, 1-parameter
subgroups of automorphisms of the algebra are equivalent
to linear dynamical systems on the underlying vector space.
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These dynamical systems should possess certain symmetries
for the algebra to have a non-abelian structure. Hence, there
is a close link between algebras, automorphic under the dynamics,
and symmetries of dynamical systems. Linear dynamical
systems find many physical applications. They form
the basis of such an important field as quantum mechanics
and underly the modern methods of integration of nonlinear
dynamical systems.
In these notes, we study the general properties of Lie algebras
that have a given linear dynamical system as its 1-
parameter group of automorphisms. We call such algebras
automorphic algebras of the dynamics. We study symmetries
of the dynamics encoded by non-abelian automorphic algebras
and their description in terms of a special limit transition
in the space of Lie algebras along the dynamical trajectories.
The latter relates automorphic algebras to the wellknown
Inönü-Wigner contraction and isospectral Lax representations.
We give a few simple examples related to applications
of this theory to classical matrix groups and nonlinear
evolution equations. These connections make automorphic
algebras a worth developing mathematical tool, useful
in the theory of both dynamical systems and Lie algebras.
We assume that the reader is familiar with the basics of
the Lie algebraic and group theory, for example, within the
classical books [1–3].
1. Automorphic algebras and symmetries
Let V be a finite-dimensional complex vector space. Let
a dynamic system Gt be given on V as a smooth 1-parameter
group of linear transformations. In other words, Gt is a
smooth representation of the additive group of real numbers
on V :
Gt : R → GL(V ), GtGs = Gt+s,
G0 = id, G
−1
t = G−t.
The exponential map
Gt = etM, M =
d
dt
Gt

t=0
identifies the trajectories
{x(t) = Gtx(0)} ⊂ V
with the solutions to the linear differential equation
x˙ = Mx, x ∈ V, (1)
to which Gt is a fundamental matrix. Note that Gt is always
a subgroup of the general linear Lie group GL(V ) (of all
transformations/automorphisms of V ) and the operator M
belongs to its Lie algebra gl(V ) (of all endomorphisms of V ).
The groupGt can be a 1-parameter subgroup of a smaller Lie
group withinGL(V ), and thenM belongs to the relevant Lie
algebra. Note also that the parameter t does not have to play
the role of the time in the usual physical sense, but can be a
more general evolution variable.
We additionally assume that the vector space V is
equipped with a Lie algebraic structure with the bracket [, ]
that satisfies the standard conditions of bilinearity, skewsymmetricity
and the Jacobi identity. We denote the corresponding
Lie algebra as a.
In this work we study the case where the group Gt acts
on the algebra a as a 1-parameter group of automorphisms,
Gt ∈ Aut(a):
Gt[x, y] = [Gtx,Gty] ∀x, y ∈ a. (2)
We call algebra a automorphic algebra of the dynamical system
Gt. By differentiation with respect to t, Eq. (2) is equivalent
to the condition that the operatorM is a derivation of a,
M ∈ der(a):
M[x, y] = [Mx, y] + [x,My] ∀x, y ∈ a. (3)
Any dynamical system Gt is uniquely defined by its operatorM.
Hence, Eqs. (2), (3) identify all dynamical systems
that have the same automorphic algebra a with the Lie algebra
der(a) of all derivations of a. The groups Gt span then
the identity component Aut(a)0 of the group Aut(a) of all
automorphisms of a. The group Aut(a) is a Lie group with
the Lie algebra der(a). Due to the Jacobi identity, the algebra
der(a) has a subalgebra (in fact, an ideal) ider(a) of inner
derivations written via the adjoint representation of a as
M = ad(y) = [y, ·], y ∈ a. Each inner derivation generates
a dynamical system Gt that belongs to the group of
inner automorphisms Inn(a) ⊆ Aut(a)0. The corresponding
Eq. (1) is of the form x˙ = [y, x]. For matrix/operator
algebras this form corresponds to Lax equations. The latter
are an important tool in the theory of nonlinear integrable
systems and quantum mechanics [7–10]. We will encounter
Lax equations again later when we consider semisimple automorphic
subalgebras and limit transitions along the group
trajectories. Note that the semisimple and nilpotent parts (in
the sense of the Jordan decomposition) of any derivation M
are also derivations of the same Lie algebra [2, 11].
A given dynamical system can have many automorphic algebras,
not isomorphic to each other. For example, abelian
algebras are automorphic for any dynamical system. Each
automorphic algebra has the same 1-parameter group Gt
of its automorphisms. Below we describe the basic general
properties of automorphic algebras (i.e., the properties common
for all automorphic algebras) of a given dynamical system
and show that they encode an important information on
its symmetries.
By an immediate observation, we come to the following
consequence of Eq. (3) and the bilinearity of the bracket [, ].
