IntroductionThe idea of symmetry and conservation laws is fundamentalin natural sciences. Mathematically, it is reduced to thestudy of algebraic properties that are invariant under groupsof transformations. From this point of view, linear objects aremuch simpler and more symmetric than nonlinear ones. Forinstance, nonlinear dynamical systems generally do not admitconservation laws (integrals of the motion) and manifoldsof their solutions are much harder to describe than those oflinear dynamical systems that always are linear spaces. It isvery tempting then to reduce nonlinear dynamical problemsto linear problems.The most remarkable success in this direction is the inversescattering method of integration of nonlinear evolutionequations. The method is based on including the nonlinearevolution into a linear operator L that satisfies a linear evolutionequation dL/dt = [M,L] such that the eigenvectorsof L satisfy the linear equation with an operator M, whilethe eigenvalues of L do not evolve. The latter property enablesa reconstruction of the nonlinear evolution using a spatialscattering theory for the operator L. For ordinary differentialequations, the isospectrality ofLis used to find conservationlaws of the nonlinear dynamics. The pairs (M,L) arecalled Lax pairs, the equations for the operator L are calledLax equations [1-7].In many cases, useful nonlinear relations exist betweensolutions to linear dynamical systems. These relations shedextra light to solutions of the relevant nonlinear problems.The simplest nonlinear extension of a linear operator is a multilinearoperator. In this work, we realise this idea in replacingthe Lax operator L by a multilinear operator that mapssolutions to the linear problem with the operatorM again tosolutions to the same problem. We call the resulting equationstensor extensions of the Lax equations.We show that the extensions thus introduced have a richalgebraic meaning, closely related to the theory of Lie algebrasand more general multilinear algebraic structures. Wereveal that the extensions we suggest are partial cases of thegeneralised Lax equations introduced by Bordemann and relatedto representations of Lie algebras other than the adjointrepresentation, on which the usual Lax equations arebased [8]. Close connections between the solutions to the extendedLax equations and Chevalley-Eilenberg cochain complexes[9, 10] are pointed out. Also, the basic constructionpresented in this work is another language for description ofisomorphic deformations of multilinear algebraic structureson vector spaces with respect to dynamical groups of transformations.In this sense, this work is a continuation of theprevious work by the author [11].Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ru51. Basic constructionLet V be a complex vector space and let Vk denote thevector space of k-linear operators L : V k → V . For anyk = 1, 2, . . ., any linear evolution equation on Vdv/dt = Mv, v ∈ V, M(t) ∈ V1 (1)generates the linear evolution equation on VkdL/dt = ρk(M)L, L∈Vk, ρk(M(t)) ∈ End(Vk) (2)such that the solution operator L is a k-symmetry of Eq. (1),i.e., maps k-tuples(v1(t), . . . , vk(t))of solutions to Eq. (1) again to solutions to Eq. (1). By multiplicationwith respect to t, we can verify thatρk(M)L(v1, . . . , vk) == ML(v1, . . . , vk) − L(Mv1, v2, . . . , vk)−−L(v1,Mv2, . . . , vk)−. . .−L(v1, v2, . . . ,Mvk). (3)It is evident thatρ1(M)L = [M,L], M,L ∈ V1,where [, ] denotes the commutator, so for k = 1 Eq. (2) is theusual Lax equation. Using the canonical injectionV k → Tk(V )of the Cartesian product V k into the k-grade of the tensoralgebra T(V ), due to the universal property of T(V ), anysolution L ∈ Vk to Eq. (2) can be uniquely identified witha linear operator ¯L : Tk(V ) → V . We call the series ofEq. (2), k = 2, 3, . . ., tensor extensions of the Lax equationfor k = 1.2. Isomorphic deformations and conservationlawsEq. (3) enables the solutions to Eqs. (2) for any k and anyinitial operator L(0) ∈ Vk to be written in the formL(t)(v1, . . . , vk) = Φ(t)L(0)(Φ−1(t)v1, . . . ,Φ−1(t)vk)(4)where Φ(t) ∈ V1 is the operator that maps any vector v ∈ Vto the solution ¯v(t) = Φ(t)v to Eq. (1) with the initial value¯v(0) = v,dΦ/dt = MΦ, Φ(0) = 1 ∈ V1. (5)Due to Eqs. (4), (5), solutions L(t) of the extended Lax equations(2) are k-multiplicative algebraic structures on V thatare isomorphic to their initial values L(0) under the evolutionof Eq. (1). Eqs. (2) describe then isomorphic deformationsof k-multiplicative algebraic structures on V . In fact, Eq. (4)is equivalent toL(t)(Φ(t)v1, . . . , Φ(t)vk) = Φ(t)L(0)(v1, . . . , vk).