Automorphic algebrasof dynamical systems and generalisedIn¨on¨u-Wigner contractionsАвтоморфные алгебрыдинамических систем и обобщенныеконтракции Иненю-ВигнераA. Karabanov А. КарабановCryogenic Ltd,London, W3 7QE, UKkarabanov@hotmail.co.ukООО «Криогеника»,г. Лондон, W3 7QE, Великобританияkarabanov@hotmail.co.ukAbstractLie algebras a with a complex underlying vector space V arestudied that are automorphic with respect to a given lineardynamical system on V , i.e., a 1-parameter subgroup Gt ⊂Aut(a) ⊂ GL(V ). Each automorphic algebra imparts aLie algebraic structure to the vector space of trajectories ofthe group Gt. The basics of the general structure of automorphicalgebras a are described in terms of the eigenspacedecomposition of the operatorM ∈ der(a) that determinesthe dynamics. Symmetries encoded by the presence of nonabelianautomorphic algebras are pointed out connected toconservation laws, spectral relations and root systems. It isshown that, for a given dynamics Gt, automorphic algebrascan be found via a limit transition in the space of Lie algebrason V along the trajectories of the group Gt itself. This proceduregeneralises the well-known Inönü-Wigner contractionand links adjoint representations of automorphic algebras toisospectral Lax representations on gl(V ). These results canbe applied to physically important symmetry groups and theirrepresentations, including classical and relativistic mechanics,open quantum dynamics and nonlinear evolution equations.Simple examples are given.АннотацияИзучаются 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-алгебраическуюструктуру векторному пространству траекторий группыGt. 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 известнуюконтракцию Иненю-Вигнера и связывает присоединенныепредставления автоморфных алгебр с изоспектральнымипредставлениями 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 динамикуи нелинейные эволюционные уравнения. Приведены про-стые примеры.Keywords:automorphic algebras, dynamical systems, generalisedIn¨on¨u-Wigner contractionsКлючевые слова:автоморфные алгебры, динамические системы, обобщен-ные контракции Иненю-ВигнераIntroductionLie groups and Lie algebras are a powerful mathematicaltool that has a variety of physical applications. The local propertiesof a Lie group are described in terms of its Lie algebra.Lie algebras have also applications, fully separate from Liegroups. This makes the theory of Lie algebras independentlyuseful. Finite-dimensional complex and real semisimple Liealgebras and their representations are fully classified [1–3].The modern theory of Lie algebras mostly concerns infinite-dimensional generalisations (with links to modern problemsof theoretical physics [4]) and geometric extensions(with links to algebraic groups and algebraic topology [5,6]). Automorphisms of Lie algebras (and adjacent algebraicstructures) describe the algebra symmetries and so play animportant role in the theory. Normally, the direct problemis tackled, i.e., the problem of finding the group of automorphismsof a given Lie algebra. We address here the inverseproblem — the problem of description of Lie algebras thathave a given group of automorphisms.The inverse problem is quite useful and intricate as well,even in the finite-dimensional case. For instance, 1-parametersubgroups of automorphisms of the algebra are equivalentto linear dynamical systems on the underlying vector space.Известия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ru 5These dynamical systems should possess certain symmetriesfor the algebra to have a non-abelian structure. Hence, thereis a close link between algebras, automorphic under the dynamics,and symmetries of dynamical systems. Linear dynamicalsystems find many physical applications. They formthe basis of such an important field as quantum mechanicsand underly the modern methods of integration of nonlineardynamical systems.In these notes, we study the general properties of Lie algebrasthat have a given linear dynamical system as its 1-parameter group of automorphisms. We call such algebrasautomorphic algebras of the dynamics. We study symmetriesof the dynamics encoded by non-abelian automorphic algebrasand their description in terms of a special limit transitionin the space of Lie algebras along the dynamical trajectories.The latter relates automorphic algebras to the wellknownInönü-Wigner contraction and isospectral Lax representations.We give a few simple examples related to applicationsof this theory to