Proposition 1. For any automorphic algebra, the bracket
of any two solutions to Eq. (1) is again a solution to Eq. (1),
x˙ = Mx, y˙ = My −→ d
dt
[x, y] = M[x, y]. (4)
In terms of Proposition 1, automorphic algebras impart an
algebraic structure to the vector space of solutions to Eq. (1)
(trajectories of the group Gt). Non-abelian automorphic algebras
enable new solutions to Eq. (1) to be generated from
known solutions that generically are not linear combinations
of the latter.
It follows from the definition that any subalgebra a0 of
an automorphic algebra a that is invariant under the dynam-
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ics,Ma0 ⊆ a0, is an automorphic algebra of the restriction
Gt

a0
. Eq. (3) and the classical results of Refs. [12, 13] immediately
imply the following statement.
Proposition 2. For any automorphic algebra a, its center,
its derived algebra, its radical and its nilradical are invariant
under the dynamics,
Mz(a) ⊆ z(a), M[a, a] ⊆ [a, a],
Mrad(a) ⊆ rad(a), Mnil(a) ⊆ nil(a),
(5)
and so are automorphic subalgebras of a.
In fact, those all are ideals of a that are characteristic ideals,
i.e., ideals invariant under any derivation M [12, 13]. The
following embeddings take place z ⊆ nil ⊆ rad. In terms of
Proposition 2, automorphic algebras enable to reduce the solutions
to Eq. (1) into smaller invariant subspaces, giving the
space of solutions additional structural properties.
More automorphic subalgebras can be constructed and
the further analysis can be carried out in terms of the
eigenspaces of the operatorM, as shown below.
Let E be the set of (distinct) eigenvalues λ of the operator
M and μλ denote the algebraic multiplicity of λ. Consider
the decomposition of the vector space V into (generalised)
eigenspaces ofM
V =
M
λ∈E
Vλ, Vλ = {x ∈ V : (λ−M)μλx = 0}. (6)
Eqs. (3), (6) imply the following classical result proved in
Ref. [11].
Proposition 3. Any automorphic algebra admits the graded
structure
∀λ, η ∈ E [Vλ, Vη] ⊆ Vλ+η, (7)
where we assume Vξ = 0 if ξ ̸∈ E.
Multiplication of the operatorM by any nonzero complex
number c = |c|eiθ homogeneously dilates and simultaneously
rotates all the eigenvalues of M with respect to the
origin on the complex plane. This however does not change
automorphic algebras. Hence, the theory of automorphic algebras
is invariant under the group of homotheties and rotations
of the complex plane of eigenvalues with respect to the
origin (the latter is a subgroup of the Möbius group of conformal
transformations of the complex plane). In particular, it
follows from Eq. (7) that, for any line that crosses the origin
on the complex plane, the sets of eigenvalues that belong to
this line, lie on one side of the line and lie on the opposite side
of the line generate subalgebras of any automorphic algebra.
Further, we distinguish two qualitatively different situations,
namely the cases where the operator M is non-degenerate,
0 ̸∈ E, and where it is degenerate, 0 ∈ E. The
non-degeneracy/degeneracy of M is equivalent to the nonexistence/
existence of non-trivial conservation laws (“integrals
of motion”), i.e., non-zero elements x ∈ V such that
Mx = 0, Gtx = x ∀t.
The classical result proved in Ref. [14] implies that in the
non-degenerate case all automorphic algebras are nilpotent.
This is a consequence of Eq. (7) and the fact that the condition
detM ̸= 0 implies the nilpotence of all adjoint representations
ad(Vλ) and so (by a generalised Engel’s theorem) the
nilpotence of a [14].
Proposition 4. If 0 ̸∈ E then all automorphic algebras are
nilpotent.
Corrollary. The existence of non-nilpotent automorphic
algebras implies the existence of nontrivial conservation
laws.
A more detailed structure of nilpotent automorphic algebras
in the non-degenerate case can be enlightened within
the following definition.
Definition 1. An eigenvalue λ ∈ E is called resonant if
λ + η − ξ = 0 for some η, ξ ∈ E. Otherwise, λ is called
non-resonant.
Proposition 5. For any automorphic algebra a, if λ is nonresonant
then the relevant eigenspace belongs to the centre
of a, Vλ ⊆ z(a). If all eigenvalues are non-resonant then all
automorphic algebras are abelian.
Proof. Let λ be non-resonant. Then λ + η − ξ ̸= 0 for
all η, ξ ∈ E. By Eq. (7), this implies [Vλ, Vη] = 0 for all
η ∈ E, as λ + η cannot be an eigenvalue. Then, for any automorphic
algebra a, we obtain [Vλ, a] = 0, i.e., Vλ ⊆ z(a).
If all eigenvalues are non-resonant then all eigenspaces Vλ
belong to the centre and a is abelian. □
Since Vλ is invariant under Gt, it is an automorphic
abelian subalgebra of a for any non-resonant λ.
