Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
aa r X i v : . [ m a t h - ph ] F e b Darboux families and the classification of real four-dimensionalindecomposable coboundary Lie bialgebras
J. de Lucas and D. Wysocki
Abstract
This work introduces a new concept, the so-called
Darboux family , which is employed to determine, to analysegeometrically, and to classify up to Lie algebra automorphisms, in a relatively easy manner, coboundary Liebialgebras on real four-dimensional indecomposable Lie algebras. The Darboux family notion can be consideredas a generalisation of the Darboux polynomial for a vector field. The classification of r -matrices and solutionsto classical Yang-Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail.Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a methodto obtain matrix representations of certain Lie algebras with a non-trivial center is developed. Lie bialgebras [15, 17, 18, 29] appeared as a tool to study integrable systems [20, 21]. A
Lie bialgebra is a Lie algebra g along with a Lie bracket on its dual space g ∗ that amounts to a cocycle in a Chevalley-Eilenberg cohomology of g . Lie bialgebras have also applications to quantum gravity [4, 8, 38, 39] and other research fields [15, 29, 43]. Indifferential geometry, they occur in the problem of classifying Poisson Lie groups [3, 15].Although much research has been devoted to the classification of Lie bialgebras up to Lie algebra automorphisms,there are still many open problems. More specifically, Lie bialgebras on two- and three-dimensional Lie algebras g have been completely classified [22, 25]. Particular instances of Lie bialgebras with dim g >
3, e.g. for a semi-simple g , have also been studied [2, 11, 14, 30, 36, 43, 44]. Employed techniques are rarely non-algebraic (cf. [35]) andthey are not very effective to analyse Lie bialgebras when dim g > coboundary Lie bialgebra is a particular type of Lie bialgebra that is characterised by means of an r - matrix ,namely a bivector on a Lie algebra g that is a solution to its modified classical Yang-Baxter equation (mCYBE)[15, 25]. This work introduces novel geometric techniques to classify, up to Lie algebra automorphisms, coboundaryLie bialgebras and their r -matrices on a fixed Lie algebra g .In particular, we introduce a generalisation, the so-called Darboux family , of the notion of Darboux polynomialfor a polynomial vector field. On a finite-dimensional vector space E , a polynomial function is a polynomialexpression on a set of linear coordinates on E . A Darboux polynomial , P , for a polynomial vector field X on E isa polynomial function on E so that XP = f P for a certain polynomial f on E , the so-called cofactor of P relativeto X [16, 33]. It is worth noting that, geometrically, one cannot define intrinsically what a polynomial on a generalmanifold is, as the explicit form of a function on a manifold depends on the chosen coordinate system.More generally, let us consider a q -dimensional Lie algebra of vector fields, let us say V := h X , . . . , X q i , on amanifold M . A Darboux family for V is an s -dimensional vector space A := h f , . . . , f s i of smooth functions on M such that X α f β = P sγ =1 g αβγ f γ for certain smooth functions g αβγ ∈ C ∞ ( M ) with α = 1 , . . . , q and β, γ = 1 , . . . , s .In this work, Darboux families are studied and employed to classify up to Lie algebra automorphisms the r -matricesand coboundary Lie bialgebras on real four-dimensional Lie algebras that are indecomposable [50], i.e. they cannotbe written as a direct sum of two proper ideals, namely ideals that are different of the zero and the total Lie algebra.Nevertheless, our methods can also be applied to any other Lie algebra.Let Aut( g ) stand for the Lie group of Lie algebra automorphisms of a Lie algebra g . Then, Aut( g ) naturally actson the space Λ g of bivectors [1] of g and, more specifically, on the space Y g ⊂ Λ g of solutions to the mCYBE on g (see [15, 35]). The classes of equivalent r -matrices (up to Lie algebra automorphisms of g ) are given by the orbitsof the action of Aut( g ) on Y g . Characterising and analysing such orbits is in general complicated even when theform of Aut( g ) is explicitly known (see [22] for a typical algebraic approach to Lie bialgebras on three-dimensionalLie algebras). Most techniques in the literature are algebraic [22, 25]. Instead, we use here a more geometricalapproach. Let us sketch our main ideas. We use a simple method (see Proposition 2.2 and Remark 2.1) to determine1he Lie algebra of fundamental vector fields, V g , of the action of Aut( g ) on Λ g . Our technique does not requireto know the explicit form of Aut( g ). Instead, it is enough to know the space of derivations on g , which can bederived by solving a linear system of algebraic equations. We show that the orbits of the connected part of Aut( g )containing its neutral element, Aut c ( g ), are the integral connected submanifolds of the generalised distribution , E g ,spanned by the vector fields of V g (see [27, 31, 41, 48, 49, 51] for general results on generalised distributions). Infact, as V g is a finite-dimensional Lie algebra, E g is integrable [31]. Finding the integral connected submanifolds ofa generalised distribution, its so-called strata , is more complicated than finding the leaves of standard distributionsbecause strata cannot always be determined by a family of common first-integrals for the elements of V g , as ithappens in the case of distributions (cf. [1, 28]). We determine them here via our Darboux families, which isrelatively easy as illustrated by our examples. Once the strata of E g in Y g have been obtained, the use of the actionon such strata of an element of each connected part of Aut( g ) allows us to determine the orbits of Aut( g ) on Y g ,which gives us the desired classification of r -matrices. Note that our method do rely on the determination of asingle element of each connected component of Aut( g ) and, more relevantly, it does not need the explicit action ofthe whole Aut( g ) on Λ g .It is known that two r -matrices for g that are not equivalent up to a Lie algebra automorphism may give rise tocoboundary Lie bialgebras that are equivalent up to a Lie algebra automorphism (cf. [15, 35]). Let (Λ g ) g be thespace of bivectors of g that are invariant relative to the action of elements of g via the algebraic Schouten bracket[15, 35]. In our work, we prove that the orbits of Aut( g ) on Y g that project onto the same orbit of the naturalaction of Aut( g ) on Λ g / (Λ g ) g (see [35]) are exactly the r -matrices that lead to equivalent (up to Lie algebraautomorphisms) coboundary Lie bialgebras on g .Our methods are computationally affordable for the study and classification of general coboundary Lie bialgebraswith three- and four-dimensional g . Our techniques are quite probably appropriate for looking into and classifyingLie bialgebras on any five-dimensional g and, possibly, for other particular instances of higher-dimensional Liealgebras g . Indeed, our procedures lead to the classification of real coboundary Lie bialgebras, up to Lie algebraautomorphisms, on any four-dimensional indecomposable Lie algebra g (see [50] for a classification of indecomposableLie algebras up to dimension six). Our results are summarised in Table 2. This is a remarkable advance relative toother classification works in the literature [2, 22, 25]. Indeed, the most complicated classification in the literatureso far is probably the classification of Lie bialgebras on symplectic four-dimensional Lie algebras [2]. Additionally,other four-dimensional Lie bialgebras have been partially studied in the mathematics and physics literature (see[4, 6, 7, 10, 12, 11, 30, 43, 44] and references therein). As a byproduct of our research, a method for the matrixrepresentation of a class of finite-dimensional Lie algebras with a non-trivial center, which cannot be representedthrough the matrices of the adjoint representation, is given. This is interesting not only for our purposes, but alsoin many other works where such a representation is employed in practical calculations, e.g. [5].The structure of the paper goes as follows. Section 2 surveys the main notions on Lie bialgebras and theirderivations, g -modules, and the notation to be used. In Section 3, the theory of generalised distributions is discussed.A method to obtain a matrix representation of a class of Lie algebras with non-trivial center is given in Section 4.Section 5 introduces Darboux families and shows how they can be used to classify r -matrices on the Lie algebra s , (according to ˇSnobl and Winternitz’s notation in [50]), up to Lie