(6)For finite values of time, the fundamental operator Φ(t) ofEq. (1) is an isomorphism between L(0) and L(t). By the action(6), the group generated by the operators Φ(t)G = gen{Φ(t), t ∈ R} ⊂ GL(V ) (7)maps any structure L(0) to structures isomorphic to L(0).Stationary solutions L(t) = L(0) to Eqs. (2) that do notexplicitly depend on time describe k-multiplicative structuresthat are automorphic with respect to the operators Φ(t) forall values of t,L(0)(Φ(t)v1, . . . , Φ(t)vk) = Φ(t)L(0)(v1, . . . , vk).The group G of Eq. (7) is then a subgroup of the automorphismsgroup of L(0),G ⊂ Aut(L(0)).For k > 1, evolutions under Eq. (1) on the vector spaceV generate symmetries of the stationary solutions to Eq. (2)as multiplicative k-linear algebraic structures on V . On theother hand, by definition, solutions to Eq. (2) are k-symmetriesof Eq. (1) as they map k-tuples of solutions to Eq. (1) againto solutions to Eq. (1) . We can say that Eqs. (1), (2) describemutual symmetries of the extended Lax pair (M,L).The operator ρk(M) : Vk → Vk as a linear function ofM defined by Eq. (3) has the property[ρk(M), ρk(N)] = ρk([M,N]) ∀M,N ∈ V1.Hence, the linear mapρk : V1 → End(Vk)is a representation of the general Lie algebra V1 = gl(V ) onthe vector space Vk, i.e., a Lie algebra homomorphismρk : gl(V ) → gl(Vk). (8)Thus, each Eq. (2) is a partial case of the generalised Laxequation suggested by Bordemann [8]. For the usual Laxequation k = 1, we have ρ1 = ad is the adjoint representation.By exponentiation, the representation ρk of the Lie algebragl(V ) generates the linear action (representation) ¯ρk ofthe general Lie group GL(V ) on the same space Vk. Thenany scalar function f : Vk → C invariant under this action,f(¯ρk(m)L) = f(L), ∀ m ∈ GL(V ), L ∈ Vk, (9)is a conservation law for Eq. (2), i.e., the values f(L(t))are time-independent and do not change along the solutionsL(t). It is directly seen by differentiation of Eq. (9) by m atthe identity element e of the group GL(V ) and the fact thatM belongs to the tangent space TeGL(V ). For k = 1 anda finite-dimensional vector space V , we haveρ1 = ad, ¯ρ1(m)L = mLm−1and the trace polynomial functionsfn(L) = Tr (Ln), n = 1, 2, . . . ,are conservation laws for the usual Lax equation.6Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ruIn fact, functions f satisfying Eq. (9) are conservation lawsfor Eq. (2) with any operator M. For k > 1, the explicit descriptionand even existence of such functions is a nontrivialproblem even if V is a finite-dimensional vector space. The“isospectrality” of Eq. (2) is closely related to symmetries ofthe operator M and manifests itself in the following observations.Let V have a finite dimension N and a basis v1, . . . , vN.Any initial operator L(0) ∈ Vk is defined by its values on thebasic vectors of the tensor k-grade Tk(V ),L(0)(vi1 , . . . , vik ) =XNs=1λ(s)i1...ikvs, (10)where the indices i1, . . . , ik independently take all valuesfrom the set {1, . . . ,N} and λ(s)i1...ikare complex coefficients,the “structure constants” of the multiplicative algebraicstructure L(0). The solution L(t) to Eq. (2) with theinitial value L(0) has the propertyL(t)(¯vi1(t), . . . , ¯vik (t)) =XNs=1λ(s)i1...ik¯vs(t), (11)where ¯vj(t) are the solutions to Eq. (1) with the initial valuesvj and the coefficients λ(s)i1...ikremain time-independent.This directly follows from Eq. (6) for any k. This does not mean(even for k = 1) that the structure constants of the initial operatorL(0) are conservation laws for the solution L(t). Infact, according to Eq. (4),L(t)(vi1 , . . . , vik ) =XNs=1¯λ(s)i1...ik(t)vs == Φ(t)L(0)(Φ−1(t)vi1 , . . . ,Φ−1(t)vik ).The expansion of the initial value by Eq. (10) generates theevolutionλ(s)i1...ik→ ¯λ(s)i1...ik(t) (12)of the structure constants of L(0) to those of L(t). This evolutionis another characteristic of the isomorphism betweenL(0) and L(t).The special case where Eq. (2) is explicitly solved iswhere the operator M is time-independent and the basisv1, . . . , vN is composed of eigenvectors ofM with eigenvaluesm1, . . . ,mN. In this case, Φ(t) = etM and the groupG defined in Eq. (7) is a 1-parameter subgroup of GL(V ):Φ(t + s) = Φ(t)Φ(s). According to Eq. (4), the evolution(12) takes the simple form¯λ(s)i1...ik(t) = etϕ(s)i1...ik λ(s)i1...ik,ϕ(s)i1...ik= ms −Xkp=1mip.