A simple example of applicability of Proposition 5 is the
case where Gt is a 1-parameter subgroup of an irreducible
representation of the (complexified) group SO(3) on V . The
group SO(3) of the rotations of the Eucledian 3-space is an
important group in physics, closely connected, for example, to
the special unitary and special linear groups SU(2), SL(2)
as well as the Möbius group of conformal transformations of
the complex plane. In this case, any operatorM is the relevant
representation of an element of the algebra so(3). The
latter can be treated as the algebra of quantum angular momentum
operators [15]. Then there exists a basis of V such
that
M = α diag (−S, −S + 1, . . . , S − 1, S), (8)
where α is some complex number, S is a positive integer or
half-integer spin number that characterises the dimension of
the representation, dim V = 2S +1. For even-dimensional
vector spaces V , the spin number S is half-integer corresponding
to fermionic representations. For odd-dimensional
V , the spin number S is integer and corresponds to bosonic
representations. For fermionic representations, since S is
half-integer, assuming α ̸= 0, it follows from Eq. (8) that
the operator M is non-degenerate and the sum of any two
eigenvalues of M is not an eigenvalue, i.e., by Definition 1,
all eigenvalues are non-resonant. By Proposition 5, all automorphic
algebras of any (nontrivial) dynamics generated by
fermionic representations of so(3) are abelian. We will return
to this example later when we consider the bosonic case.
In terms of Proposition 5, for the existence of non-abelian
automorphic algebras it is necessary that the resonance condition
λ + η − ξ = 0 is satisfied for some λ, η, ξ ∈ E. In
the non-degenerate case, it means that the operator M has
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at least two distinct eigenvalues. Since 2-dimensional nonabelian
algebras are non-nilpotent, we obtain then dim a =
dim V ⩾ 3. The minimal example is the 3-dimensional
Heisenberg algebra h3
[vλ, vη] = vξ, [vξ, vλ] = [vξ, vη] = 0.
Here λ, η are resonant and ξ is non-resonant, so
span {vξ} = z and the centre is invariant under M in
accordance with Proposition 2.
Nilpotent algebras are constructed as successive central
extensions of abelian algebras, so any nilpotent algebra always
has a nontrivial centre. Nilpotent algebras are solvable.
All subalgebras and homomorphic images of nilpotent algebras
are nilpotent. The Killing form on nilpotent algebras is
zero. The adjoint representations of nilpotent algebras consist
of nilpotent operators. Nilpotent algebras have outer automorphisms
and outer derivations. So far, no general approach
has been found to classification of nilpotent Lie algebras.
Let us now assume 0 ∈ E, i.e.,M is a degenerate operator,
detM = 0. In this case, we can write λ + 0 − λ = 0,
so, by Definition 1, all eigenvalues λ of the operator M are
resonant. Eq. (7) immediately implies the following result.
Proposition 6. If 0 ∈ E then the subspace V0 is a
nonzero subalgebra of any automorphic algebra that contains
a nonzero subalgebra ¯ V0 = {x ∈ V0 : Mx = 0} of conservation
laws. The adjoint representation of the subalgebra ¯ V0
acts on the space of solutions to Eq. (1): for any solution y(t)
to Eq. (1) within any automorphic algebra, the linear transformation
y
′
(t) = ad(x)y(t) = [x, y(t)], x ∈ ¯ V0, (9)
gives again a solution to Eq. (1).
Since V0, ¯ V0 are invariant under Gt, they are automorphic
subalgebras. The operator M is nilpotent on V0, so the
restrictionGt

V0
is polynomial in t. Eq. (9) is a partial case of
Eq. (4) that shows that, besides their conservative character,
within automorphic algebras, nontrivial conservation laws of
Eq. (1) play an important role in the structure of solutions.
To extend the result for the non-degenerate case to the
degenerate case, it is natural to consider automorphic algebras
as extensions of algebras that contain V0 by nilpotent
ideals. This can be done as follows.
Definition 2. The set E of eigenvalues of the operator M
is called split if E = E0 ∪ ¯ E with the properties:
i) E0 ∩ ¯ E = ∅;
ii) 0 ∈ E0;
iii) ¯ E ̸= ∅;
iv) for any λ0, η0 ∈ E0, ¯λ, ¯η ∈ ¯ E
either λ0 + η0 ̸∈ E or λ0 + η0 ∈ E0,
either ¯λ + ¯η ̸∈ E or ¯λ + ¯η ∈ ¯ E,
either λ0 + ¯λ ̸∈ E or λ0 + ¯λ ∈ ¯ E.
(10)
Proposition 7. If the set E is split then any automorphic
algebra is a semidirect sum
a = a0 + ¯a,
a0 =
M
λ∈E0
Vλ, ¯a =
M
λ∈¯ E
Vλ,
where a0 ⊇ V0 is a subalgebra and ¯a is a nilpotent ideal.
Proof. Indeed, by Eqs. (7), (10), we have
[a0, a0] ⊆ a0, [¯a, ¯a] ⊆ ¯a, [a0, ¯a] ⊆ ¯a.
Hence, a0 is a subalgebra and ¯a is an ideal of any automorphic
algebra. We have 0 ∈ E0, so V0 is a subalgebra of a0.
