IntroductionThe substantial interest to graded Lie algebras aroseabout 60 years ago, in the context of similarity between deformationsof complex-analytic structures on compact manifoldsand deformations of associative algebras and Lie algebras[1–4], in combination with the relevant cohomologicaltheories [5, 6]. In these algebras, the interplay of “even”and “odd” subspaces carrying skew-symmetric and symmetricmultiplication laws plays the crucial role. Later, the newinterest to these structures arose in theoretical physics, inthe context of supergauge symmetries relating particles ofbosonic and fermionic statistics. Although supersymmetryhas not been experimentally discovered, these studies stimulatedan interesting new mathematics [7–10].At the same time, the advent of the inverse scatteringmethod gave a boost to the studies of Lie groups and Lie algebrasin mathematical physics, in the context of integrabilityof nonlinear evolution equations. In such studies, the nonlineardynamics is encoded in the evolution under the “semilinear”Lax equations possessing trace polynomial integralsof motion or revealing the isospectrality of the evolving operators[11–16].In this work, the standard construction of the Lax equationson Lie algebras is extended to Lie superalgebras, theZ2-graded Lie algebras of supersymmetry. The extendedequations possess the canonical trace polynomial integrals ofmotion and so can be applied in a similar manner to nonlinearproblems. It is shown that the odd subspace admits extrapolynomial integrals of motion independent of the canonicalintegrals. The geometry of the relevant orbit spaces is studiedrevealing a nontrivial homological algebra. Thus, an algebraiclink is established between integrable evolution equations,supersymmetry and the deformation theory. This workcan be regarded as a continuation of the previous work by theauthor [17].It is assumed that the reader is familiar with the basicsof the theory of Lie groups and Lie algebras and their representationsas well as the basics of algebraic geometry andhomological algebra.1. Lie superalgebrasThe algebra of supersymmetry comes from theoreticalphysics as an attempt to combine into one unified theory twostatistically different types of particles, bosons and fermions.According to the method of second quantization, the (complexfinite-dimensional) vector state spaces of these two types areseparated by parity, the one being represented in the evenspace l0, the other in the odd space l1. To relate these spaces,one assumes that the same symmetry (connected) Lie groupG linearly acts on both spaces. The group actions are repre-Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru5sented by the group homomorphismsTk : G → GL(lk), k = 0, 1.The even action is assumed to be simply the adjoint action ofG, T0 = Ad, i.e. l0 is the Lie algebra of the Lie groupG. Thebilinear skew-symmetric bracket in l0[, ]0 : l0 × l0 → l0, [x, y]0 = −[y, x]0is the standard Lie bracket. The differential ad of T0ad : l0 → End(l0), ad(x)y = [x, y]0, x, y ∈ l0 (1)represents the adjoint linear action of l0 on itself. It is furtherassumed that the odd action T1 is tensorially intertwined withT0. This means that a symmetric bilinear bracket[, ]1 : l1 × l1 → l0, [x, y]1 = [y, x]1is defined on l1 with values in l0 such that[T1(g)x, T1(g)y]1 = T0(g)[x, y]1,x, y ∈ l1, g ∈ G. (2)Using the brackets [, ]k, k = 0, 1, and the differential ofthe action T1ρ : l0 → End(l1), (3)a bilinear bracket [, ] on the direct suml = l0 ⊕ l1can be defined as[x, y] =[x, y]0, x, y ∈ l0,[x, y]1, x, y ∈ l1,ρ(x)y, x ∈ l0, y ∈ l1.With this bracket, the graded vector space l becomes a(complex) Lie superalgebra, i.e., a Z2-graded algebra whosebracket satisfies the conditions[x, y] ⊆ lξ+η, [x, y] = −(−1)ξη[y, x],(−1)ξν[x, [y, z]] + (−1)ξη[y, [z, x]]++(−1)ην[z, [x, y]] = 0∀ x ∈ lξ, y ∈ lη, z ∈ lν, ξ, η, ν = 0, 1.