classical matrix groups and nonlinearevolution equations. These connections make automorphicalgebras a worth developing mathematical tool, usefulin the theory of both dynamical systems and Lie algebras.We assume that the reader is familiar with the basics ofthe Lie algebraic and group theory, for example, within theclassical books [1–3].1. Automorphic algebras and symmetriesLet V be a finite-dimensional complex vector space. Leta dynamic system Gt be given on V as a smooth 1-parametergroup of linear transformations. In other words, Gt is asmooth representation of the additive group of real numberson V :Gt : R → GL(V ), GtGs = Gt+s,G0 = id, G−1t = G−t.The exponential mapGt = etM, M =ddtGtt=0identifies the trajectories{x(t) = Gtx(0)} ⊂ Vwith the solutions to the linear differential equationx˙ = Mx, x ∈ V, (1)to which Gt is a fundamental matrix. Note that Gt is alwaysa subgroup of the general linear Lie group GL(V ) (of alltransformations/automorphisms of V ) and the operator Mbelongs to its Lie algebra gl(V ) (of all endomorphisms of V ).The groupGt can be a 1-parameter subgroup of a smaller Liegroup withinGL(V ), and thenM belongs to the relevant Liealgebra. Note also that the parameter t does not have to playthe role of the time in the usual physical sense, but can be amore general evolution variable.We additionally assume that the vector space V isequipped with a Lie algebraic structure with the bracket [, ]that satisfies the standard conditions of bilinearity, skewsymmetricityand the Jacobi identity. We denote the correspondingLie algebra as a.In this work we study the case where the group Gt actson 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 systemGt. By differentiation with respect to t, Eq. (2) is equivalentto 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 systemsthat have the same automorphic algebra a with the Lie algebrader(a) of all derivations of a. The groups Gt span thenthe identity component Aut(a)0 of the group Aut(a) of allautomorphisms of a. The group Aut(a) is a Lie group withthe Lie algebra der(a). Due to the Jacobi identity, the algebrader(a) has a subalgebra (in fact, an ideal) ider(a) of innerderivations written via the adjoint representation of a asM = ad(y) = [y, ·], y ∈ a. Each inner derivation generatesa dynamical system Gt that belongs to the group ofinner automorphisms Inn(a) ⊆ Aut(a)0. The correspondingEq. (1) is of the form x˙ = [y, x]. For matrix/operatoralgebras this form corresponds to Lax equations. The latterare an important tool in the theory of nonlinear integrablesystems and quantum mechanics [7–10]. We will encounterLax equations again later when we consider semisimple automorphicsubalgebras and limit transitions along the grouptrajectories. Note that the semisimple and nilpotent parts (inthe sense of the Jordan decomposition) of any derivation Mare 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, abelianalgebras are automorphic for any dynamical system. Eachautomorphic algebra has the same 1-parameter group Gtof its automorphisms. Below we describe the basic generalproperties of automorphic algebras (i.e., the properties commonfor all automorphic algebras) of a given dynamical systemand show that they encode an important information onits symmetries.By an immediate observation, we come to the followingconsequence of Eq. (3) and the bilinearity of the bracket [, ].Proposition 1. For any automorphic algebra, the bracketof any two solutions to Eq. (1) is again a solution to Eq. (1),x˙ = Mx, y˙ = My −→ ddt[x, y] = M[x, y]. (4)In terms of Proposition 1, automorphic algebras impart analgebraic structure to the vector space of solutions to Eq. (1)(trajectories of the group Gt). Non-abelian automorphic algebrasenable new solutions to Eq. (1) to be generated fromknown solutions that generically are not linear combinationsof the latter.It follows from the definition that any subalgebra a0 ofan automorphic algebra a that is invariant under the dynam-6Известия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ruics,Ma0 ⊆ a0, is an automorphic algebra of the restrictionGta0. Eq. (3) and the classical results of Refs. [12, 13] immediatelyimply the following statement.Proposition 2. For any automorphic algebra a, its center,its derived algebra, its radical and its nilradical are invariantunder 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]. Thefollowing embeddings take place z ⊆ nil ⊆ rad. In terms ofProposition 2, automorphic algebras enable to reduce the solutionsto