Since 0 ̸∈ ¯ E, by Proposition 4, the ideal ¯a is nilpotent, as
it is invariant under the operator M (forming then an automorphic
subalgebra) where this operator is non-degenerate.
The subalgebra a0 acts on ¯a by derivations, so the short exact
sequence
¯a −→ a −→ a0
defines a split extension of a0, i.e., the semidirect sum a =
a0 + ¯a. □
In terms of Proposition 7, both a0, ¯a are invariant under
the dynamics and so a0, ¯a are automorphic subalgebras.
Proposition 5 can be applied then to the ideal ¯a in terms of
Definition 1 and the set ¯ E — to specify the centre of ¯a. If several
splittings of E exist, from the point of view of the Levi
decomposition, in Proposition 7 the splitting with the minimal
possible subset E0 should be chosen.
Since ¯a ̸= 0, under the condition of Proposition 7, all automorphic
algebras are non-semisimple. The important case
where Proposition 7 is directly applicable is the semidissipative
case
∀λ ∈ E Re λ ⩽ 0, ∃η ∈ E Re η < 0. (11)
In this case, the set E is split into the subsets (remind 0 ∈ E0)
E0 = {λ ∈ E : Re λ = 0}, ¯ E = {λ ∈ E : Re λ < 0}.
This gives for any automorphic algebra the following semidirect
sum of a subalgebra and a nilpotent ideal
a = a0 + ¯a,
a0 =
M
Re λ=0
Vλ, ¯a =
M
Reλ<0
Vλ. (12)
Due to the invariance under rotations of eigenvalues with respect
to the origin (see the comments after Proposition 3), the
same situation occurs where the eigenvalues of the operator
M are split into eigenvalues that belong to one side of a line
that crosses the origin and eigenvalues that belong to this
line.
We aim now to describe the situations where there exist
non-solvable automorphic algebras, i.e., automorphic algebras
with semisimple subalgebras. This is closely connected
to projections of root systems of semisimple Lie algebras to
the complex plane of eigenvalues of the operatorM. The root
systems of semisimple complex Lie algebras are fully classified
[1–3].
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Definition 3. A set of complex numbers ρ(g) ⊂ C is
called a root projection for a complex semisimple Lie algebra
g if there exists an element m ∈ h of a maximal toral
subalgebra h ⊂ g such that ρ(g) = {α(m) : α ∈ Φ}
where Φ ⊂ h∗ is the set of roots of g corresponding to h.
Proposition 8. Let a root projection of a semisimple complex
Lie algebra g exist such that ρ(g) ⊂ E. Let μ′
λ ⩾ ¯μλ
if 0 ̸= λ ∈ ρ(g). Let μ′
0 ⩾ r + ¯μ0 if 0 ∈ ρ(g) where
r = rank g. Here μ′
λ is the geometric multiplicity of λ in
E and ¯μλ is the multiplicity of λ in ρ(g). Then there exists
an automorphic algebra a with a semisimple subalgebra a0
isomorphic to g.
Proof. Consider the vector space
a0 = ¯ V0 ⊕
M
λ∈ρ(g)
¯ Vλ,
¯ V0 =
Mr
k=1
v(k)
0
⊆ V0, ¯ Vλ =
Mμ¯λ
q=1
v(q)
λ
⊆ Vλ,
where v(k)
0 , v(q)
λ are eigenvectors of the operatorM corresponding
respectively to the eigenvalues 0 and λ. Such eigenvectors
exist by the conditions imposed on the multiplicities
of the eigenvalues. By Eq. (7) and the root space decomposition
of g, the vector space a0 is a semisimple automorphic
subalgebra of an automorphic algebra a with the restriction
M0 ≡ M

a0
= ad(m). To get an automorphic algebra
a on the full space V , it is sufficient to consider the trivial
extension of the subalgebra a0 by the abelian subalgebra
h = V \ a0. □
In terms of Proposition 8, the semisimple subalgebra a0
is invariant under the dynamics, so it is an automorphic subalgebra.
The set ρ(g) is centrally symmetric with respect to
the origin on the complex plane. This implies TrM0 = 0.
Hence, for the restriction G0t
= Gt

a0
we obtain detG0t
=
exp(tTrM0) = 1 ∀t and G0t
belongs then to the special
linear Lie group SL(a0). As a result, G0t
preserves the volume
and orientation of the vector space a0. Note also that the
restrictionM0 is always a semisimple operator and an inner
derivation of a0. The restriction of Eq. (1) on a0 is of the Lax
type x˙ = [m, x].
The simplest example of applicability of Proposition 8 is
the case 0, ±α ∈ E with an arbitrary complex number α.
In this case, there exists an automorphic subalgebra isomorphic
to A1 = sl(2) = so(3). For instance, getting back to
Eq. (8), this situation is realised for bosonic representations
of the algebra so(3) where the spin number S is integer and
so any operator M has the eigenvalues 0, ±α. By Proposition
8, unlike the case of fermionic representations where all
automorphic algebras are abelian (see the comments after
Proposition 5), any dynamics generated by bosonic representations
of so(3) has an automorphic subalgebra isomorphic
to so(3) and so has a non-solvable automorphic algebra. By
Propositions 5 and 8, a strict algebraic difference exists between
fermions and bosons in terms of automorphic algebras.