(4)The skew-symmetry between l0 and l1 and the graded Jacobiidentity are externally imposed on x, y, z ∈ l1 (to naturallyextend the representation theory) while the rest of the conditionsfollow the intrinsic properties of the construction above.The combined action T = (T0, T1) of the Lie group Gon the Lie superalgebra l = (l0, l1) generates the structuralgroup of automorphisms of l,[T(g)x, T(g)y] = T(g)[x, y], x, y ∈ l, g ∈ G. (5)The differential (ad, ρ) of this action generates a representationof the Lie algebra l0 on l.2. RepresentationsRepresentations of Lie superalgebras are Lie superalgebrahomomorphismsϕ : l → L, ϕ([x, y]) = [ϕ(x), ϕ(y)]L (6)into operator Lie superalgebras L. The latter are constructedas follows. For a Z2-graded (complex finite-dimensional)vector spaceV = V 0 ⊕ V 1,let L0,L1 be the spaces of linear operators V → V of homogeneousdegrees 0,1. This means that operators from L0act on the grades while those from L1 permute the grades,L0V k ⊆ V k, k = 0, 1, L1V 0,1 ⊆ V 1,0.On the Z2-graded vector spaceL = L0 ⊕ L1define a bracket [, ]L by the rule[X, Y ]L = XY − (−1)ξηY X,X ∈ Lξ, Y ∈ Lη, ξ, η = 0, 1.(7)With this bracket, L is a Lie superalgebra (the graded Jacobiidentity follows from Eq. (7)). Representations of l are homomorphismsof Eqs. (6), (7) such thatϕ(lk) ⊆ Lk, k = 0, 1. (8)In particular, the restrictionsϕ0 = ϕ|l0 (9)to the even subspace are representations of the Lie algebral0.Nontrivial representations of Lie superalgebras alwaysexist. For example, the homomorphisml → Ider(l), x → ∂x ≡ [x, ·] (10)to the space of inner derivations of l satisfies the requirement.This representation generalizes the adjoint representation ofa Lie algebra. The existence of faithful representations (in amore general context of graded Lie algebras over commutativerings) has been proved in Ref. [3]. Each faithful representationof l0 (guaranteed by Ado’s theorem) can be extended toa faithful representation of l.3. Invariants and Lax equationsThe group action T on l admits a set of canonical invariants,the (complex) trace polynomial functions on lIs[ϕ](x) = Tr ([ϕ(x)]s) , x ∈ l (11)taken for any power s ⩾ 0 and any representation ϕ of l.In fact, according to Eq. (7), for X ∈ L0 and Y ∈ L, thebracket [X, Y ]L is the commutator of operators. Hence Tacts on operators of algebra representations by conjugationand so preserves the traces of their powers. The restrictions6Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ruof Eq. (9) generate the set of canonical invariants in the evensubspace,I0s [ϕ0](x) = Is[ϕ0](x), x ∈ l0. (12)These are the standard trace polynomial invariants generatedby the Lie bracket in l0.The intertwining of Eq. (2) enables extra invariants to bebuilt for the group action T1 on l1. Precisely, for any invariantf of the action T0, the functionI1[f](x) = f([x, x]), x ∈ l1, (13)is an invariant of the action T1. In fact, the mapw : l1 → l0, w(x) = [x, x] (14)gives a (nonlinear) intertwining of T1 with T0. Its compositionfw with any invariant f of T0 is an invariant of T1. Theinvariants given by Eq. (13) are called derived invariants.For each k = 0, 1, the canonical invariants Is[ϕ] ofEq. (11) are integrals of motion (conservation laws) of evolutionequations of the formdl/dt = [m, l], m ∈ l0, l ∈ lk (15)where m = m(t) is any time-independent or (continuously)time-dependent magnitude. In fact, for any m, including thecase where m depends on l, the trajectories of the solutionsto Eq. (15) in the subspaces lk belong to orbits of the group actionsTk determined by the initial values. Eqs. (15) are calledLax equations on the Lie superalgebra l.In the subspace l1, Eq. (15) is