Eq. (1) into smaller invariant subspaces, giving thespace of solutions additional structural properties.More automorphic subalgebras can be constructed andthe further analysis can be carried out in terms of theeigenspaces of the operatorM, as shown below.Let E be the set of (distinct) eigenvalues λ of the operatorM and μλ denote the algebraic multiplicity of λ. Considerthe decomposition of the vector space V into (generalised)eigenspaces ofMV =Mλ∈EVλ, Vλ = {x ∈ V : (λ−M)μλx = 0}. (6)Eqs. (3), (6) imply the following classical result proved inRef. [11].Proposition 3. Any automorphic algebra admits the gradedstructure∀λ, η ∈ E [Vλ, Vη] ⊆ Vλ+η, (7)where we assume Vξ = 0 if ξ ̸∈ E.Multiplication of the operatorM by any nonzero complexnumber c = |c|eiθ homogeneously dilates and simultaneouslyrotates all the eigenvalues of M with respect to theorigin on the complex plane. This however does not changeautomorphic algebras. Hence, the theory of automorphic algebrasis invariant under the group of homotheties and rotationsof the complex plane of eigenvalues with respect to theorigin (the latter is a subgroup of the Möbius group of conformaltransformations of the complex plane). In particular, itfollows from Eq. (7) that, for any line that crosses the originon the complex plane, the sets of eigenvalues that belong tothis line, lie on one side of the line and lie on the opposite sideof 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. Thenon-degeneracy/degeneracy of M is equivalent to the nonexistence/existence of non-trivial conservation laws (“integralsof motion”), i.e., non-zero elements x ∈ V such thatMx = 0, Gtx = x ∀t.The classical result proved in Ref. [14] implies that in thenon-degenerate case all automorphic algebras are nilpotent.This is a consequence of Eq. (7) and the fact that the conditiondetM ̸= 0 implies the nilpotence of all adjoint representationsad(Vλ) and so (by a generalised Engel’s theorem) thenilpotence of a [14].Proposition 4. If 0 ̸∈ E then all automorphic algebras arenilpotent.Corrollary. The existence of non-nilpotent automorphicalgebras implies the existence of nontrivial conservationlaws.A more detailed structure of nilpotent automorphic algebrasin the non-degenerate case can be enlightened withinthe following definition.Definition 1. An eigenvalue λ ∈ E is called resonant ifλ + η − ξ = 0 for some η, ξ ∈ E. Otherwise, λ is callednon-resonant.Proposition 5. For any automorphic algebra a, if λ is nonresonantthen the relevant eigenspace belongs to the centreof a, Vλ ⊆ z(a). If all eigenvalues are non-resonant then allautomorphic algebras are abelian.Proof. Let λ be non-resonant. Then λ + η − ξ ̸= 0 forall η, ξ ∈ E. By Eq. (7), this implies [Vλ, Vη] = 0 for allη ∈ E, as λ + η cannot be an eigenvalue. Then, for any automorphicalgebra a, we obtain [Vλ, a] = 0, i.e., Vλ ⊆ z(a).If all eigenvalues are non-resonant then all eigenspaces Vλbelong to the centre and a is abelian. □Since Vλ is invariant under Gt, it is an automorphicabelian subalgebra of a for any non-resonant λ.A simple example of applicability of Proposition 5 is thecase where Gt is a 1-parameter subgroup of an irreduciblerepresentation of the (complexified) group SO(3) on V . Thegroup SO(3) of the rotations of the Eucledian 3-space is animportant group in physics, closely connected, for example, tothe special unitary and special linear groups SU(2), SL(2)as well as the Möbius group of conformal transformations ofthe complex plane. In this case, any operatorM is the relevantrepresentation of an element of the algebra so(3). Thelatter can be treated as the algebra of quantum angular momentumoperators [15]. Then there exists a basis of V suchthatM = α diag (−S, −S + 1, . . . , S − 1, S), (8)where α is some complex number, S is a positive integer orhalf-integer spin number that characterises the dimension ofthe representation, dim V = 2S +1. For even-dimensionalvector spaces V , the spin number S is half-integer correspondingto fermionic representations. For odd-dimensionalV , the spin number S is integer and corresponds to bosonicrepresentations. For fermionic representations, since S ishalf-integer, assuming α ̸= 0, it follows from Eq. (8) thatthe operator M is non-degenerate and the sum of any twoeigenvalues of M is not an eigenvalue, i.e., by Definition 1,all eigenvalues are non-resonant. By Proposition 5, all automorphicalgebras of any (nontrivial) dynamics generated byfermionic representations of so(3) are abelian. We will returnto this