The situation is somewhat similar in the case whereGt ⊂
SO(N). For the even series of the orthogonal groupsDn =
SO(2n) andM ̸= 0 all automorphic algebras are nilpotent,
while for the odd series Bn = SO(2n + 1) automorphic
subalgebras exist, isomorphic to sl(2).
The lowest-dimensional bosonic case S = 1 corresponds
to the standard 3-dimensional representation of so(3). In
this case, as a consequence of the fact {0, ±α} = E and
Eq. (7), the dynamics Gt generated by any operator M from
this representation (plane uniform rotations around a fixed
coordinate line) has three non-abelian automorphic algebras
that are not isomorphic to each other:
so(3) = sl(2) : [v0, v±] = ±v±, [v+, v−] = v0,
h3 : [v0, v±] = 0, [v+, v−] = v0,
e(2) : [v0, v±] = ±v±, [v+, v−] = 0.
Here v0, v± are the eigenvectors ofM corresponding to the
eigenvalues 0, ±α. These algebras are respectively simple
(so(3)), nilpotent (the Heisenberg algebra h3) and non-nilpotent
solvable (the Euclidean algebra e(2)).
At the end of this section, we point out that the spectral
problem (6) for the operatorM on V generates a symmetric
spectral problem for the operator ad(M) on gl(V ) in terms
of the adjoint representations of automorphic algebras.
Proposition 9. For any automorphic algebra a, for each
λ ∈ E and each v ∈ Vλ, the adjoint representation ad(v) of
the element v ∈ a satisfies the spectral problem
(λ − ad(M))μλad(v) = 0. (13)
Proof. Indeed, in terms of the operators ad(x) = [x, ·],
Eq. (3) is written as
ad(Mx)−[M, ad(x)] = ad(Mx)−ad(M)ad(x) = 0.
For v ∈ Vλ, we have (λ−M)μλv = 0. Utilizing the Jordan
form of M on Vλ, we can choose a basis {v1, . . . , vμλ
} in
Vλ such that
Mvk = λvk +
kX−1
s=1
cksvs, k = 1, . . . , μλ
for some complex constants cks. This implies
ad(Mvk) = λ ad(vk) +
kX−1
s=1
cksad(vs)
and we come to the fact that the operators ad(vk) all satisfy
Eq. (13). Precisely,
(λ − ad(M))kad(vk) = 0, k = 1, . . . , μλ. □
In terms of decomposition (6) and Proposition 9, if the operatorM
is semisimple on Vλ then cks = 0 and both spectral
problems (6), (13) are split on Vλ:
(λ −M)vk = 0, (λ − ad(M))ad(vk) = 0. (14)
We have M, ad(vk) ∈ der(a) ⊂ gl(V ), so Proposition 9
and Eqs. (13), (14) reduce the procedure of finding automorphic
algebras to the usual linear algebra.
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2. Automorphic algebras and generalised
Inönü-Wigner contractions
In this section we show that automorphic algebras of a
given dynamical system Gt can be produced from non-automorphic
algebras by a special limit transition along the trajectories
of Gt. This limit procedure generalises the wellknown
Inönü-Wigner contraction [16] that finds a variety of
physical applications [10, 17–23].
Let an algebra Lie a with a bracket [, ] (generically nonautomorphic
for Gt) be given on the vector space V . Consider
the bilinear operation on V
[x, y]t = G−t[Gtx,Gty], t ∈ R, x, y ∈ V. (15)
For all t, the bracket [, ]t inherits the bilinearity, skew-symmetricity
and Jacobi identity of the bracket [, ] of the algebra
a. Hence, each bracket [, ]t defines a Lie algebra at on V .
Proposition 10. Let for all x, y ∈ V there exist the finite
limit (in the standard topology of the vector space V = Cn)
[x, y]
′
= lim
t→+∞
[x, y]t. (16)
Then the limit algebra a′ with the bracket [, ]′ is automorphic
for Gt.
Proof. By differentiation of Eq. (15) with respect to t, we
get for all t, x, y
d
dt
[x, y]t = −M[x, y]t + [Mx, y]t + [x,My]t. (17)
The existence of the finite limit (16) implies both left-hand and
right-hand sides of Eq. (17) to vanish at t → +∞. The operator
M becomes a derivation of the limit algebra a′. The
latter is then automorphic for the dynamical system Gt. □
Due to the relation [Gtx,Gty] = Gt[x, y]t, for each finite
t, the intermediate algebra at is isomorphic to a. If a is
automorphic for Gt then [x, y]t = [x, y] is independent of t
and the intermediate algebras at and the limiting algebra a′
all coincide with a. Otherwise, the limit a′ is a Lie algebra that
is (in general) not isomorphic to a, although a and a′ have the
same underlying vector space V .