algebra automorphisms thereof. Section 6 analysesseveral geometric properties of solutions to mCYBEs and it also provides some hints on the use Darboux familiesto study the equivalence up to Lie algebra automorphisms of r -matrices and coboundary Lie bialgebras. In Section7, Darboux families are applied to studying and classifying coboundary Lie bialgebras on real four-dimensionalindecomposable Lie algebras. Section 8 summarises our results and presents some further work in progress. Let us provide a brief account on the notions of Lie bialgebras, Schouten brackets, and g -modules to be used hereafter(see [15, 29, 51] for further details). Some simple results on the geometric properties of the action of Aut( g ) on spacesof k -vectors are provided. Our approach is more geometric than in standard works on Lie bialgebras. We hereafterassume that g and E are a finite-dimensional real Lie algebra and a finite-dimensional real vector space, respectively.Meanwhile, GL ( E ) and gl ( E ) stand for the Lie group of automorphisms and the Lie algebra of endomorphisms on E , respectively.Let V m M be the vector space of m -vector fields on a manifold M . The Schouten-Nijenhuis bracket [37, 51] on V M := ⊕ m ∈ Z V m M is the unique bilinear map [ · , · ] : V M × V M → V M satisfying that[ f, g ] = 0 , X ∧ . . . ∧ X s , f ] := ι df X ∧ . . . ∧ X s := s X i =1 ( − i +1 ( X i f ) X ∧ . . . ∧ c X i ∧ . . . ∧ X s , [ X ∧ . . . ∧ X s , Y ∧ . . . ∧ Y l ]:= X i =1 ,...,sj =1 ,...,l ( − i + j [ X i , Y j ] ∧ X ∧ . . . ∧ b X i ∧ . . . ∧ X s ∧ Y ∧ . . . ∧ b Y j ∧ . . . ∧ Y l , (2.1)where f, g ∈ C ∞ ( M ), s, l ∈ N , the X , . . . , X s , Y , . . . , Y l are arbitrary vector fields on M , an omitted vector field X is denoted by the hat symbol b X , and [ X i , Y j ] is the Lie bracket of X i and Y j (see [37] for more details). Remarkably,[ X , Y ] ∈ V s + l − for X ∈ V s M and Y ∈ V l M . The space, V L G , of left-invariant elements of V G for a Lie group G is closed relative to [ · , · ]. In particular, left-invariant vector fields on G , i.e. V L G , span a finite-dimensionalLie algebra called the Lie algebra of G , which can be identified with T e G [1]. Vice versa, every abstract finite-dimensional Lie algebra g can be thought of as the Lie algebra of left-invariant vector fields of a Lie group [19].Meanwhile, V L G can be identified with the Grassmann algebra Λ g , namely the algebra relative to the exteriorproduct spanned by all the multivectors of the Lie algebra, g , of G [26]. Moreover, [ · , · ] can be restricted to V L G leading to the algebraic Schouten bracket on Λ g [35, 51]. For simplicity, we will call it the Schouten bracket on Λ g .Recall that the Grassmann algebra Λ E of a vector space E satisfies that Λ E = L m ∈ Z Λ m E , where Λ m E is thevector space of m -vectors of E .A Lie bialgebra is a pair ( g , δ ), where g admits a Lie bracket [ · , · ] g , whilst δ : g → Λ g , the cocommutator , is alinear map, its transpose δ ∗ : Λ g ∗ → g ∗ is a Lie bracket on g ∗ , and δ ([ v , v ] g ) = [ v , δ ( v )] + [ δ ( v ) , v ] , ∀ v , v ∈ g . (2.2)A Lie bialgebra homomorphism is a Lie algebra homomorphism φ : g → h between Lie bialgebras ( g , δ g ) and( h , δ h ) such that ( φ ⊗ φ ) ◦ δ g = δ h ◦ φ . A coboundary Lie bialgebra is a Lie bialgebra ( g , δ r ) such that δ r ( v ) := [ v, r ]for every v ∈ g and some r ∈ Λ g , a so-called r -matrix . To characterise r -matrices, we use the following notionsand Theorem 2.1. The standard identification of an abstract Lie algebra g with the Lie algebra of left-invariantvector fields on a Lie group G allows us to understand the tensor algebra T ( g ) of g as the tensor algebra, T L ( G ),of left-invariant tensor fields on G . This gives rise to a Lie algebra representation ad : v ∈ g ad v ∈ gl ( T ( g )),where ad v ( w ) := L v w for every w ∈ T ( g ) and L v w is the Lie derivative of w relative to v , which are understood aselements in T L ( G ) in the natural way. Note that the geometric notation L v w is conciser than algebraic ones (cf.[15]). An element q ∈ T ( g ) is called g -invariant if L v q = 0 for all v ∈ g . We write ( T ( g )) g for the set of g -invariantelements of T ( g ). The map ad : g → gl ( T ( g )) admits a restriction ad : g → gl (Λ m g ). We recall that (Λ m g ) g standsfor the space of g -invariant m -vectors. Let us recall the following well-known result. Theorem 2.1.
The map δ r : v ∈ g [ v, r ] ∈ Λ g , for r ∈ Λ g , is a cocommutator if and only if [ r, r ] ∈ (Λ g ) g . We call [ r, r ] ∈ (Λ g ) g the modified classical Yang-Baxter equation (mCYBE) of g , while [ r, r ] = 0 is referred toas the classical Yang-Baxter equation (CYBE) of g and its solutions amount to left-invariant Poisson bivectors onany Lie group G with Lie algebra isomorphic to g [51].Note that two r -matrices r , r ∈ Λ g satisfy that δ r = δ r if and only if r − r ∈ (Λ g ) g . Then, what reallymatters to the determination of coboundary Lie bialgebras is not r -matrices, but their equivalence classes in thequotient space Λ R g := Λ g / (Λ g ) g .A g -module is a pair ( V, ρ ), where ρ : v ∈ g ρ v ∈ gl ( V ) is a Lie algebra morphism. A g -module ( V, ρ ) willbe represented just by V , while ρ v ( x ), for any v ∈ g and x ∈ V , will be written simply as vx if ρ is understoodfrom the context. If ad : v ∈ g [ v, · ] g ∈ gl ( g ) stands for the adjoint representation of g , then the fact thateach [ v, · ] g , with v ∈ g , is a derivation of the Lie algebra g (relative to its Lie bracket [ · , · ] g ) allows us to ensurethat ( g , ad) is a g -module [24]. The map ad can be considered as a mapping of the form ad : g → der ( g ), where der ( g ) is the Lie algebra of derivations on g . As a second relevant example of g -module, consider the Lie groupAut( g ) of the Lie algebra automorphisms of g (see [45] for details on its Lie group structure) and its Lie algebra,which is denoted by aut ( g ). The tangent map at the identity map on g , let us say id g ∈ Aut( g ), to the injection ι : Aut( g ) ֒ → GL ( g ) induces a Lie algebra morphism c ad : aut ( g ) ≃ T id g Aut( g ) → gl ( g ) ≃ T id g GL ( g ) and ( g , c ad)becomes an aut ( g )-module.In view of the properties of the algebraic Schouten bracket, each g gives rise to a g -module (Λ g , ad), wheread : v ∈ g [ v, · ] ∈ gl (Λ g ) (cf. [51]). This fact can be viewed as a consequence of [35, Proposition 2.1]. Tograsp this result and related ones, recall that every T ∈ gl ( E ) gives rise to the mappings Λ m T : λ ∈ Λ m E ∈ Λ m E for m ≤
0, and the maps Λ m T ∈ gl (Λ m E ), for m >
0, given by the restriction to Λ m E of Λ m T := We denote similar structures with the same symbol, but their meaning is clear from the context. ⊗ id ⊗ . . . ⊗ id( m − operators) + . . . + id ⊗ . . . ⊗ id ⊗ T ( m − operators) , where id is the identity on E . Moreover,Λ T := L m ∈ Z Λ m T ∈ gl (Λ E ). If T is considered as an element of GL ( E ), we define Λ m T := T ⊗ . . . ⊗ T ( m − operators)for m ≥ m T is the identity on Λ m E for m ≤
0. Finally, Λ T := L m ∈ Z Λ m T .The Lie group Aut( g ) gives rise to a Lie group action T ∈ Aut( g ) Λ m T ∈ GL (Λ m g ). Moreover, this givesrise to an infinitesimal Lie group action ρ : aut ( g ) → gl (Λ m g ) such that[ ρ ( d )]( w ) := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 [Λ m exp( td )]( w ) = (Λ m d )( w ) , ∀ d ∈ aut ( g ) , ∀ w ∈ Λ m g . We write V g for the Lie algebra of fundamental vector fields of this Lie group action for m = 2. Recall that aut ( g )is indeed the space of derivations on g (cf. [35]). The following result was proved in [35]. Proposition 2.1.
The dimension of the orbit O w of the action of Inn( g ) on Λ m g through w ∈ Λ m g is dim Im Θ mw , where Θ mw : v ∈ inn ( g ) [ v, w ] ∈ Λ m g . In this work, we will use the following rather straightforward generalisation of Proposition 2.1.
Proposition 2.2.
The dimension of the orbit O w of the action of Aut( g ) on Λ m g through w ∈ Λ m g is dim Im Υ mw , where Υ mw : d ∈ der ( g ) (Λ m d )( w ) ∈ Λ m g .Proof. The orbit of w ∈ Λ m g relative to Aut( g ) is given by the points (Λ m T )( w ) for every T ∈ Aut( g ). Defineexp( td ) =: T t , with t ∈ R , for d ∈ der ( g ). Then, the tangent space at w of O w is spanned by the tangent vectors ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 [Λ m exp( td )]( w ) = (Λ m d )( w ) ∈ T w Λ m g ≃ Λ m g . (2.3)Then, dim O w = dim der ( g ) − dim g w , where G w is the isotropy group of w ∈ Λ m g relative to the action of Aut( g )and g w is the Lie algebra of G w . Moreover, g w is given by those d ∈ der ( g ) such that (Λ m d )( w ) = 0. This amountsto d ∈ ker Υ mw . Hence, dim O w = dim der ( g ) − dim g w = dim Im Υ mw . Remark 2.1.