example later when we consider the bosonic case.In terms of Proposition 5, for the existence of non-abelianautomorphic algebras it is necessary that the resonance conditionλ + η − ξ = 0 is satisfied for some λ, η, ξ ∈ E. Inthe non-degenerate case, it means that the operator M hasИзвестия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ru 7at least two distinct eigenvalues. Since 2-dimensional nonabelianalgebras are non-nilpotent, we obtain then dim a =dim V ⩾ 3. The minimal example is the 3-dimensionalHeisenberg algebra h3[vλ, vη] = vξ, [vξ, vλ] = [vξ, vη] = 0.Here λ, η are resonant and ξ is non-resonant, sospan {vξ} = z and the centre is invariant under M inaccordance with Proposition 2.Nilpotent algebras are constructed as successive centralextensions of abelian algebras, so any nilpotent algebra alwayshas a nontrivial centre. Nilpotent algebras are solvable.All subalgebras and homomorphic images of nilpotent algebrasare nilpotent. The Killing form on nilpotent algebras iszero. The adjoint representations of nilpotent algebras consistof nilpotent operators. Nilpotent algebras have outer automorphismsand outer derivations. So far, no general approachhas 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 areresonant. Eq. (7) immediately implies the following result.Proposition 6. If 0 ∈ E then the subspace V0 is anonzero subalgebra of any automorphic algebra that containsa nonzero subalgebra ¯ V0 = {x ∈ V0 : Mx = 0} of conservationlaws. The adjoint representation of the subalgebra ¯ V0acts on the space of solutions to Eq. (1): for any solution y(t)to Eq. (1) within any automorphic algebra, the linear transformationy′(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 automorphicsubalgebras. The operator M is nilpotent on V0, so therestrictionGtV0is polynomial in t. Eq. (9) is a partial case ofEq. (4) that shows that, besides their conservative character,within automorphic algebras, nontrivial conservation laws ofEq. (1) play an important role in the structure of solutions.To extend the result for the non-degenerate case to thedegenerate case, it is natural to consider automorphic algebrasas extensions of algebras that contain V0 by nilpotentideals. This can be done as follows.Definition 2. The set E of eigenvalues of the operator Mis 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 ∈ ¯ Eeither 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 automorphicalgebra is a semidirect suma = a0 + ¯a,a0 =Mλ∈E0Vλ, ¯a =Mλ∈¯ EVλ,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 automorphicalgebra. We have 0 ∈ E0, so V0 is a subalgebra of a0.Since 0 ̸∈ ¯ E, by Proposition 4, the ideal ¯a is nilpotent, asit is invariant under the operator M (forming then an automorphicsubalgebra) where this operator is non-degenerate.The subalgebra a0 acts on ¯a by derivations, so the short exactsequence¯a −→ a −→ a0defines a split extension of a0, i.e., the semidirect sum a =a0 + ¯a. □In terms of Proposition 7, both a0, ¯a are invariant underthe dynamics and so a0, ¯a are automorphic subalgebras.Proposition 5 can be applied then to the ideal ¯a in terms ofDefinition 1 and the set ¯ E — to specify the centre of ¯a. If severalsplittings of E exist, from the point of view of the Levidecomposition, in Proposition 7 the splitting with the minimalpossible subset E0 should be chosen.Since ¯a ̸= 0, under the condition of Proposition 7, all automorphicalgebras are non-semisimple. The important casewhere Proposition 7 is directly applicable is the semidissipativecase∀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 semidirectsum of a subalgebra and a nilpotent ideala = a0 + ¯a,a0 =MRe λ=0Vλ, ¯a =MReλ<0Vλ. (12)Due to the invariance under rotations of eigenvalues with respectto the origin (see the comments after Proposition 3), thesame situation occurs where the eigenvalues of the operatorM are split into eigenvalues that belong to one side of a linethat crosses the origin and eigenvalues that belong to thisline.We aim now to describe the situations where there existnon-solvable automorphic algebras, i.e., automorphic algebraswith semisimple subalgebras. This is closely connectedto projections of root systems of semisimple Lie algebras tothe complex plane of eigenvalues of the operatorM. The rootsystems of semisimple complex Lie algebras are fully classified[1–3].8Известия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ruDefinition 3. A set of complex numbers ρ(g) ⊂ C iscalled a root projection for a complex semisimple Lie algebrag