In terms of the decomposition (6) into eigenspaces ofM,
for the limit of Eq. (16) to exist, it is sufficient to generalise the
graded structure of Eq. (7) to
[Vλ, Vη] ⊂
M
Vξ,
ξ = λ + η or Re ξ > Re λ + Re η.
(18)
The limit (16) transforms the grading (18) to the grading (7).
The limit transition (15), (16) enables to describe automorphic
algebras of a dynamical system Gt in a self-consistent
way, as limit cases of any algebras, satisfying Eq. (18), along
the trajectories of the groupGt itself. Similarly to Eq. (16), we
can consider the limit
[x, y]
′
− = lim
t→−∞
[x, y]t. (19)
Provided the latter exists, we again come to a new Lie algebra
a′
− that is automorphic forGt. The two limits (16), (19) are
mutually connected by inversion of the signs of the eigenvalues
of M. If both limits (16), (19) simultaneously exist then
a′ = a′
−.
For semidissipative dynamical systems (11), the procedure
(15), (16) is equivalent to the Inönü-Wigner contraction. In this
case, the vector space V shrinks (contracts) along the trajectories
of Gt. As per Proposition 7 and Eq. (12), limit algebras
of the Inönü-Wigner contraction are always non-semisimple.
They are split extensions of the subalgebra a0 spanned by the
eigenspaces ofM corresponding to purely imaginary eigenvalues
by the nilpotent ideal ¯a spanned by the eigenspaces
ofM corresponding to eigenvalues with negative real parts.
The restriction of Eq. (15) onto a0 is either compact for all t
or has terms that polynomially grow with t, so for the limit
(16) to exist, the bracket [x, y]t should be independent of t for
x, y ∈ a0. Then the limit algebra keeps the initial bracket on
a0. As a result, there exists the homomorphism
G
′
: a
′ → a, kerG
′
= ¯a, fixG
′
= a0 (20)
that realises the aforementioned split extension (the short exact
sequence)
¯a −→ a
′ −→ a0.
For example, in the original setting [16, 17], the Inönü-
Wigner contraction corresponds to the case
Mx = 0, My = λy, Re λ < 0,
Gtx = x, Gty = eλty, x ∈ a0, y ∈ h,
(21)
where a0 ⊂ a is a subalgebra, h is the complementary subspace.
In the limit t → +∞, according to Eqs. (15), (16), (18),
we come to the new Lie bracket on V that keeps a0 as a subalgebra
and makes h an abelian ideal,
[a0, a0]′ = [a0, a0] ⊆ a0,
[a0, h]′ ⊆ h, [h, h]′ = 0.
(22)
Eqs. (15), (16) generalise the Inönü-Wigner procedure to
any, not only semidissipative dynamics satisfying Eq. (18). Unlike
Eq. (20), we do not require the limit algebra to be a split
extension of a nonzero algebra. The limit algebra a′ can be
semisimple in certain cases where the initial algebra a is
semisimple.
To give a simple example, qualitatively different from
Eqs. (21), (22), consider the 3-dimensional algebra a spanned
by vectors v−, v0, v+ with
Mvξ = ξλvξ, ξ = −, 0, +, Re λ > 0,
[v−, v0] = 2v− + αv0 + βv+,
[v+, v−] = v0 + αv+, [v0, v+] = 2v+,
(23)
where α, β are arbitrary complex numbers. The bracket [, ]
satisfies the Jacobi identity and so defines a Lie algebra. Taking
the limit (16), we come to the new bracket
[v−, v0]′ = 2v−, [v+, v−]′ = v0,
[v0, v+]′ = 2v+
(24)
that is the bracket of the algebra a′ = sl(2) that is a simple
Lie algebra. The aforedefined operator M is a derivation of
10
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the new algebra, so a′ is automorphic forGt. This is however
not the case for the initial algebra a unless α = β = 0.
This example illustrates also Proposition 8. Here M =
ad(λv0/2) (v0 spans the maximal toral subalgebra) with the
eigenvalue 0 and the two non-zero eigenvalues ±λ generated
by the set of roots for sl(2).
In the example (23), (24), the limit algebra a′ = sl(2) is
isomorphic to the initial one a. Examples of non-dissipative
dynamics generating non-isomorphic limit algebras also can
be easily given. For example, it is sufficient to modify Eq. (23)
as
Mvξ = λξvξ, ξ = −, 0, +,
λ0 = 0, Re λ+ > 0,
Re λ− < −Re λ+ < 0,
[v−, v0] = 2v− + αv0 + βv+,
[v+, v−] = v0 + αv+, [v0, v+] = 2v+.
(25)
After the limit (16), we obtain
[v−, v0]′ = 2v−, [v+, v−]′ = 0,
[v0, v+]′ = 2v+.