As a byproduct, Proposition 2.2 shows that the fundamental vector fields of the natural Lie groupaction of Aut( g ) on Λ g are spanned by X v ( w ) = P si =1 [Υ w ] i ∂ x i , where x , . . . , x s is any linear coordinate systemon Λ g , the w is any point in Λ g , and v belongs to der ( g ). Moreover, the coordinates of such vector fields are thecoordinates of the extensions to Λ g of the derivations of g , which can easily be obtained as the solutions, d , to thelinear problem d ([ e , e ]) = [ d ( e ) , e ] + [ e , d ( e )] , ∀ e , e ∈ g . In this work we want to show that the problem of determining equivalent r -matrices up to Lie algebra automorphismscan be significantly simplified by studying the strata of the so-called generalised distributions [31, 48, 49, 51]. Letus detail some useful facts on these geometric entities. Unless otherwise stated, we assume all objects to be smoothand globally defined. Hereafter, X ( M ) stands for the Lie algebra of vector fields on M .A generalised distribution (also called a Stefan–Sussmann distribution ) on a manifold M is a correspondence D attaching each x ∈ M to a subspace D x ⊂ T x M . We call rank of D at x the dimension of D x . A generaliseddistribution need not have the same rank at every point of M . If D has the same rank at every point of M , then D is said to be regular or D is simply called a distribution . Otherwise, D is said to be singular . A generaliseddistribution D on M is involutive if every two vector fields taking values in D satisfy that their Lie bracket takesvalues in D as well.A stratification , let us say F , on a manifold M is a partition of M into connected disjoint immersed submanifolds {F k } k ∈ I , where I is a certain set of indices, i.e. M = S k ∈ I F k and the submanifolds {F k } k ∈ I satisfy F k ∩ F k ′ = ∅ for k = k ′ and k, k ′ ∈ I . The connected immersed submanifolds F k , with k ∈ I , are called the strata of thestratification. A stratification is regular if its strata are immersed submanifolds of the same dimension, whilst it is singular otherwise. Regular stratifications are called foliations and their strata are called leaves . The tangent spaceto a stratum, F k , of a stratification passing through a point x ∈ M is a subspace D x ⊂ T x M . All the subspaces D x ⊂ T x M for every point x ∈ M give rise to a generalised distribution D := S x ∈ M D x on M . All the leaves of afoliation have the same dimension and, therefore, the generalised distribution formed by the tangent spaces at everypoint to its leaves is regular. Meanwhile, a singular stratification gives rise to a singular generalised distribution.4n this work, we are specially interested in generalised distributions generated by finite-dimensional Lie algebrasof vector fields, the so-called Vessiot–Guldberg Lie algebras [34]. More specifically, let V be a Vessiot–Guldberg Liealgebra, the vector fields of V span a generalised distribution D V given by D Vx := { X x : X ∈ V } ⊂ T x M, ∀ x ∈ M. Since the space of vector fields tangent to the strata of a stratification are closed under Lie brackets, the Liebracket of vector fields on M taking values in a distribution D can be restricted to each one of its strata. Ageneralised distribution D on M is integrable if there exists a stratification F on M such that each stratum F k thereof satisfies T x F k = D x for every x ∈ F k . A relevant question is whether a generalised distribution on M isintegrable of not. For regular distributions, the Frobenius theorem holds [23, 31]. Theorem 3.1. If D is a distribution on a manifold M , then D is integrable if and only if it is involutive. Involutivity is a natural necessary condition for a generalised distribution to be integrable because the set ofvector fields tangent to any stratum of a stratification is involutive. Nevertheless, if a generalised distribution isnot regular, its involutiveness does not necessarily implies its integrability. If a generalised distribution D on M is analytical , i.e. for every x ∈ M there exists a family of analytical vector fields taking values in D and spanning D x ′ for every x ′ in an open neighbourhood of x , one has the following proposition. Theorem 3.2. (Nagano [41] and [31])
Let M be a real analytic manifold, and let V be a sub-Lie algebra ofanalytic vector fields on M . Then, the induced analytic distribution D V is integrable. If a generalised distribution is not analytical, the Stefan-Sussmann’s theorem establishes additional conditionsto involutivity to ensure integrability. We are more interested in the following result that ensures integrability in aparticular case of relevance to us.
Theorem 3.3. (Hermann [27] and [31])
Let M be a smooth manifold. If V is a finite-dimensional Lie subalgebraof X ( M ) , then the distribution D V is integrable. One relevant problem concerns the determination of the form of the strata of a stratification. If a stratificationis regular on an open subset U ⊂ M , then one can define locally a set of functionally independent functions whoselevel sets are the strata of the stratification. Indeed, this amounts to the integration of a standard distribution. Ifa stratification is singular, the previous statement is not longer true and other methods are needed. The followingfacts are useful to understand further parts of this work. First, given a point x ∈ M and a Vessiot–Guldberg Liealgebra V on M , the stratum F k of the generalised distribution D V passing through x is given by the points of theform [48] exp( X ) ◦ exp( X ) ◦ . . . ◦ exp( X s ) x, (3.1)where s is any natural number, X , . . . , X s are any vector fields of V , and each exp( X ) stands for the localdiffeomorphism on M induced by the vector field X on M . Let f be a function on M such that if X ∈ V , thenthere exists a function f X ∈ C ∞ ( M ) such that Xf = f X f . This can be considered as a not necessarily polynomialanalogue of a Darboux function for a vector field. Let us assume that f = 0 at a point of F k . It immediately followsfrom (3.1) that f vanishes on the whole F k . Hence, two points x , x ∈ M such that f ( x ) = f ( x ) cannot belongto the same F k . As shown in following sections, f does not need to be a constant of motion of the vector fields in V . This is specially relevant when determining the strata of an integrable singular generalised foliation, as theirstrata cannot always be determined locally as the zeroes of a family of common first-integrals of the vector fields of V . Meanwhile, this can always be achieved locally for the leaves of an integrable distribution [1, 51]. If a Lie algebra g has a nontrivial center, then the image of the map ad : g ∋ v [ v, · ] ∈ gl ( g ) is a matrix Liealgebra that is not isomorphic to g . Hence, the image of ad does not give a matrix algebra representation of g ,which may give rise to a problem as matrix Lie algebra representation are quite practical in computations [5]. Theaim of this section is to provide a method to obtain a matrix Lie algebra isomorphic to g when its center, Z ( g ), isnot trivial, namely Z ( g ) = 0, and g satisfy some additional conditions. Our method is to be employed in the restof our work during calculations. 5et { e , . . . , e s , e s +1 , . . . , e n } be a basis of g such that { e , . . . , e s } form a basis for Z ( g ). Let c kij , with i, j, k =1 , . . . , n , be the structure constants of g in the given basis, i.e. [ e i , e j ] = P nk =1 c kij e k for i, j = 1 , . . . , n . Definethen ˜ g := h e , . . . , e n , e i . Let us determine the conditions ensuring that ˜ g is a Lie algebra relative to a Lie bracket,[ · , · ] ¯ g , that is an extension of the Lie bracket on g , i.e. [ e ′ , e ′′ ] ¯ g = [ e ′ , e ′′ ] g for every e ′ , e ′′ ∈ g , and [ e, e i ] ¯ g = α i e i for i = 1 , . . . , n so that α · α · . . . · α s = 0 . (4.1)The meaning of the last condition will be clear in a while. If [ · , · ] ¯ g is to be a Lie bracket, its Jacobi identity leadsto the following conditions[ e, [ e i , e j ] ¯ g ] ¯ g = [[ e, e i ] ¯ g , e j ] ¯ g + [ e i , [ e, e j ] ¯ g ] ¯ g ⇐⇒ n X k =1 α k c kij e k = n X k =1 ( α i + α j ) c kij e k , ∀ i, j = 1 , . . . , n, [ e, [ e, e j ] ¯ g ] ¯ g = [[ e, e ] ¯ g , e j ] ¯ g + [ e, [ e, e j ] ¯ g ] ¯ g = [ e, [ e, e j ] ¯ g ] ¯ g , ∀ j = 1 , . . . , n. (4.2)Thus, previous conditions can be reduced to requiring that, for those indices i, j, k satisfying c kij = 0, one gets α i + α j = α k . If the latter condition is satisfied, a new Lie algebra ˜ g arises. Note that g is an ideal of ˜ g . This leads to a Liealgebra morphism R : v ∈ g R v := [ v, · ] ˜ g ∈ gl (˜ g ). If R v = 0, then [ v, e i ] ˜ g = 0 for i = 1 , . . . , n , which means that v ∈ Z ( g ). Thus, v = P si =1 λ i e i for certain constants λ , . . . , λ s and 0 = [ e, v ] ˜ g = P si =1 λ i α i e i . Then, condition(4.1) yields λ = . . . = λ s = 0 and it turns out that v = 0. Hence, the elements R v , with v ∈ g , span a matrix Liealgebra of endomorphisms on ˜ g isomorphic to g .It is clear that conditions (4.1) and (4.2) do not need to be satisfied for a general Lie algebra g with a nontrivialcenter. Nevertheless, we used in this work that this method works for all indecomposable real four-dimensionalLie algebras with non-trivial center (see [50] for a complete list of such Lie algebras). This will be enough for ourpurposes.Let us illustrate our method. Consider the Lie algebra s = h e , e , e , e i with non-vanishing commutationrelations [ e , e ] = e , [ e , e ] = e . Let us construct a new Lie algebra ˜ g = h e , e , e , e , e i following (4.1) and (4.2).This gives rise to the following system of equations: α + α = α , α + α = α , α = 0 . We have that Z ( s ) = h e i . The previous system, under the corresponding restriction (4.1), has a solution α =0 , α ∈ R , α = α , α ∈ R \{ } . In particular, set α = α = 1 and α = 0. Thus, the endomorphisms R e i on ˜ g read R e = −
10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 , R e = − −