if there exists an element m ∈ h of a maximal toralsubalgebra 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 complexLie algebra g exist such that ρ(g) ⊂ E. Let μ′λ ⩾ ¯μλif 0 ̸= λ ∈ ρ(g). Let μ′0 ⩾ r + ¯μ0 if 0 ∈ ρ(g) wherer = rank g. Here μ′λ is the geometric multiplicity of λ inE and ¯μλ is the multiplicity of λ in ρ(g). Then there existsan automorphic algebra a with a semisimple subalgebra a0isomorphic to g.Proof. Consider the vector spacea0 = ¯ V0 ⊕Mλ∈ρ(g)¯ Vλ,¯ V0 =Mrk=1v(k)0⊆ V0, ¯ Vλ =Mμ¯λq=1v(q)λ⊆ Vλ,where v(k)0 , v(q)λ are eigenvectors of the operatorM correspondingrespectively to the eigenvalues 0 and λ. Such eigenvectorsexist by the conditions imposed on the multiplicitiesof the eigenvalues. By Eq. (7) and the root space decompositionof g, the vector space a0 is a semisimple automorphicsubalgebra of an automorphic algebra a with the restrictionM0 ≡ Ma0= ad(m). To get an automorphic algebraa on the full space V , it is sufficient to consider the trivialextension of the subalgebra a0 by the abelian subalgebrah = V \ a0. □In terms of Proposition 8, the semisimple subalgebra a0is invariant under the dynamics, so it is an automorphic subalgebra.The set ρ(g) is centrally symmetric with respect tothe origin on the complex plane. This implies TrM0 = 0.Hence, for the restriction G0t= Gta0we obtain detG0t=exp(tTrM0) = 1 ∀t and G0tbelongs then to the speciallinear Lie group SL(a0). As a result, G0tpreserves the volumeand orientation of the vector space a0. Note also that therestrictionM0 is always a semisimple operator and an innerderivation of a0. The restriction of Eq. (1) on a0 is of the Laxtype x˙ = [m, x].The simplest example of applicability of Proposition 8 isthe case 0, ±α ∈ E with an arbitrary complex number α.In this case, there exists an automorphic subalgebra isomorphicto A1 = sl(2) = so(3). For instance, getting back toEq. (8), this situation is realised for bosonic representationsof the algebra so(3) where the spin number S is integer andso any operator M has the eigenvalues 0, ±α. By Proposition8, unlike the case of fermionic representations where allautomorphic algebras are abelian (see the comments afterProposition 5), any dynamics generated by bosonic representationsof so(3) has an automorphic subalgebra isomorphicto so(3) and so has a non-solvable automorphic algebra. ByPropositions 5 and 8, a strict algebraic difference exists betweenfermions 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) automorphicsubalgebras exist, isomorphic to sl(2).The lowest-dimensional bosonic case S = 1 correspondsto the standard 3-dimensional representation of so(3). Inthis case, as a consequence of the fact {0, ±α} = E andEq. (7), the dynamics Gt generated by any operator M fromthis representation (plane uniform rotations around a fixedcoordinate line) has three non-abelian automorphic algebrasthat 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 theeigenvalues 0, ±α. These algebras are respectively simple(so(3)), nilpotent (the Heisenberg algebra h3) and non-nilpotentsolvable (the Euclidean algebra e(2)).At the end of this section, we point out that the spectralproblem (6) for the operatorM on V generates a symmetricspectral problem for the operator ad(M) on gl(V ) in termsof the adjoint representations of automorphic algebras.Proposition 9. For any automorphic algebra a, for eachλ ∈ E and each v ∈ Vλ, the adjoint representation ad(v) ofthe 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 asad(Mx)−[M, ad(x)] = ad(Mx)−ad(M)ad(x) = 0.For v ∈ Vλ, we have (λ−M)μλv = 0. Utilizing the Jordanform of M on Vλ, we can choose a basis {v1, . . . , vμλ} inVλ such thatMvk = λvk +kX−1s=1cksvs, k = 1, . . . , μλfor some complex constants cks. This impliesad(Mvk) = λ ad(vk) +kX−1s=1cksad(vs)and we come to the fact that the operators ad(vk) all satisfyEq. (13). Precisely,(λ − ad(M))kad(vk) = 0, k = 1, . . . , μλ. □In terms of decomposition (6) and Proposition 9, if the operatorMis semisimple on Vλ then cks = 0 and both spectralproblems (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 9and Eqs. (13), (14) reduce the procedure of finding automorphicalgebras to the usual linear algebra.