(26)
The operator M is a derivation of a′, so the limit algebra is
indeed automorphic. Here the initial algebra is isomorphic to
sl(2) while the limit algebra (isomorphic to the Lie algebra
e(2) of the Euclidean group E(2)) is solvable and so nonisomorphic
to sl(2).
Example (25), (26) also illustrates Proposition 7. Choosing
E0 = {0}, ¯ E = {λ+, λ−}, we see that the set E is split.
Hence, indeed, the subspace ¯a = span (v−, v+) is a nilpotent
(abelian in this case) ideal in a′. This subspace coincides
with the derived algebra, ¯a = [a′, a′]. This illustrates Proposition
2, as ¯a is indeed invariant underGt. In accordance with
Propositions 6, 7, the 1-dimensional subspace spanned by v0
is a subalgebra that contains conservation laws.
As the final result of this study, we point out that the spectral
problem (14) for semisimple operatorsM can be linked to
the limit procedure (15), (16) via a Lax representation in gl(V ).
In fact, in terms of the adjoint representation adt(vλ) in the
intermediate algebras at, Eq. (17) is recast as
d
dt
adt(vλ) = (λ − ad(M))adt(vλ). (27)
We used the fact thatM is semisimple, so the eigenspace Vλ
is split into a set of eigenvectors vλ, thus splitting the spectral
problem (13) into Eq. (14). Eq. (27) easily implies the following
result.
Proposition 11. The operator
Lλ(t) = e
−λtadt(vλ) ∈ gl(V ) (28)
satisfies the isospectral Lax representation
d
dt
Lλ(t) = [Lλ(t),M]. (29)
In particular, the eigenvalues of Lλ(t) and analytical functions
of them are conservation laws of Eq. (29).
In terms of proposition 11, the limit (15), (16) is equivalent
to the limit along the trajectories of Eq. (27)
ad0(vλ) → ad
′
(vλ), t → +∞,
where ad0(vλ), ad
′
(vλ) are the adjoint representations in
the initial and the limit algebras a, a′. The trajectories are
found by the transformation (28) and the Lax representation
of Eq. (29). It is worth mentioning that the adjoint representations
adt(v0) corresponding to conservation laws of Eq. (1)
directly satisfy the Lax representation of Eq. (29) without the
transformation of Eq. (28). Proposition 11 implies the following
statement.
Proposition 12. Let the finite limit (15), (16) exist. Then, for
Re λ ⩾ 0, λ ̸= 0, the operators adt(vλ) are nilpotent for
all t. For Re λ < 0 the limit operator ad
′
(vλ) is nilpotent.
The eigenvalues of the operator adt(v0) that corresponds to
λ = 0 (and so all their analytical functions) are conservation
laws of Eqs. (27), (29).
Proof. By Proposition 11, the eigenvalues of the operator
Lλ(t) of Eq. (28) are conservation laws of Eq. (29). Then
any eigenvalue αt(λ) of the operator adt(vλ) has the form
αt(λ) = eλtα0(λ), where α0(λ) is an eigenvalue of the
operator ad0(vλ) of the initial algebra. Hence, if Re λ ⩾ 0,
λ ̸= 0, for the limit (15), (16) to exist, it is necessary α0(λ) =
0, so αt(λ) = 0 for all t, i.e., adt(vλ) should be nilpotent
for all t. If Re λ < 0 then αt(λ) → 0, t → +∞, i.e.,
the limit operator ad
′
(vλ) is nilpotent. For λ = 0, we have
L0(t) = adt(v0), so the eigenvalues of adt(v0) are conservation
laws of Eqs. (27), (29). □
It follows from Proposition 12 that, for any λ ̸= 0, the adjoint
representation ad
′
(vλ) in the limit algebra a′ is a nilpotent
operator. In fact, it follows from Eq. (7) that, for any automorphic
algebra, for all v ∈ Vλ with λ ̸= 0, the operator
ad(v) is nilpotent (the condition of semisimplicity of M can
be lifted). Remarkably, the conservation laws v0 of Eq. (1)
on automorphic algebras on V generate conservation laws
Tr [adt(v0)m] of Eq. (29) on the algebra gl(V ).
Propositions 10-12 along with Proposition 9 illustrate the
remarkable algebraic role of the limit procedure (15), (16) for
description of adjoint representations of automorphic algebras.
Conclusion
Automorphic Lie algebras of linear dynamical systems
have been introduced as Lie algebraic structures on the space
of their trajectories. We have formulated the basic general
properties of automorphic Lie algebras of a given dynamical
system in terms of the eigenspace decomposition of the
dynamics. We have pointed out the symmetries that are encoded
by the presence of non-abelian automorphic algebras.
In particular, non-nilpotent automorphic algebras are related
to conservation laws of the dynamics. In the presence of a
semisimple automorphic subalgebra, there is a natural correspondence
between the set of roots related to the subalgebra
to the set of eigenvalues of the dynamical system. We have
shown that automorphic algebras can be found by a limit transition
along the trajectories of the dynamics, a procedure that
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generalises the well-known Inönü-Wigner contraction. We
have demonstrated that, in terms of the adjoint representation,
the limit transition is naturally reduced to an isospectral
Lax representation. We have given simple examples related
to applications of the developed theory to classical matrix
groups. This suggests that automorphic algebras are worth
developing tool in the theory of both dynamical systems and
Lie algebras.