10 0 0 0 00 0 0 0 00 0 0 0 0 , R e = − , R e = . As desired, the above matrices have the same non-vanishing commutation relations as s , i.e.[ R e , R e ] = R e , [ R e , R e ] = R e . Let us prove that not every Lie algebra g with non-trivial center admits an extended Lie algebra ˜ g of the formrequired by our method. Consider the Lie algebra s , = h e , . . . , e i with nonzero commutation relations (see[50] for details):[ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e . Thus, Z ( s , ) = h e i . The corresponding system of equations (4.2) reads α + α = α , α + α = α , α + α = α , α + α = α , α + α = α , α + α = α . Then, α = 0 , α = 0 and α = α = 0, which contradicts the assumption of our method (4.2), namely α = 0.Consider the 7-dimensional Lie algebra g := h e , . . . , e i with Z ( g ) = h e i and nonzero commutation relations(cf. [47, p. 492, case 7 I ])[ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e . α + α = α , (ii) α + α = α , (iii) α + α = α , (iv) α + α = α , (v) α + α = α , (vi) α + α = α , (vii) α + α = α , (viii) α + α = α , (ix) α + α = α . From (vii) and (viii), we get α = α ; from (vi) and (ix), we get α = α ; and from (iv) and (viii), we get α = α .Thus, α = α = α . From (i), it follows that α = 0. Thus, α = α = 0. From (ii), we get α = 0 and (iii) yields α = 0. From (iv), one gets α = α . From (ix), we get α = 0, and thus α = 0. Therefore, α = . . . = α = 0 isthe only possibility, but it does not satisfy (4.1). s s up to its Lie algebra automorphisms. As a byproduct,we introduce a generalisation of the concept of Darboux polynomial of a vector field.The space of derivations on s can straightforwardly be obtained. It is indeed the solution of a linear algebraproblem that can be easily solved by hand calculation and/or via any mathematical computation program. The samecould be achieved for any four, five, or even some other higher-dimensional Lie algebra. In particular, derivationson s take the form der ( s ) = µ µ µ µ µ µ µ : µ , µ , µ , µ , µ , µ ∈ R (5.1)in the basis { e , e , e , e } appearing in Table 1. By Proposition 2.2 and, more specifically, Remark 2.1, the Liealgebra V s of fundamental vector fields of the natural Lie group action of Aut( s ) on Λ s is spanned by the basis(over the reals) X = 2 x ∂ x + x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x ,X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x . (5.2)As X , . . . , X close on a finite-dimensional Lie algebra, Theorem 3.3 shows that they span an integrable generaliseddistribution D V s on Λ s . In view of (3.1) the strata of D V s are the orbits of the action of the connected part ofthe identity of Aut( g ), let us say Aut c ( s ), on Λ s .We define M ( p ) to be a matrix whose entry ( i, j ) is the j -coefficient of X i at p ∈ Λ s in the basis ∂ x | p , . . . , ∂ x | p ,namely M ( p ) := x x x x x x x − x − x x − x x x x x x , p := ( x , . . . , x ) ∈ Λ s . The rank of D V s at p ∈ Λ s is equal to the rank of M ( p ). It is simple to calculate that the mCYBE for s is givenby the common zeroes of the functions on Λ s of the form f := x x , f := x x , f := x . (5.3)It is immediate that the space of solutions to the mCYBE, let us say Y s , is not a submanifold in Λ s . Nevertheless, X i f j = X k =1 λ kij f k , i = 1 , . . . , , j = 1 , , , (5.4)for certain constants λ kij , with i = 1 , . . . , j, k = 1 , ,
3. Relations (5.4) show that the integral curves of anyvector field in V s passing through w ∈ Y s is contained in Y s . In fact, the derivative of the functions f , f , f along an integral curve, Ψ( t ), of a vector field X j ∈ V s such that Ψ(0) ∈ Y s reads df i (Ψ( t )) dt = ( X j f i )(Ψ( t )) = X k =1 λ kji f k (Ψ( t )) , i = 1 , , . f i (Ψ( t )), with i = 1 , ,
3, can be understood as the solutions to a linear system of first-orderordinary differential equations in normal form with constant coefficients and zero initial condition since f i (Ψ(0)) = 0for i = 1 , ,
3. Hence, f , f , f vanish along Ψ( t ) and, since (3.1) shows that the integral curves for all the vectorfields in V connect all the points within the same strata of D V s , one obtains that the functions f , f , f are zeroon any strata of D V s containing a point where f , f , f vanish. It is worth noting that f , f , f are not constantsof motion common to all the vector fields of V s . We hereafter call h f , f , f i a Darboux family for the Lie algebraof vector fields h X , . . . , X i on Λ s . More generally, we propose the following definition and we extend previousresults to a more general realm. Definition 5.1.
We call an s -dimensional linear space of functions A := h f , . . . , f s i on a manifold M a Darbouxfamily for a Vessiot–Guldberg Lie algebra V on M if, for every X ∈ V and f j , with j = 1 , . . . , s , one can write Xf j = s X i =1 h ijX f i , for a certain family of smooth functions, h ijX , with i = 1 , . . . , s , on M , the so-called co-factors of f j relative to X and the basis { f , . . . , f s } . The subset S A := { p ∈ M : f ( p ) = 0 , ∀ f ∈ A} is called the locus of the Darboux family A . It is worth stressing that we require the functions f , . . . , f s and h ijX , for i, j = 1 , . . . , s and X ∈ V , in theabove definition to be smooth. If all the functions h ijX are equal to zero for every X ∈ V and i = 1 , . . . , s , then f j becomes a constant of motion for the vector fields of V on M . If the h ijX are constants for every X ∈ V and i, j = 1 , . . . , s , then we say that the Darboux family is linear . In this case, V gives rise to a Lie algebra representation ρ : X ∈ V D X ∈ End( A ), where D X stands for the action of the vector field X on the space of functions A .The vector fields of V span a generalised distribution, which is integrable and leads to a stratification of M bysome disjoint immersed submanifolds which may have different dimensions. Darboux families are interesting to usbecause the integral curves of vector fields in V passing through the set S A of common zeroes for the elementsof A remain within it. Let us prove and analyse this fact, which represents a rather simple generalisation of theargument given for s . Consider a point in the locus S A and consider a basis Y , . . . , Y q of the Lie algebra V ofvector fields. Consider an integral curve Ψ( t ) of a vector field of V passing through a point Ψ(0) ∈ S A . Then, thetime derivative of the functions f (Ψ( t )) , . . . , f s (Ψ( t )) is df i (Ψ( t )) dt = ( Xf i )Ψ( t ) = s X i =1 h jiX (Ψ( t )) f j (Ψ( t )) , i = 1 , . . . , s. Hence, the values of the f i (Ψ( t )) can be understood as the solutions to the linear system of first-order ordinarydifferential equations in normal form with t -dependent coefficients du i dt = s X j =1 h jiX (Ψ( t )) u j , i = 1 , . . . , s, with initial condition u = . . . = u s = 0. By the existence and uniqueness theorem, the solution to the previousCauchy problem is u ( t ) = . . . = u s ( t ) = 0. Therefore, the functions f , . . . , f s vanish on the integral curves of X . From this and the decomposition (3.1), we can infer that the functions f , . . . , f s vanish on the strata of thedistribution D V containing a point within S A . In other words, the strata of D V containing some point of S A arefully contained in S A .In view of the above, Darboux families can be used to reduce the determination of the strata in M of thegeneralised distribution D V to determining the strata within S A and out of S A . This will be specially interesting toobtain the strata of D V at points where the generalised distribution is not regular and it may not exist a constantof motion common to all the vector fields in V that can be used to obtain the strata of D V . Note also that if h f , . . . , f s i form a Darboux family for V , then the set of common zeroes of h f , . . . , f s i must be the sum (as subsetsof M ) of a collection of strata of the generalised distribution D V .Let us study how to use Darboux families to study solutions to mCYBEs and CYBEs. Let us start by a generalresult. Theorem 5.2.
Let A (3) g be a Darboux family for V (3) g on Λ g . Then, the space of functions A := n g ∈ C ∞ (Λ g ) : g = f ◦ [ · , · ] , f ∈ A (3) g o (5.5) is a Darboux family for V g on Λ g . If A (3) g is a linear Darboux family, then A is also a linear Darboux family. roof. Since each X (2) ∈ V g is a fundamental field of the action of Aut( g ) on Λ g , its flow is given by the one-parametric group of diffeomorphisms Λ T t induced by a certain one-parametric group of Lie algebra automorphisms T t ∈ Aut( g ), t ∈ R . Every g ∈ A is of the form g ( r ) = f ([ r, r ]) for some f ∈ A (3) g and every r ∈ Λ g . Then,( X (2) g )( r ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 g (Λ T t ( r )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ([Λ T t ( r ) , Λ T t ( r )])= ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f (Λ T t [ r, r ]) = ( X (3) f )([ r, r ]) = r X i =1 h i ([ r, r ]) f i ([ r, r ]) , where f , . . . , f s form a basis of the Darboux family A (3) g , the functions h , . . . , h s are the cofactors of f relative to X (3) and the basis { f , . . . , f s } , and X (3) is the fundamental vector field of the action of Aut( g ) on Λ g induced bythe one-parameter group { T t } t ∈ R of Lie algebra automorphisms of g . Since f ([ r, r ]) , . . . , f s ([ r, r ]) belong to A and h ([ r, r ]) , . . . , h s ([ r, r ]) are functions on Λ g , the A forms a Darboux family for V g on Λ g .If A (3) g is a linear Darboux family, then h , . . . , h s are constants. Therefore, the cofactors h ([ r, r ]) , . . . , h s ([ r, r ])are also constants, which makes A into a linear Darboux family for V g on Λ g .In short, the following proposition shows that the solutions to the mCYBE on g can be considered as the locusof a Darboux family relative to V g . A similar result can be applied to the CYBE on any Lie algebra g . Proposition 5.1.