Известия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ru 92. Automorphic algebras and generalisedInönü-Wigner contractionsIn this section we show that automorphic algebras of agiven dynamical system Gt can be produced from non-automorphicalgebras by a special limit transition along the trajectoriesof Gt. This limit procedure generalises the wellknownInönü-Wigner contraction [16] that finds a variety ofphysical applications [10, 17–23].Let an algebra Lie a with a bracket [, ] (generically nonautomorphicfor Gt) be given on the vector space V . Considerthe 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-symmetricityand Jacobi identity of the bracket [, ] of the algebraa. Hence, each bracket [, ]t defines a Lie algebra at on V .Proposition 10. Let for all x, y ∈ V there exist the finitelimit (in the standard topology of the vector space V = Cn)[x, y]′= limt→+∞[x, y]t. (16)Then the limit algebra a′ with the bracket [, ]′ is automorphicfor Gt.Proof. By differentiation of Eq. (15) with respect to t, weget for all t, x, yddt[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 andright-hand sides of Eq. (17) to vanish at t → +∞. The operatorM becomes a derivation of the limit algebra a′. Thelatter is then automorphic for the dynamical system Gt. □Due to the relation [Gtx,Gty] = Gt[x, y]t, for each finitet, the intermediate algebra at is isomorphic to a. If a isautomorphic for Gt then [x, y]t = [x, y] is independent of tand the intermediate algebras at and the limiting algebra a′all coincide with a. Otherwise, the limit a′ is a Lie algebra thatis (in general) not isomorphic to a, although a and a′ have thesame 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 thegraded structure of Eq. (7) to[Vλ, Vη] ⊂MVξ,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 automorphicalgebras of a dynamical system Gt in a self-consistentway, as limit cases of any algebras, satisfying Eq. (18), alongthe trajectories of the groupGt itself. Similarly to Eq. (16), wecan consider the limit[x, y]′− = limt→−∞[x, y]t. (19)Provided the latter exists, we again come to a new Lie algebraa′− that is automorphic forGt. The two limits (16), (19) aremutually connected by inversion of the signs of the eigenvaluesof M. If both limits (16), (19) simultaneously exist thena′ = a′−.For semidissipative dynamical systems (11), the procedure(15), (16) is equivalent to the Inönü-Wigner contraction. In thiscase, the vector space V shrinks (contracts) along the trajectoriesof Gt. As per Proposition 7 and Eq. (12), limit algebrasof the Inönü-Wigner contraction are always non-semisimple.They are split extensions of the subalgebra a0 spanned by theeigenspaces ofM corresponding to purely imaginary eigenvaluesby the nilpotent ideal ¯a spanned by the eigenspacesofM corresponding to eigenvalues with negative real parts.The restriction of Eq. (15) onto a0 is either compact for all tor has terms that polynomially grow with t, so for the limit(16) to exist, the bracket [x, y]t should be independent of t forx, y ∈ a0. Then the limit algebra keeps the initial bracket ona0. As a result, there exists the homomorphismG′: a′ → a, kerG′= ¯a, fixG′= a0 (20)that realises the aforementioned split extension (the short exactsequence)¯a −→ a′ −→ a0.For example, in the original setting [16, 17], the Inönü-Wigner contraction corresponds to the caseMx = 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 subalgebraand 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 toany, not only semidissipative dynamics satisfying Eq. (18). UnlikeEq. (20), we do not require the limit algebra to be a splitextension of a nonzero algebra. The limit algebra a′ can besemisimple in certain cases where the initial algebra a issemisimple.To give a simple example, qualitatively different fromEqs. (21), (22), consider the 3-dimensional algebra a spannedby vectors v−, v0, v+ withMvξ = 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. Takingthe 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 simpleLie algebra. The aforedefined operator M is a derivation of10Известия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ruthe new algebra, so a′ is automorphic forGt. This is howevernot 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 theeigenvalue 0 and the two non-zero eigenvalues ±λ generatedby the set of roots for sl(2).In the example (23), (24), the limit algebra a′ = sl(2) isisomorphic to the initial one a. Examples of non-dissipativedynamics generating non-isomorphic limit algebras also canbe easily given. For example, it is sufficient to modify Eq. (23)asMvξ = 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 isindeed automorphic. Here the initial algebra is isomorphic tosl(2) while the limit algebra (isomorphic to the Lie algebrae(2) of the Euclidean group E(2)) is solvable and so nonisomorphicto