(13)It follows from Eq. (13) that the structural constants of the initialoperator L(0) that satisfy the conditionϕ(s)i1...ikλ(s)i1...ik= 0do not change under the evolution L(t), i.e., are conservationlaws of Eq. (2). In particular, the zero structural constantsare always conserved. A nonzero structural constant λ(s)i1...ikis conserved if the “resonance” ϕ(s)i1...ik= 0 takes place betweenthe eigenvalues m1, . . . ,mN of the operatorM.The stationary solutions L(0) to Eq. (2) that are automorphicwith respect to the groupG are defined then by the conditionϕ(s)i1...ikλ(s)i1...ik= 0 ∀s, i1, . . . , ik.It follows, for instance, that if all the eigenvalues are “nonresonant”ϕ(s)i1...ik̸= 0 ∀s, i1, . . . , ikthen all stationary solutions to Eq. (2) are trivial L(0) = 0.Note that the case k = 2 with skew-symmetric bilinearoperators L corresponds to Lie algebraic structures if additionallythe Jacobi identity is satisfied. The finite limit transitions¯λ(s)i1i2(t) → ˜λ(s)i1i2 , t → ±∞,are closely related to Inönü-Wigner contractions and lead tostationary solutions to Eq. (2), automorphic with respect tothe “dynamical” group G. This situation has been consideredin more detail in the previous work by the author [11].For k = 1 (regardless of whetherM is time-independentor not), eigenvectors of the operator L(t) ∈ V1 that evolvesunder the usual Lax equation are solutions to Eq. (1) and therelevant eigenvalues are time-independent (being eigenvaluesof the initial operator L(0)). This underlies the inversescattering method of integration of nonlinear evolution equations[1-7].3. Exterior algebras and cochain symmetriesIt can be verified that, for any k = 1, 2, . . ., if L′ ∈ V1and L ∈ Vk are solutions to Eq. (2) then the operator compositionL′L ∈ Vk is also a solution to Eq. (2). In this sense, theleft multiplication by the solutions to the usual Lax equationis a symmetry of the extended Lax equations (2).For anyM and any k, the operator ρk(M) : Vk → Vk isinvariant under the action of the symmetric group Sk on Vk.For any permutation σ ∈ Sk of the indices 1, . . . , k,ρk(M)σ(L) = σ(ρk(M)L),σ(L)(v1, . . . , vk) ≡ L(vσ(1), . . . , vσ(k)).(14)Thus, Sk is a symmetry group for Eq. (2). For any solutionL(t) and any permutation σ ∈ Sk, the “braided” operatorσ(L(t)) is also a solution.This symmetry and the idea of considering only k-tuplesof linearly independent solutions to Eq. (1) leads to the restrictionfrom the infinite-dimensional tensor algebra T(V ) to thefinite-dimensional exterior (Grassmann) algebraV(V ) thatis a quotient of the tensor algebra with respect to the leftrightideal generated by the tensors of the form v ⊗ v. Interms of Eq. (2), it means that only alternating k-linear operatorsL are to be considered, i.e., those withσ(L) = sgn(σ)L, σ ∈ Sk.Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ru7The vector spaces Vk will denote now the vector spaces ofalternating operators L : V k → V . Each such operator canbe identified with a linear operator from the k-grade of theexterior algebra, ¯L :Vk(V ) → V . We assume that thevector space V is finite-dimensional, dim V = N.The construction related to the representation (8) can beextended to a representation of any Lie algebra. In fact, leta be a Lie algebra and letρ : a → gl(V ) (15)be its representation on V . Then the compositionπk = ρkρ : a → gl(Vk)is a representation of a on Vk. The extended Lax equations(2) are written then asdL/dt = πk(a)L, a ∈ a.Let now the underlying vector space of the Lie algebraa be V and the representation ρ in Eq. (15) be the adjointrepresentation. This enables the Chevalley-Eilenberg cochaincomplex to be built,V −→δ V1−→δ V2−→δ . . . −→δ VNwhere δ : Vk−1 → Vk, δ2 = 0, is the exterior derivative(δL)(v1, . . . , vk) ==Xks=1(−1)s+1ρ(vs)L(v1, . . . , ˆvs, . . . , vk)++Xs<s′(−1)s+s′L([vs, vs′ ], v1, . . . , ˆvs, . . . , ˆvs′ , . . . vk),k > 1, (δL)v = ρ(v)L, L ∈ V.Here [, ] is the Lie bracket in a and the hat means that the relevantvariable should be omitted [9,10]. The solutions L ∈ Vkto Eq. (2) are then naturally identified with (time-dependent)k-cochains of this complex.It can be verified that the exterior derivative δ is a symmetryof the set of the extended Lax equations (2). In fact,if L ∈ Vk is a solution in the k-grade then δL ∈ Vk+1 isa solution in the next (k + 1)-grade. We call this symmetrycochain symmetry. In the case of the exterior algebra,according to Eqs. (4), (5), for k = 1,N the extended Laxequations (2) are solved explicitly asL(t) = Φ(t)L(0)Φ−1(t), k = 1,L(t) =Φ(t)det Φ(t)L(0), k = N.4. ConclusionIt has been demonstrated that the classical Lax equations,important in the integrability theory and quantum dynamics,can be extended in a manner closely related to symmetriesof multilinear algebraic structures and representations of Liealgebras other than the adjoint.