Let ((Λ g ) g ) ◦ be the annihilator of (Λ g ) g , i.e. the subspace of elements of (Λ g ) ∗ vanishing on (Λ g ) g . The functions f υ : r ∈ Λ g υ ([ r, r ]) ∈ R , υ ∈ ((Λ g ) g ) ◦ , (5.6) span a linear Darboux family, A , for the Lie algebra V g of fundamental vector fields of the action of Aut( g ) on Λ g .Its locus is the set of solutions to the mCYBE for g . Moreover, the components of the CYBE in a coordinate systemgiven by a basis of Λ g ∗ span also a linear Darboux family for V g on Λ g .Proof. Let us prove that the annihilator of (Λ g ) g is a Darboux family for V (3) g on Λ g . In fact, if X (3) ∈ V (3) g , thenits flow is given by a one-parameter group of diffeomorphisms of the form Λ T t for certain Lie algebra automorphisms T t ∈ Aut( g ) with t ∈ R . Consider a function f ∈ Λ g ∗ such that f ( w ) = 0 for every w ∈ (Λ g ) g , i.e. f ∈ ((Λ g ) g ) ◦ .Then, ( X (3) f )( w ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ((Λ T t )( w )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( w t ) = 0 , where we have used that (Λ g ) g is closed under the extension to Λ g of Lie algebra automorphisms of g and thus(Λ T t )( w ) =: w t ∈ (Λ g ) g . Hence, X (3) f vanishes on (Λ g ) g . Since X (3) is a linear vector field on Λ g and f is alinear function on Λ g , one obtains that X (3) f is a linear function on Λ g vanishing on (Λ g ) g . Hence, X (3) f is alinear combination of elements of a basis of ((Λ g ) g ) ◦ and ((Λ g ) g ) ◦ becomes a linear Darboux family of V (3) g onΛ g . By Theorem 5.2, the space of functions (5.6) becomes a linear Darboux family for V g on Λ g . The locus of(5.6) is the space of solutions of the mCYBE for g .Since every X (3) ∈ V (3) g is linear vector field on a linear coordinate system on Λ g , the space Λ g ∗ is also aDarboux family for V (3) g on Λ g . By Theorem 5.2, one has that the induced by (5.5) family of functions on Λ g isa linear Darboux family relative to V g on Λ g . The Darboux family is indeed spanned by the components of theCYBE of g , i.e. [ r, r ] = 0, and its locus is its set of solutions.A natural question is how to determine Darboux families. Here, we provide several results about them that willbe useful to as so as to derive new Darboux families from known ones. Proposition 5.2. If A and A are Darboux families for the same Vessiot–Guldberg Lie algebra V on a manifold M , then the sum A + A is a Darboux family for V on M as well. The proof of the previous result is immediate. Obviously, the locus of A + A is contained in the locus of A and A . This will be interesting in next sections to distinguish between different strata of the generalised distributionspanned by the vector fields of V g on Y g .In next sections, it will be relevant to study the linear Darboux families for a certain V g given by a one-dimensional A ⊂ (Λ g ) ∗ . This is due to the fact that, as seen in Section 7, the sum of such Darboux families asvector spaces will generate new useful Darboux families. We will call such a linear one-dimensional Darboux familya brick . Bricks are one-dimensional representations of V g on Λ g ∗ .9ote that, for every linear Darboux family for V g , bricks are easy to obtain since, in view of (2.3), they are givenby the intersection of an eigenvector space for each endomorphism of the form Λ d ∈ gl (Λ g ) with d ∈ der ( g ).Let us give another interesting proposition, which is an immediate consequence of the fact that the vector fieldsof V g are linear relative to a linear coordinate set on Λ g . Proposition 5.3.
The space Λ g ∗ is a linear Darboux family of functions on Λ g relative to the Lie algebra V g . Since Λ g ∗ is a linear Darboux family of V g , there exists a linear representation of aut ( g ) on Λ g ∗ . Its irreduciblerepresentations also give rise to Darboux families, which can be summed (as linear spaces) to the Darboux family,or potentially elements thereof, associated with the mCYBE to determine new Darboux families. The locus ofsuch sums will be interested to as when they contain elements of Y g . This will be employed to obtain the strataof V g within Y g and, therefore, inequivalent r -matrices relative to the action of Aut c ( g ) on Λ g . This in turnwill be used to obtain families of inequivalent r -matrices and coboundary cocommutators for real four-dimensionalindecomposable Lie algebras [50] in Section 7. Let us study several details on the geometry of the space of solutions to modified and non-modified classical Yang-Baxter equations, the problem of classification up to Lie algebra automorphisms of coboundary Lie bialgebras, andDarboux families. As previously, we hereafter write V g for the Lie algebra of fundamental vector fields of the actionof Aut( g ) on Λ g , whilst E g stands for the distribution on Λ g spanned by the vector fields of V g . We recall thatwe have defined Λ R g := Λ g / (Λ g ) g and π g : w ∈ Λ g [ w ] ∈ Λ R g stands for the canonical projection onto thequotient space Λ R g . As previously, we hereafter write Y g for the space of r -matrices of g .The action of Aut( g ) on Λ g induces an action of Aut( g ) on Λ R g of the form (see [35, Lemma 8.3] for details)Ψ : T ∈ Aut( g ) [Λ T ] ∈ GL (Λ R g ) , [Λ T ]([ w ]) := [Λ T ( w )] , ∀ w ∈ Λ g . The above result leads to the following consequence π g ((Λ T )( w )) = [Ψ( T )]( π g ( w )) , ∀ w ∈ Λ g , ∀ T ∈ Aut( g ) . (6.1)Recall that two actions of a Lie group G , let us say Φ i : G × N i → N i with i = 1 ,
2, are equivariant relative to φ : N → N if φ ◦ Φ ( g, x ) = Φ ( g, φ ( x )) for every g ∈ G and x ∈ N . It can be proved that if Φ and Φ areequivariant, the Lie algebra of fundamental vector fields of Φ projects, via φ ∗ , onto the Lie algebra of fundamentalvector fields of Φ . Moreover, the orbits of Φ project, via φ , onto the orbits of Φ (see [1] for details).Expression (6.1) shows that the actions of Aut( g ) on Λ g and Λ R g are equivariant relative to π g : Λ g → Λ R g .Then, the vector fields of V g project via π g ∗ onto the Lie algebra of fundamental vector fields of the action ofAut( g ) on Λ R g . We write E R g for the generalised distribution spanned by the fundamental vector fields of the actionof Aut( g ) on Λ R g . This means that the strata of the generalised distribution E g project onto the strata of thegeneralised distribution E R g .As the elements of Aut( g ) map solutions to the mCYBE of g onto solutions of the same equation, the orbitsof Aut( g ) containing a solution to the mCYBE consist of solutions to the mCYBE. Similarly, the orbits of theaction of Aut( g ) on Λ g containing a solution to the CYBE consist of solutions to the CYBE. Two r -matrices areequivalent up to Lie algebra automorphisms of g if and only if they belong to the same orbit of Aut( g ) within Y g .Recall that Aut( g ) / Aut c ( g ), where Aut c ( g ) is the connected component of the identity of the Lie group Aut( g ),is discrete and therefore countable [13]. In other words, the Lie group Aut( g ) is a numerable sum (as subsets) ofdisjoint and connected subsets of Aut( g ) diffeomorphic to Aut c ( g ). The strata of E g coincide with the orbits ofAut c ( g ). Therefore, the orbits of Aut( g ) are given by a numerable sum of strata of E g . Moreover, each particularorbit of Aut( g ) is an immersed submanifold in Λ g of a fixed dimension. Hence, each one of the orbits of Aut c ( g ),whose sum gives rise to an orbit of Aut( g ), must have the same dimension. Similarly, the orbits of Aut( g ) on Λ R g are given by a numerable sum (as subsets) of strata of E R g of the same dimension, which are orbits of the action ofAut c ( g ) on Λ R g .It can be proved that two coboundary cocommutators can be equivalent up to Lie algebra automorphisms evenwhen their associated r -matrices are not. For instance, the zero cocommutator δ : g → g ∧ g can take the form0 = δ = [ r, · ] for r = 0 or for any other r ∈ (Λ g ) g \{ } . Nevertheless, the 0 and r cannot be connected by theaction of an element of Aut( g ) on Λ g , since the action of Aut( g ) on Λ g is linear and 0 ∈ Λ g is an orbit. Inspite of that, the cocommutators generated by 0 and r are the same. Let us give the conditions ensuring that twocoboundary cocommutators on g are equivalent up to a Lie algebra automorphism of g .10 roposition 6.1. Two coboundary cocommutators δ i : v ∈ g [ r i , v ] ∈ g ∧ g , with i = 1 , and r i ∈ Y g , areequivalent under a Lie algebra automorphism of g if and only if there exists T ∈ Aut( g ) such that (Λ T )( r ) and r belong to the same equivalence class in Λ R g .Proof. It is clear that the transformation of δ by T reads, by the properties of the Schouten bracket, as followsΛ T ◦ δ ◦ T − = Λ T ◦ [ r , T − ( · )] = [Λ T r , · ] . Hence, δ = Λ T ◦ δ ◦ T − ⇔ [Λ T r , · ] = [ r , · ] ⇔ [Λ T r − r , · ] = 0 , and the last condition amounts to the fact that Λ T r − r ∈ (Λ g ) g , which implies that Λ T r , r belong to thesame equivalence class in Λ R g .Note that two r -matrices are equivalent (relative to the action of an element of Aut( g ) on Λ g ) if and only ifthey belong to the same family of strata of the distribution E g giving rise to an orbit of Aut( g ) in Y g . Nevertheless,two r -matrices give rise to two equivalent coboundary cocommutators if and only if they belong to the same orbitof Aut( g ) on Λ R g . In other words, we have proven the following proposition. Proposition 6.2.
There exists a one-to one correspondence between the r -matrices for g that are equivalent up to anelement of Aut c ( g ) and the strata of the generalised distribution E g within Y g . Moreover, there exists a one-to-onecorrespondence between the families of equivalent (up to Lie algebra automorphisms of g ) coboundary cocommutatorsof a Lie algebra g and the orbits of the action of Aut( g ) on π g ( Y g ) ⊂ Λ R g . Every such an orbit in π g ( Y g ) is thesum (as subsets) of a numerable collection of strata of the same dimension of the generalised distribution E R g . In practice, we shall obtain each orbit of Aut( g ) on Y g as a numerable family of strata of E g in Y g . Such orbitsrepresent the families of r -matrices that are equivalent up to an element of Aut( g ). Then, we will derive all stratain Y g that map onto the same space in Λ R g . This last task will give us, along with the orbits of Aut( g ) on Y g , theequivalent classes of coboundary cocommutators on g up to Lie algebra automorphisms of g .Let us study how Darboux families can be employed to obtain the orbits of Aut( g ) on Λ g and Λ R g . In particular,Darboux families are employed here to identify the strata of E g . Proposition 6.3.