sl(2).Example (25), (26) also illustrates Proposition 7. ChoosingE0 = {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 coincideswith the derived algebra, ¯a = [a′, a′]. This illustrates Proposition2, as ¯a is indeed invariant underGt. In accordance withPropositions 6, 7, the 1-dimensional subspace spanned by v0is a subalgebra that contains conservation laws.As the final result of this study, we point out that the spectralproblem (14) for semisimple operatorsM can be linked tothe limit procedure (15), (16) via a Lax representation in gl(V ).In fact, in terms of the adjoint representation adt(vλ) in theintermediate algebras at, Eq. (17) is recast asddtadt(vλ) = (λ − ad(M))adt(vλ). (27)We used the fact thatM is semisimple, so the eigenspace Vλis split into a set of eigenvectors vλ, thus splitting the spectralproblem (13) into Eq. (14). Eq. (27) easily implies the followingresult.Proposition 11. The operatorLλ(t) = e−λtadt(vλ) ∈ gl(V ) (28)satisfies the isospectral Lax representationddtLλ(t) = [Lλ(t),M]. (29)In particular, the eigenvalues of Lλ(t) and analytical functionsof them are conservation laws of Eq. (29).In terms of proposition 11, the limit (15), (16) is equivalentto the limit along the trajectories of Eq. (27)ad0(vλ) → ad′(vλ), t → +∞,where ad0(vλ), ad′(vλ) are the adjoint representations inthe initial and the limit algebras a, a′. The trajectories arefound by the transformation (28) and the Lax representationof Eq. (29). It is worth mentioning that the adjoint representationsadt(v0) corresponding to conservation laws of Eq. (1)directly satisfy the Lax representation of Eq. (29) without thetransformation of Eq. (28). Proposition 11 implies the followingstatement.Proposition 12. Let the finite limit (15), (16) exist. Then, forRe λ ⩾ 0, λ ̸= 0, the operators adt(vλ) are nilpotent forall t. For Re λ < 0 the limit operator ad′(vλ) is nilpotent.The eigenvalues of the operator adt(v0) that corresponds toλ = 0 (and so all their analytical functions) are conservationlaws of Eqs. (27), (29).Proof. By Proposition 11, the eigenvalues of the operatorLλ(t) of Eq. (28) are conservation laws of Eq. (29). Thenany eigenvalue αt(λ) of the operator adt(vλ) has the formαt(λ) = eλtα0(λ), where α0(λ) is an eigenvalue of theoperator 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 nilpotentfor all t. If Re λ < 0 then αt(λ) → 0, t → +∞, i.e.,the limit operator ad′(vλ) is nilpotent. For λ = 0, we haveL0(t) = adt(v0), so the eigenvalues of adt(v0) are conservationlaws of Eqs. (27), (29). □It follows from Proposition 12 that, for any λ ̸= 0, the adjointrepresentation ad′(vλ) in the limit algebra a′ is a nilpotentoperator. In fact, it follows from Eq. (7) that, for any automorphicalgebra, for all v ∈ Vλ with λ ̸= 0, the operatorad(v) is nilpotent (the condition of semisimplicity of M canbe lifted). Remarkably, the conservation laws v0 of Eq. (1)on automorphic algebras on V generate conservation lawsTr [adt(v0)m] of Eq. (29) on the algebra gl(V ).Propositions 10-12 along with Proposition 9 illustrate theremarkable algebraic role of the limit procedure (15), (16) fordescription of adjoint representations of automorphic algebras.ConclusionAutomorphic Lie algebras of linear dynamical systemshave been introduced as Lie algebraic structures on the spaceof their trajectories. We have formulated the basic generalproperties of automorphic Lie algebras of a given dynamicalsystem in terms of the eigenspace decomposition of thedynamics. We have pointed out the symmetries that are encodedby the presence of non-abelian automorphic algebras.In particular, non-nilpotent automorphic algebras are relatedto conservation laws of the dynamics. In the presence of asemisimple automorphic subalgebra, there is a natural correspondencebetween the set of roots related to the subalgebrato the set of eigenvalues of the dynamical system. We haveshown that automorphic algebras can be found by a limit transitionalong the trajectories of the dynamics, a procedure thatИзвестия Коми научного центра УрО РАН, серия «Физико-математические науки» № 5 (57), 2022www.izvestia.komisc.ru 11generalises the well-known Inönü-Wigner contraction. Wehave demonstrated that, in terms of the adjoint representation,the limit transition is naturally reduced to an isospectralLax representation. We have given simple examples relatedto applications of the developed theory to classical matrixgroups. This suggests that automorphic algebras are worthdeveloping tool in the theory of both dynamical