rewritten asdl/dt = ρ(m)l, l ∈ l1 (16)where ρ is the representation of l0 on l1 given by the differentialof the group action T1 (see Eq. (3)). This is a generalizationof the standard Lax equation on the Lie algebra l0 toanother representation subspace l1. Similar generalizations(outside the Lie superalgebras theory) have been considered,for example, in Ref. [16]. The derived invariants I1[f] givenby Eq. (13) are integrals of motion of Eq. (16) additional to thecanonical invariants.The property of the Lax equations to have the “m-universal”conservation laws is very useful. It enables one to integratenonlinear evolution equations (15) generated by any(continuous) dependences of m on l and t.4. Geometry of orbitsSince the invariants are integrals of motion, the trajectoriesof evolution under Eq. (15) belong to the intersections ofintegral surfaces, on which the invariants take constant valuesdetermined by the initial states. Each such intersectionis filled with orbits of the group action T on l. The form ofthe canonical invariants Is[ϕ] suggests their strong dependenceon the representations ϕ. The representations (on thesame vector space) are subdivided into equivalence classeswith respect to the canonical invariants,∃g ∈ G : ϕ′ = ϕT(g) −→ Is[ϕ′] = Is[ϕ].Besides this, it is hard to formulate anything general aboutthe integral surfaces created by the invariants Is[ϕ].The derived invariants I1[f] on the odd subspace l1 aredifferent. They are written as compositions of any T0-invariantwith the map w of Eq. (14) that is independent of representationsof l. By Eq. (5), we have[T1(g)x, T1(g)x] = T0(g)[x, x],∀ g ∈ G, x ∈ l1. (17)Hence, if the vector v = [x0, x0] ∈ l0 is fixed under thegroup action T0,T0(g)v = v ∀g ∈ G, (18)then the trajectory of the solution l(t) to Eq. (16) starting fromx0 is completely contained in the setSv = {x ∈ l1 : [x, x] = v}. (19)In fact, in this case, any constant function f is suitable for thederived invariant I1[f]. The space of the vectors v defined byEq. (18) is the zeroth cohomology group h0(G, l0) of the groupG with coefficients in l0. This space also forms the centre ofthe Lie algebra l0.For any v, the relation that defines the set Sv is quadraticallypolynomial with respect to the coordinates in l1, so theset Sv is an (affine) algebraic variety. By Hilbert’s Nullstellensatz,it is defined by the zero locus of a proper ideal in thepolynomial ring C[l1] containing these quadratic polynomials.There is an obvious link of Eq. (19) to the classical problemof intersections of quadrics. The variety Sv is symmetricunder the reflection with respect to the origin x → −x.It is non-compact in general: the homotheties v → λv,x →√λ x (λ ̸= 0) make the varieties Sv and Sλv isoomorphic.In the case v = 0, removing the trivial orbit x = 0, Svbecomes compact as a projective variety.The special property of the variety Sv of Eq. (19) is thatit lies in the intersection of integral surfaces of all canonicalpolynomial invariants passing through the point x0. In fact,for any representation ϕ of l and any x ∈ l1, in accordancewith Eqs. (6), (7), (8),ϕ([x, x]) = 2[ϕ(x)]2and any odd power of the operator ϕ(x) permutes the evenand odd subspaces and so has a zero trace. Hence, we obtainfor any integer s > 0 and any representation ϕI2s[ϕ](x) = 2−sTr ([ϕ([x, x])]s) ,I2s−1[ϕ](x) = 0, x ∈ l1.(20)According to Eqs. (17), (18), the set Sv is filled with orbitsof the group action T1 on l1. This generates the orbit spaceSv/G that classifies points of Sv. Two points belong to thesame equivalence class if they belong to the same orbit. Inthe case v ̸= 0, we will assume that the group action T1 isirreducible on l1.The classification problem Sv/G can be approached asfollows. For any x ∈ Sv, let ∂x be the inner derivation definedby the homomorphism of Eq. (10). In other words,∂xy = [x, y], x ∈ Sv, y ∈ l.Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ru7It follows from the Jacobi identity (see Eq. (4)) that2∂x∂xy = [v, y]. (21)Eq. (18) is equivalent to the condition[m, v] = 0 ∀m ∈ l0.By Eq. (21), this implies∂x∂xl0 = 0. (22)Also, we have[v, y] = [[y, y], y] = 0 ∀y ∈ Svwhere we again used the Jacobi identity. Hence, Sv is a subsetof the centralizer of v in l1. This centralizer isG-invariantbecause v is G-fixed. For v ̸= 0, we assumed that the T1-action is irreducible, so the whole l1 centralizes v (otherwise,there would exist a smaller invariant subspace of T1),[l1, v] = 0. (23)By Eqs. (21), (23), we conclude then that∂x∂xl1 = 0. (24)Combination of Eqs. (22), (24) gives∂x∂xl = 0.Considering the restrictions on the even and odd subspaces∂kx = ∂x|lk , k = 0, 1,we have on l∂1x∂0x = ∂0x∂1x = 0.This enables the Lie superalgebra l to be represented as the“loop” chain complexl0∂0x−↽−−−−⇀−∂1xl1with respect to the differential ∂x. Introducing the kernelsand images (the cycles and boundaries)Zkx = ker ∂kx, B0,1x = im ∂1,0x ,we assign to each point x ∈ Sv the even and odd homologygroups as the quotientsHkx = Zkx/Bkx, k = 0, 1. (25)The groups Hkx , Hkx′ are isomorphic if x, x′ belong to thesame G-orbit.Introducing the vector spacesZx = Z0x⊕ Z1x, Bx = B0x⊕ B1x, Hx = H0x⊕ H1x,we see that Zx is a Lie superalgebra that is a subalgebra ofl, Bx is an ideal in Zx and so Hx = Zx/Bx also becomes aLie superalgebra.The subspaces Z1x,B1xare respectively the tangent spaceto Sv and the tangent space to the orbit of x at the point x.If the odd homology group is trivial, H1x = 0, then the orbitof x covers a whole neighbourhood of the point x in Sv.All small deformations of x within Sv areG-orbit equivalent.Such points x are called rigid. For Z1x = 0 (for v ̸= 0), theset Sv consists of one point x (which in this case is fixed underthe group action, B1x= 0). If H1x̸= 0 then the orbit of xtends to lie strictly inside Sv.The subspace Z0x is the Lie subalgebra of l0 that centralizesx: [Z0x, x] = 0. The subspace B0xis the image of xunder the odd inner derivations: B0x= [l1, x]. By the Jacobiidentity and Eqs. (19), (23), it is a Lie subalgebra (actually anideal) of Z0x. If the even homology group is trivial, H0x = 0,then x is a simple point of Sv. In fact, let H0x = 0 and letx + u ∈ Sv be a deformation of the point x in Sv. Then usatisfies the deformation equation2∂1xu + [u, u] = 0. (26)We can write the solution to Eq. (26) as a formal power seriesu = zu1 + z2u2 + . . . (27)in some (complex) scalar parameter z. The first term2∂1xu1 = 0 −→ u1 ∈ Z1xcan be chosen arbitrarily. To find the higher terms, the followinginduction can be applied. Let the first q terms be known.Then they satisfy the equations2∂1xur + Jr = 0,Jr =Σr−1p=1[up, ur−p], r = 1, . . . , q.(28)To find the (q + 1)th term, the following equation should besolved2∂1xuq+1 + Jq+1 = 0. (29)Letu(q) =Σqr=1zrurbe the qth partial sum. Using the Jacobi identity, we have[x + u(q), [x + u(q), x + u(q)]] = 0.Taking the (q+1)th power of z, with the use of Eqs. (23), (24),we obtain[x, Jq+1] +Σqr=1[uq+1−r, 2[x, ur] + Jr] = 0.By Eqs. (28), this gives∂0xJq+1 = 0.This means that Jq+1 ∈ Z0x and so Jq+1 ∈ B0xbecause weassumed H0x = 0. Then Eq. (29) can be resolved for uq+1,uniquely if we take the zero projection to Z1x. Hence, all theterms of the power series of Eq. (27) can be uniquely found.This series converges for any u1 as long as |z| is sufficientlysmall. We obtain that the point x ∈ Sv can be analyticallydeformed within Sv in any direction given by the space Z1x oftangent vectors to Sv. Thus, for H0x = 0, the point x is simple.A structure of a complex manifold on Sv can be definedin a neighbourhood of x. The situation is very similar to thatdescribed in Ref. [1].8Известия Коми научного центра Уральского отделения Российской академии наук № 5 (71), 2024Серия «Физико-математические науки»www.izvestia.komisc.ruThe consideration of dimensions gives the following relationsdimHkx = dim lk −Σp=0,1dimBpx,dimZkx = dim lk − dimB(k+1)x , k = 0, 1.