The locus, ℓ g , of a Darboux family A E g relative to the Lie algebra V g on Λ g is a sum (as subsets)of the orbits of the action of Aut c ( g ) on Λ g containing a point in ℓ g . If ℓ g is a connected submanifold in Λ g ofdimension given by the rank of the generalised distribution E g , then ℓ g is a strata of the generalised distribution E g .Proof. For each point p in ℓ g the orbit O p of Aut c ( g ) passing through p has a tangent space spanned by the vectorfields of V g . By the definition of a Darboux family for V g and the fact that the functions of A E g vanish at p , onehas that all functions of A E g are equal to zero on O p (see Section 5). Hence, O p is contained in ℓ g . Then, ℓ g is thesum (as subsets) of orbits of Aut c ( g ) passing through the points of ℓ g .Recall that the strata of E g are the orbits of Aut c ( g ) acting on Λ g . Hence, ℓ g is a sum of strata of E g . If therank of E g on ℓ g is equal to dim ℓ g , where ℓ g is assumed to be a connected submanifold, then ℓ g is locally generatedaround any point p ∈ ℓ g by the action of Aut c ( g ) on that point. Since ℓ is connected, it is wholly generated by theaction of Aut c ( g ) and it becomes a strata of E g .Note that we are interested only in those loci of Darboux families of V g contained in the space Y g of solutionsto the mCYBE of g .It is interesting that necessary conditions for the existence of a Lie algebra automorphism connecting two r -matrices can be given using Λ g . Consider, for instance, the following proposition, whose proof is straightforward. Proposition 6.4. If r and r are two equivalent r -matrices for a Lie algebra g , then, [ r , r ] and [ r , r ] belong tothe same orbit of Aut( g ) acting on (Λ g ) g . The necessary condition in Proposition 6.4 is not sufficient. For instance, it may happen that two r -matricesare solutions to the classical Yang-Baxter equation without being equivalent (as seen in Table 2). From a practicalpoint of view, Proposition 6.4 shows that the determination of the orbits of the action of Aut( g ) on Λ g may beof interest. Following the reasoning for studying the equivalence of r -matrices in Λ g , the orbits of the action ofAut( g ) on Λ g can be obtained by obtaining the strata of the distribution, E (3) g , spanned by the fundamental vectorfields V (3) g of the action of Aut c ( g ) on Λ g . Such strata can be obtained through Darboux families for V (3) g in ananalogous way to the method employed to study E g . Then, the action of one element of each connected componentof Aut( g ) on the strata of V (3) g in (Λ g ) g gives the orbits of Aut( g ) within (Λ g ) g .11 roposition 6.5. If two r -matrices r , r are equivalent, then the rank of r , r , as bilinear mappings on g ∗ , arethe same.Proof. Recall that r -matrices can be considered as the antisymmetric bilinear mappings on g ∗ associated withthem. The action of Aut( g ) on an r -matrix does not change its rank as a bilinear antisymmetric mapping. Hence,equivalent r -matrices must have the same rank.Although r -matrices whose bilinear antisymmetric mappings on g ∗ have different rank are not equivalent up toLie algebra automorphisms of g , it may happen that r -matrices related to the same rank are not equivalent. Infact, although there always exists in this latter case an A ∈ GL ( g ) acting on Λ g mapping one element of this spaceinto another of the same order ( A can be obtained by mapping both bivectors into a canonical form), A does notnecessarily belong to Aut( g ). Hence, both bivectors need not be equivalent up to Lie algebra automorphisms of g .Many examples of this will appear in Table 2. A Lie algebra g is decomposable when it can be written as the direct sum of two of its proper ideals. Otherwise,we say that g is indecomposable . Winternitz and ˇSnobl classified in [50, Part 4] all indecomposable Lie algebrasup to dimension six (see [50, p. 217] for comments on previous classifications) . The aim of this section is topresent the classification up to Lie algebra automorphisms of coboundary Lie bialgebras on real four-dimensionalindecomposable Lie algebras via Darboux families and the Winternitz–ˇSnobl (WˇS) classification. Moreover, theequivalence of r -matrices and solutions to CYBEs for four-dimensional real indecomposable Lie algebras are alsoanalysed.We hereafter assume g to be a real four-dimensional and indecomposable Lie algebra. For the sake of complete-ness, the structure constants for every g in a basis { e , e , e , e } are given in Table 1. We hereafter endow Λ g with a basis { e := e ∧ e , e := e ∧ e , e := e ∧ e , e := e ∧ e , e := e ∧ e , e := e ∧ e } , while itsdual basis is denoted by { x , x , x , x , x , x } . Additionally, Λ g is endowed with a basis { e , e , e , e } .Moreover, due to the bilinearity and symmetry of the Schouten bracket on Λ g , it is enough to know [ e ij , e kl ], with1 ≤ i < k ≤ i = k ∈ { , . . . , } , ≤ j ≤ l ≤
4, to determine the Schouten bracket on the whole Λ g . In Tables3–5, we summarise several relevant Schouten brackets between the basis elements of g , Λ g , and Λ g that can easilybe obtained via Table 1 and, eventually, by means of a symbolic computation program. Anyway, we add suchcalculations to easily follow the proofs of our following results. The classes of Lie algebras in Table 1 may containseveral non-isomorphic Lie algebras depending on several parameters, e.g. α, β . To simplify the notation, we willnot detail specific values of the parameters when we refer to properties of a whole class of Lie algebras and/or it isclear the particular case we are discussing from the context.It is interesting to discuss some physical and mathematical applications of the Lie bialgebras given in our listand their relations to some previous works. For instance, the nilpotent Lie algebra n corresponds to the so-calledGalilean Lie algebra, which is deformed in the study of XYZ models with the quantum Galilei group appearing in[12]. Lie bialgebras defined on Lie algebras of the type n also occur in [42, 43]. The coboundary Lie bialgebrason the Lie algebra s , which corresponds to the so-called harmonic oscillator algebra , are obtained in [6] withoutstudying the equivalence up to Lie algebra automorphisms. In that work, the quantisations and R -matrices of sucha sort of Lie bialgebras are also studied. In particular, equation (3.7) in [6] matches the mCYBE obtained in Section7.6. The work [7] proves that all Lie bialgebras on the harmonic oscillator algebra are coboundary ones. This showsthat our classification of Lie bialgebras on s finishes the classification of all Lie bialgebras on the harmonic oscillatoralgebra. On the other hand, the Lie algebra s with α = − / A ¯ G (1) in [42]. TheLie algebra s provides a central extension of Caley-Klein Lie algebras analysed in [10, eq. (3.13)]. Other cases ofreal four-dimensional indecomposable Lie bialgebras can be found in [4, 44], where, for instance, two-dimensionalcentral extended Galilei algebras are studied. s The structure constants for Lie algebra s are given in Table 1, while relevant Schouten brackets between theelements of the bases of s , Λ s , Λ s to be used hereafter are displayed in Tables 3–5. In particular, one sees fromsuch tables that (Λ s ) s = h e i and (Λ s ) s = 0. 12ie algebra [ e , e ] [ e , e ] [ e , e ] [ e , e ] [ e , e ] [ e , e ] Parameters s − e − e s − e − e − e − e − e s − e − αe − βe < | β | ≤ | α | ≤ α, β ) = ( − , − s − e − e − e − αe α ∈ R \{ } s − αe − βe + e − e − βe α > , β ∈ R s e − e e s e e − e s − (1 + α ) e e − e − αe α ∈ ] − , \{ } s − αe e − αe + e − e − αe α > s − e e − e − e − e s − e e − e s − e e − e − e n e e Table 1: Structure constants for real four-dimensional indecomposable Lie algebras according to the WˇS classifica-tion. Such Lie algebras are denoted by s ,k , n , for 1 ≤ k ≤
12 in the WˇS classification. Nevertheless, the subindexconcerning the dimension of the Lie algebra will be removed in our work to simplify the notation. The letters n and s are used to denote nilpotent and solvable (but not nilpotent) Lie algebras, respectively. To avoid duplicities ofLie algebras, the parameters α, β of s must satisfy additional restrictions that were not detailed explicitly in [50](see [50, p. 228]). Such restrictions will be determined in Subsection 7.3.To obtain the classification of r -matrices for s up to Lie algebra automorphisms thereof, we will rely on thedetermination of the strata of the distribution E s (see Proposition 6.2) via Darboux families for V s and a fewelements of Aut( s ).By Proposition 6.3, the locus of a Darboux family for a Lie algebra V s is the sum (as subsets) of strata of thegeneralised distribution E s spanned by the vector fields of V s . In particular, we are interested in those loci beinga submanifold within Y s of dimension matching the rank of E s on them. This is due to the fact that, in viewof Proposition 6.3, their connected components are the orbits of Aut c ( s ) on Y s . With this aim, we derive thefundamental vector fields of the action of Aut( s ) on Λ s . By Remark 2.1, the derivations of s take the form(5.1), and a basis of V s is given by (5.2).For an element r ∈ Λ s , we get[ r, r ] = 2( − x x + x x − x x ) e − x e + 2( x − x ) x e + 2 x x e . This expression can easily be derived by using the properties of the algebraic Schouten bracket and the structureconstants for s . A similar computation would not be difficult