systems andLie algebras.Inönü-Wigner contractions, in the dynamical setting ofEqs. (15), (16), have been applied before to dissipative dynamicalsystems [20, 22, 23]. Other dynamical deformations of Liealgebras, both related and unrelated to Inönü-Wigner contractions,have also been discussed [24, 25]. We are unawareof whether the automorphic character of limit algebras hasever been noticed. We are also unaware of an earlier use ofnon-dissipative dynamical systems for constructing non-isomorphicalgebras on the same vector space. As far as weknow, the link of the limit transition of the Inönü-Wigner typeto the Lax representations has not been made before.As direct physical applications of the methodology developedin this work, we would expect first of all cases wherethe groups Gt have additional special properties: for example,belong to various physically important symmetry groupsand their representations. Among them, we can find classicaland relativistic mechanics (the classical matrix groups,the Galilean, Lorentz and Poincaré groups [26]) and quantumapplications such as, for example, Lindblad equations ofopen quantum dynamics (completely positive quantum semigroups[27]). Considering finite and discrete groups that generatediscrete 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 , especiallyfunctional spaces, or finite-dimensional Lie algebras overfunctional rings. In the latter cases, it might be expected thatthe linear systems (1), the graded structures (7) and the Laxrepresentations (29) are related to some integrable nonlinearevolution equations of mathematical physics [7–10]. Accordingto Proposition 4, for an open set of dynamical systems, allautomorphic algebras are nilpotent. Nilpotent algebras playan important role in the representation theory, especially inthe orbit method and geometric quantisation [28]. It would becurious to build links of these modern theories to the theoryof automorphic algebras we developed.Some of the results we obtained can be reformulated forarbitrary (not necessarily Lie) algebras, making this subjectuseful in a wider algebraic context.To give one simple example in relation to the last twoparagraphs, consider the vector space V of smooth complexfunctions x : Ξ → C given on a smooth real manifold Ξ ofa dimension n with (local) coordinates ξ = (ξ1, . . . , ξn).Let a smooth vector field F(ξ) = (F1(ξ), . . . , Fn(ξ)) begiven on Ξ and let Eq. (1) be generated by the operatorM ofdifferentiation along the field F:x˙ = Mx ≡ 〈F,∇x〉 =Xnk=1Fk∂x∂ξk. (30)Then the solutions x(t, ξ) are given by evolution of the initialvalue x(0, ξ) = x0(ξ) along the flow on Ξ generated by thevector field F: x(t, ξ) = x0(η(t, ξ)) whereη˙ = F(η), η(0, ξ) = ξ (31)(the group Gt is a realisation of such evolution). The operatorM is a differentiation “from the left”, so M is a derivationof the associative algebra on V generated by the usualproduct x, y → xy. This algebra is automorphic for the dynamicsof Eq. (30). In particular, the product of any two solutionsx(t, ξ)y(t, ξ) is again a solution. The conservationlawsMx(ξ) = 0 of Eq. (30) are in a one-to-one correspondencewith the conservation laws 〈F,∇x〉 = 0 of Eq. (31). Insome cases, this observation helps to find conservation lawsfor the nonlinear dynamics of Eq. (31) from the linear dynamicsof Eq. (30).The classical case is the Hamiltonian dynamics where themanifold Ξ is even-dimensional, n = 2m, and F is a Hamiltonianvector field:F(ξ) = J∇h(ξ), J =0 Im−Im 0.Here h(ξ) ∈ V is the Hamiltonian, J is the matrix of a symplecticbilinear form on Ξ (Im is them×munit matrix). TheHamiltonian h(ξ) is always a conservation law for Eq. (31).In fact, Mx = {h, x}, so Mh = {h, h} = 0 where thePoisson bracket{x, y} = 〈J∇x,∇y〉defines a Lie algebraic structure on the functional vectorspace V . In some cases, the condition {h, x}=0 provides additionalconservation laws x of Eq. (31). Along with the associativealgebra generated by the usual product, the Lie algebrawith the Poisson bracket is also automorphic for thedynamics of Eq. (30). This imparts the relevant Lie algebraicstructure to the space of solutions to Eq. (30). HereM = ad(h) is an inner derivation, so in the Hamiltoniancase Eq. (30) x˙ = {h, x} is of the Lax type. The symplecticstructure makes the manifold Ξ a symplectic manifold. Anysymplectic manifold can be realised as an orbit of the coadjointrepresentation of some Lie group [28]. Extensions relatedto partial differential equations and quantum mechanicsare possible [9, 28].