Inönü-Wigner contractions, in the dynamical setting of
Eqs. (15), (16), have been applied before to dissipative dynamical
systems [20, 22, 23]. Other dynamical deformations of Lie
algebras, both related and unrelated to Inönü-Wigner contractions,
have also been discussed [24, 25]. We are unaware
of whether the automorphic character of limit algebras has
ever been noticed. We are also unaware of an earlier use of
non-dissipative dynamical systems for constructing non-isomorphic
algebras on the same vector space. As far as we
know, the link of the limit transition of the Inönü-Wigner type
to the Lax representations has not been made before.
As direct physical applications of the methodology developed
in this work, we would expect first of all cases where
the groups Gt have additional special properties: for example,
belong to various physically important symmetry groups
and their representations. Among them, we can find classical
and relativistic mechanics (the classical matrix groups,
the Galilean, Lorentz and Poincaré groups [26]) and quantum
applications such as, for example, Lindblad equations of
open quantum dynamics (completely positive quantum semigroups
[27]). Considering finite and discrete groups that generate
discrete dynamical systems would be curious as well
(for example, within the theory of Ref. [5]).
It would be interesting, to our mind, to consider also infinite-
dimensional underlying vector spaces V , especially
functional spaces, or finite-dimensional Lie algebras over
functional rings. In the latter cases, it might be expected that
the linear systems (1), the graded structures (7) and the Lax
representations (29) are related to some integrable nonlinear
evolution equations of mathematical physics [7–10]. According
to Proposition 4, for an open set of dynamical systems, all
automorphic algebras are nilpotent. Nilpotent algebras play
an important role in the representation theory, especially in
the orbit method and geometric quantisation [28]. It would be
curious to build links of these modern theories to the theory
of automorphic algebras we developed.
Some of the results we obtained can be reformulated for
arbitrary (not necessarily Lie) algebras, making this subject
useful in a wider algebraic context.
To give one simple example in relation to the last two
paragraphs, consider the vector space V of smooth complex
functions x : Ξ → C given on a smooth real manifold Ξ of
a dimension n with (local) coordinates ξ = (ξ1, . . . , ξn).
Let a smooth vector field F(ξ) = (F1(ξ), . . . , Fn(ξ)) be
given on Ξ and let Eq. (1) be generated by the operatorM of
differentiation along the field F:
x˙ = Mx ≡ ⟨F,∇x⟩ =
Xn
k=1
Fk
∂x
∂ξk
. (30)
Then the solutions x(t, ξ) are given by evolution of the initial
value x(0, ξ) = x0(ξ) along the flow on Ξ generated by the
vector field F: x(t, ξ) = x0(η(t, ξ)) where
η˙ = F(η), η(0, ξ) = ξ (31)
(the group Gt is a realisation of such evolution). The operator
M is a differentiation “from the left”, so M is a derivation
of the associative algebra on V generated by the usual
product x, y → xy. This algebra is automorphic for the dynamics
of Eq. (30). In particular, the product of any two solutions
x(t, ξ)y(t, ξ) is again a solution. The conservation
lawsMx(ξ) = 0 of Eq. (30) are in a one-to-one correspondence
with the conservation laws ⟨F,∇x⟩ = 0 of Eq. (31). In
some cases, this observation helps to find conservation laws
for the nonlinear dynamics of Eq. (31) from the linear dynamics
of Eq. (30).
The classical case is the Hamiltonian dynamics where the
manifold Ξ is even-dimensional, n = 2m, and F is a Hamiltonian
vector field:
F(ξ) = J∇h(ξ), J =

0 Im
−Im 0

.
Here h(ξ) ∈ V is the Hamiltonian, J is the matrix of a symplectic
bilinear form on Ξ (Im is them×munit matrix). The
Hamiltonian h(ξ) is always a conservation law for Eq. (31).
In fact, Mx = {h, x}, so Mh = {h, h} = 0 where the
Poisson bracket
{x, y} = ⟨J∇x,∇y⟩
defines a Lie algebraic structure on the functional vector
space V . In some cases, the condition {h, x}=0 provides additional
conservation laws x of Eq. (31). Along with the associative
algebra generated by the usual product, the Lie algebra
with the Poisson bracket is also automorphic for the
dynamics of Eq. (30). This imparts the relevant Lie algebraic
structure to the space of solutions to Eq. (30). Here
M = ad(h) is an inner derivation, so in the Hamiltonian
case Eq. (30) x˙ = {h, x} is of the Lax type. The symplectic
structure makes the manifold Ξ a symplectic manifold. Any
symplectic manifold can be realised as an orbit of the coadjoint
representation of some Lie group [28]. Extensions related
to partial differential equations and quantum mechanics
are possible [9, 28].

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