even for many higher-dimensional Lie algebras.Since (Λ s ) s = 0, the mCYBE and the CYBE for s are equal and read x x = 0 , x x = 0 , x = 0 . It is interesting that, since (Λ s ) s = 0, every r -matrix for s amounts to a left-invariant Poisson bivector on s ∗ (cf. [15]).Let us apply to s the following procedure to obtain the orbits of Aut c ( g ) on Y g for a Lie algebra g of arbitrarydimension, i.e. the strata of E g within Y g . We start by considering a locus ℓ of a Darboux family of V g on Λ g .As we are interested in determining the strata of E g within Y g , we look for an ℓ containing some solution to themCYBE, namely ℓ ∩ Y g = ∅ . Recall that the strata of E g are completely contained within the locus or completelycontained off the locus of every Darboux family for V g . This divides the strata of E g on Y g into those within oroutside ℓ . If ℓ ∩ Y g is a submanifold of dimension given by the rank of E g on it, its connected parts are the strataof E g in ℓ ∩ Y g . Otherwise, to obtain the strata within ℓ ∩ Y g , we look for a new Darboux family for V g containingthe previous Darboux family so that its new locus, let us say ℓ , will satisfy ℓ ∩ Y g = ∅ . This process is repeateduntil we obtain a locus, ℓ k , that is a submanifold of dimension equal to the rank of E g on it. In other words, theconnected components of the locus ℓ k are the orbits of Aut c ( g ) in ℓ k .To determine all the strata of E g on Y g ∩ ( ℓ k ) c , where ( ℓ k ) c stands for the complementary subset of ℓ k in Λ g ,which is always open, one considers a subspace of the final Darboux family and the previous procedure is applied13gain iteratively to the obtained Darboux family on Λ g \ ℓ k . At the end, we obtain another family of strata of E g and the procedure can be repeated to search for remaining strata of E g in Y g . The above-described process, for allfour-dimensional indecomposable Lie algebras, allows us to obtain the orbits of Aut c ( g ) on Y g .To represent schematically our procedure, we use a diagram, which is hereafter called a Darboux tree . EveryDarboux family of the above-mentioned method is described by a set of boxes going from an edge of the diagramon the left to one of the edges on the right. The squared boxes of the type f = 0, for a certain function f , givethe generating functions of the Darboux family while the squared boxes of the type f = 0, give the conditionsrestricting the manifold in Λ g where the Darboux family and V g are restricted to. The connected parts of theloci of all the Darboux families represented in a Darboux tree give rise to a decomposition of Y g into orbits ofAut c ( g ). To help understanding the calculations, some oval boxes with additional information, e.g. explaining whyother possibilities are not considered, are given. It is worth noting that bricks where employed to generate Darbouxfamilies. In particular, the Darboux tree of s is given below. In this case, it is immediate that x and x are bricksfor V s . We use this fact to generate Darboux families of the following Darboux tree. x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 No solutions x = 0 x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVVVIVIIVIII The connected components of the loci of the Darboux families of the previous Darboux tree give the orbits ofAut c ( s ) on Y s . Results are given in Table 2. Let us study the equivalence of r -matrices up to action of the wholeAut( s ). To do so, we use that the group of automorphisms of s readsAut( s ) = T T T T T T T : T , T ∈ R \{ } , T , T , T , T ∈ R . This Lie group is easy to be derived from the structure constants of s . In reality, it is enough for our purposes toconsider an element of each connected component of Aut( s ). It is remarkable that Lie algebra automorphism groupscan rather easily be obtained for Lie algebras of relatively high dimension and/or satisfying special properties, e.g.for semisimple Lie algebras [40]. In our case, it is immediate that Aut( s ) has four such components. One elementof each connected component of Aut( s ) and its extension to Λ s are given by T λ ,λ := λ λ λ
00 0 0 1 = ⇒ Λ T λ ,λ := λ λ λ λ λ λ
00 0 0 0 0 λ , λ , λ ∈ {± } . (7.1)The orbits of Aut( s ) in Y s are given by the action of Λ T λ ,λ on the orbits of Aut c ( s ) on Y s . By using allmappings Λ T λ ,λ , we can verify whether some of the orbits of Aut c ( s ) on Y s can be connected among themselvesby a Lie algebra automorphism of s . Our results are summarised in Table 2.14ecall that (Λ s ) s = h e i . Consequently, all orbits of Aut( s ) on Y s mapping onto the same space in Λ R s via π s give equivalent coboundary coproducts up to the action of elements of Aut( s ). In particular, we get fiveclasses of coboundary coproducts induced by the following classes of r -matrices: a) I ± , b) II, III ± , c) IV, V ± , d)VI, and e) VII, VIII ± . We recall that all given r -matrices are solutions to the CYBE. s Let us apply the formalism given in the previous section to Lie algebra s . Lie algebra s has structure constantsgiven in Table 1. Relevant Schouten brackets between the bases of elements of s , Λ s , and Λ s are given inTables 3–5. Remarkably, (Λ s ) s = 0 and (Λ s ) s = 0.By Remark 2.1, it follows that der ( s ) = µ µ µ µ µ µ µ µ µ : µ , µ , µ , µ , µ , µ ∈ R , which gives rise, by lifting these derivations to Λ s , to the basis of V s of the form X = 2 x ∂ x + 2 x ∂ x + x ∂ x + 2 x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x + x ∂ x + x ∂ x ,X = − x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x . Meanwhile, for r ∈ Λ s , one has[ r, r ] = 2(2 x x − x x + x x + 2 x x − x x ) e + 2( x x − x ) e − x x e − x e . Since (Λ s ) s = 0, the mCYBE and the CYBE are the same and read x x = 0 , x = 0 , x = 0 . With the previous information, we are ready to obtain the classification of orbits of the action of Aut c ( s ) onthe subset of r -matrices Y s ⊂ Λ s by using Darboux families. We shall start by using bricks. The only brick of s is x . The procedure is accomplished as in the previous section and it is summarised in the following Darbouxtree. x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIV As commented in Subsection 7.1, to obtain the orbits of Aut( s ) on Y s , it is enough to derive the extension toΛ s of a single element of each connected component of Aut( s ). The automorphisms of the Lie algebra s caneasily be obtained. They readAut( s ) = T T T T T T T T T : T ∈ R \{ } , T , T , T , T , T ∈ R . s ) for each of its connected components and their extensions to Λ s read T λ := λ λ λ
00 0 0 1 = ⇒ Λ T λ = λ λ
00 0 0 0 0 λ , λ ∈ {± } . (7.2)The nine connected parts of the submanifolds 0, I ± , II ± , III, IV ± are the orbits of Aut c ( s ) on Y s . It is simpleto see which of them are further connected through an element of Aut( s ). In result, the orbits of Aut( s ) on Y s are given by the eight submanifolds 0 , I − , I + , II − , II + , III , IV − , IV + given in Table 2. Since (Λ s ) s = 0, eachsuch a submanifold gives a family of equivalent coboundary coproducts that is non-equivalent to r -matrices withinremaining submanifolds. This gives all possible classes of non-equivalent coboundary coproducts. As in the previoussubsection, all r -matrices are solutions to the CYBE in s . s The structure constants of the Lie algebras of type s are given in Table 1. It was noted in [50, p. 228] thatadditional restrictions must be imposed on the parameters α, β given in Table 1 to avoid the repetition of isomorphicLie algebras within the Lie algebra class s . Such restrictions were not explicitly detailed in [50], but it is immediatethat isomorphic cases within Lie algebras of the class s can be classified by the adjoint action of e on h e , e , e i .In particular, if | α | = | β | , then the Lie algebras with parameters ( α, β ) and ( β, α ) are isomorphic relative to the Liealgebra isomorphism that interchanges e with e and leaves e invariant. It will be relevant in what follows thatthe lift of this Lie algebra automorphism to Λ s interchanges x with x , x with x , it maps x to − x , and itleaves x invariant. Due to the previous isomorphism, we restrict ourselves to the case α ≥ β when | α | = | β | .Using the structure constants given in Table 1 and the induced Schouten brackets between the relevant toour purposes basis elements of s , Λ s , and Λ s in Tables 3–5, one gets that (Λ s ) s ⊂ { e , e , e } . Morespecifically, e ∈ (Λ s ) s if and only if α = −
1; while e ∈ (Λ s ) s if and only if β = −