(30)Here in addition to Eq. (25) we used the isomorphismsB0,1x≃ l1,0/Z1,0x .Also, since v ∈ B0x⊆ Z0x, we obtainv ̸= 0 −→ dimZ0x ⩾ dimB0x⩾ 1 (31)(in particular, if Sv ̸= ∅ for v ̸= 0 then the group action T1on l1 cannot be free). It immediately follows from Eq. (30) thatif the subspaces l0,1 are not isomorphic, dim l1 ̸= dim l0(i.e., the representation spaces of ad and ρ are not isomorphicas vector spaces), then the groups H0,1x are not simultaneouslytrivial and are not isomorphic, dimH1x̸= dimH0x.This is valid for each point x ∈ Sv. This means, for instance,that neither point x ∈ Sv can be simultaneouslya simple point of the variety Sv and have its orbit coveringthe whole neighbourhood of x in Mv. In particular, fordim l1 ̸= dim l0, the variety Sv cannot be a (nontrivialsmooth) homogeneous space of the G-action.Eqs. (30), (31) enable an estimation of possible orbitclasses in the space Sv/G to be made. The existence of functionson l1 that separate orbits in Sv and their links to thehomology on Sv are interesting open questions. In addition toEq. (20), note that, for the “adjoint representation” of Eq. (10),the canonical integrals take the zero values on Sv,ϕ(x) = ∂x −→ Is[ϕ](x) = 0, x ∈ Sv, s > 0.In fact, for x ∈ Sv, we have ∂2x = 0, so the operator ∂x isnilpotent and its all positive powers have a zero trace.5. ConclusionWe have shown that the well-known construction of theLax equations on Lie algebras can be extended to Lie superalgebras,important in mathematics and theoretical physicsin their relation to the deformation theory and supersymmetry.Like the usual Lax equations, the extended ones admit thecanonical trace polynomial integrals of motion which can beused in the integrability theory for nonlinear evolution equations.Besides the canonical integrals, the extra set of derivedintegrals occurs in the odd subspace, as a result of tensorialintertwining with the even subspace. This new feature is dueto the symmetric character of multiplication within the oddsubspace. The orbit spaces generated by constant values ofthe derived integrals [x, x] = v, where v belongs to the 0thcohomology group of the underlying Lie group action, possessthe natural (co)homological structure with respect to the innerderivations ∂x. This structure is generically nontrivial,giving obstacles for the integral surfaces to be locally homogeneousspaces. These results algebraically relate the integrabilitytheory of evolution equations with supersymmetryand the deformation theory.The future work can be focused on possible connectionsof the orbit space Sv/G with the “intrinsic properties” of thealgebraic variety Sv independent of its embedding into theodd subspace l1 (say, in the spirit of the Zariski and Mumfordtheories). An extension of the described algebraic structuresto the general graded Lie algebras should be possiblein terms of their natural grading into the even and odd subspaces.From the point of view of physical applications, it canbe interesting to relate the above constructions to integrablenonlinear dynamics and supersymmetry (for instance, to connectEqs. (19), (26) to symplectic geometry and Hamiltoniandynamics as well as to the extended supersymmetry theory,say, for the Poincaré algebra). Possible relations of the deformationEq. (26) to the Maurer-Cartan formalism and thegauge theories can also be interesting.The author declares no conflict of interest.