1. Finally, e ∈ (Λ s ) s just when α + β = 0. Moreover, (Λ s ) s = h e i if α + β = − s ) s = 0, otherwise.Since the Lie algebra class s contains so many subcases that relevant properties may change from one to anothersubcase with given parameters ( α, β ), e.g. the Lie algebra automorphism group, we will develop here a modificationof our method consisting of applying the Darboux family method to the fundamental vector fields of the action ofthe Lie group Aut all ( s ) of all common Lie algebra automorphisms for all parameters α, β and analysing its relationto the Lie algebra automorphisms for each case, s α,β , to obtain our final classification. To simplify the notation, weskip the parameters α, β in s α,β when they are not needed to understand what we are talking out, e.g. if we jestcare about the dimension of the linear space Λ s .By Remark 2.1, one obtains the following common Lie algebra of derivations of s for all the values of α and β : der all ( s ) = µ µ µ µ µ µ : µ , µ , µ , µ , µ , µ ∈ R , and, after lifting the elements of a basis of der all ( s ) to Λ s , we obtain a basis of V all s , namely the Lie algebra offundamental vector fields of the action of the Lie algebra automorphism group Aut all ( s ) common to all α, β actingon Λ s , of the form X = x ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + x ∂ x ,X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x . It is worth stressing that if α = β and/or one on the coefficients α, β are equal to one, then the Lie algebra ofderivations of the particular s α,β is larger due to the existence of Lie algebra automorphisms of s α,β leaving invariantthe eigenspaces of ad e acting on h e , e , e i (more detailed calculations can be seen at the end of this subsection).By dealing with der all ( s ), we shall derive the orbits of the connected part of the identity of the group of common Liealgebra automorphisms for all α, β , i.e. Aut all ( s ), on each Y s α,β via Darboux families. To obtain the equivalenceof r -matrices up to the action of Aut( s α,β ) for each pair ( α, β ), we will use the action of elements of Aut( s α,β ) notcontained in Aut all , c ( s ) for each particular pair of parameters ( α, β ).16or an element r ∈ Λ s , we get[ r, r ] = 2[(1 + α ) x x − (1 + β ) x x + ( α + β ) x x ] e + 2( α − x x e + 2( β − x x e + 2( β − α ) x x e . For 1 + α + β = 0, we have (Λ s α,β ) s α,β = 0. Then, the mCYBE and the CYBE are the same in this case andthey read(1 + α ) x x − (1 + β ) x x + ( α + β ) x x = 0 , ( α − x x = 0 , ( β − x x = 0 , ( β − α ) x x = 0 . If 1 + α + β = 0, then (Λ s α, − − α ) s α, − − α = h e i and the mCYBE reads( α − x x = 0 , ( β − x x = 0 , ( β − α ) x x = 0 . Since x , x , x are bricks of s α,β for every pair ( α, β ), our Darboux tree for the Darboux families of V all s startswith cases x i = 0 and x i = 0, i ∈ { , , } . The full Darboux tree is presented below. x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 α =1 x = 0 x = 0 x = 0 x = 0 α + β ∈{ , − } x = 0 x = 0 β = − α + β = − x = 0 x x − x x = 0 β = − x x − x x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVVVIVIIVIIIIX α + β ∈{ , − } XXI β = − α + β = − XII α =1 XIII α = − β =1 = 0 x = 0 α = β x = 0 x = 0 β =1 x = 0 x = 0 α =1 x = 0 x = 0 α = − α + β = − x = 0 x x + x x = 0 Isomorphic to XIII α = − β =1 α = − , x x + x x = 0 x x − x x = 0 α = β = − / x x − x x = 0 x x − x x + x x = 0 No solutions x x − x x + x x = 0 XIVXV α = − α + β = − XVI β =1 XVII α = β XVIII α = β = − / XIX α = β =1 We now study four subcases: a) α = β = 1, b) α = 1 = β , c) α = β = 1, d) remaining non-isomorphic cases. Case d) : In this case, ad e acts on h e , e , e i having three different eigenvalues and this leads to the fact thatthe only derivations are those common to all α, β , i.e. der ( s α,β ) = der all ( s ). The connected parts of the loci ofthe Darboux families of the above Darboux tree can be found in Table 2. Such connected parts are the orbits ofAut c ( s α,β ) in Y s α,β for α, β, r -matrices up to elementsof Aut( s α,β ). On the one hand, the Lie algebra automorphism group readsAut( s α,β ) = T T T T T T : T , T , T ∈ R \{ } , T , T , T ∈ R . As in previous sections, we only need for our purposes one element of each connected part of Aut( s α,β ), which haseight ones. An element of each connected component and its lift to Λ s are given by T λ ,λ ,λ := λ λ λ
00 0 0 1 ⇒ Λ T λ ,λ ,λ := λ λ λ λ λ λ λ λ
00 0 0 0 0 λ , λ , λ , λ ∈ {± } . (7.3)By using (7.3), we can verify whether some of the strata of E s in Y s α,β are still connected by a Lie algebraautomorphism of s α,β . The results are summarised in Table 2. To obtain the equivalence classes of coboundarycoproducts for each α, β , it is enough to identify the orbits in Table 2 whose elements are the same up to an elementof (Λ s α,β ) s α,β . In particular, we have the following subcases: • Case α = − , β = 1. Hence, (Λ s − ,β ) s − ,β = h e i . By analysing Table 2, we obtain the coboundarycoproduct classes a ) I , b ) II , III , c ) IV , V , d ) VI , VII ± , e ) VIII , f ) X , g ) XIV , XV α = − . • Case α = − β = −
1. Since (Λ s α,β ) s α,β = 0, the classes of equivalent coboundary coproducts are givenby the orbits of Aut( s α,β ) within Y s α,β . In this case, we have the classes of equivalent coboundary coproductsgiven by I , II , III , IV , V , VI , VII + , VII − , VIII , IX α + β ∈{ , − } , X , XI α + β = − , XIV , XV α + β = − . ase c): This time α = β = 1. Then, we haveAut( s α,α ) = T T T T T T T T : T , T T − T T ∈ R \{ } , T , T , T , T , T , T , T , T ∈ R . To obtain the orbits of Aut( s α,α ) on Y s α,α from the orbits of Aut all ,c ( s ), it is necessary to write Aut( s α,α ) asa composition of Aut all ,c ( s ) with certain Lie algebra automorphisms of s α,α so that their composition generatesAut( s α,α ). This can be done by using the Lie algebra automorphisms of s α,α of the form T A := Id ⊗ A ⊗ Id, for A ∈ GL (2 , R ). Then, Λ T A = A ⊗ Id ⊗ (det A )Id ⊗ A . By taking the action of these Λ T A on the strata of thedistribution spanned by V all s in Y s α,α , we obtain the orbits of Aut( s α,α ) on Y s α,α . Our results are summarised inTable 2.To obtain the equivalence classes of coboundary coproducts for each α = β , it is enough to identify the orbitsin Table 2 whose elements are the same up to an element of (Λ s α,α ) s α,α = 0. Therefore, the result is given by theclasses of equivalent r -matrices detailed in Table 2. Case b):
This time α = 1 = β . Therefore,Aut( s ,β ) = T T T T T T T T : T , T T − T T ∈ R \{ } , T , T , T , T , T , T , T ∈ R . To derive the orbits of Aut( s ,β ) on Y s ,β from the orbits of Aut all ,c ( s ), we again write Aut( s ,β ) as a compositionof Aut all ,c ( s ) with certain Lie algebra automorphisms of s ,β so that their composition generates Aut( s ,β ). Thiscan be done by employing the Lie algebra autmorphisms of s ,β given by T A := A ⊗ Id ⊗ Id, for A ∈ GL (2 , R ). ThenΛ T A = (det A )Id ⊗ ( τ ◦ A ⊗ A ◦ τ ) ⊗ Id, where τ is the permutation of coordinates three and four in Λ s ,β .By taking the action of these Λ T A on the strata of the distribution spanned by V all s in Y s ,β , we obtain the orbitsof Aut( s ,β ) on Y s ,β . Our results are summarised in Table 2.To obtain the equivalence classes of coboundary coproducts for each β , it is enough to identify the equivalenceclasses of r -matrices in Table 2 whose elements are the same up to an element of (Λ s ,β ) s ,β . If β = −
1, these arejust the induced by the orbits of Aut( s ,β ) on Y s ,β . Otherwise, (Λ s , − ) s , − = h e , e i and a ) I , I I I , b ) I I (zero − class) , c ) I V , d ) V β = − e ) V I . Case a):
In this case, it is easier to derive the Darboux families for the Lie algebra of derivations. der all ( s ) = (cid:26)(cid:18) A v (cid:19) : A ∈ gl (3 , R ) , v ∈ R (cid:27) . Due to the larger family of symmetries for α = β = 1, the Lie algebra V s , is spanned by X = x ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + x ∂ x ,X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x ,X = x ∂ x + x ∂ x , X = − x ∂ x + x ∂ x , X = x ∂ x + x ∂ x , X = x ∂ x + x ∂ x ,X = − x ∂ x + x ∂ x , X = x ∂ x + x ∂ x . The Darboux tree is very simple and becomes x x − x x + x x = 0 No solution x x − x x + x x = 0 x = 0 , x = 0 , x = 0 x + x + x = 0 x = 0 , x = 0 , x = 0 x + x + x = 0 0iii
19t is immediate that the above gives rise to two non-zero coboundaries given in Table 2. Since in this case(Λ s , ) s , = 0, each class of equivalent r -matrices amounts to a class of equivalent coboundary coproducts. s Tables 1, 3–5 contain the necessary information on the structure constants of s and several Schouten brackets toprove our following results. Recall that α ∈ R \{ } . First, (Λ s ) s = h e i for 2 + α = 0 and it is zero for remainingallowed values of α . Meanwhile, (Λ s ) s = 0 for α = − s ) s = h e i for α = − s depend on a parameter α ∈ R \{ } , the space of derivations depend on α . Itis indeed the same for all values of α ∈ R \{ , } , and it becomes larger for α = 1. Due to this, We shall proceed asin Subsection 7.3. By Remark 2.1, the space of derivations of s for all possible values of α read der all ( s ) := µ µ µ µ µ µ µ : µ , µ , µ , µ , µ , µ ∈ R . By extending the previous derivations to Λ s and using Remark 2.1, we obtain a basis of V all s of the form X = 2 x ∂ x + x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x ,X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x . We recall that these vector fields span the Lie algebra of fundamental vector fields of the action on Λ s of the Liealgebra automorphisms Aut all ( s ) that are common for all values of α ∈ R \{ } .For an element r ∈ Λ s , we get[ r, r ] = 2[2 x x − (1 + α ) x x + (1 + α ) x x − x x ] e − x e + 2[ − (1 − α ) x x − x x ] e + 2( α − x x e . If α = −
2, we have (Λ s ) s = 0. Thus, the mCYBE and the CYBE are equal in this case and they read2 x x + (1 + α ) x x = 0 , (1 − α ) x x = 0 , x = 0 . For the case α + 2 = 0, we have (Λ s ) s = h e i . Thus, the mCYBE reads(1 − α ) x x = 0 , x = 0 . Since x , x are the bricks for s , our Darboux tree for the Darboux families starts with the cases x i = 0 and x i = 0, i ∈ { , } . The full Darboux tree is presented below. x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 α = − ∨ α = − x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVVVIVII α ∈{− , − } = 0 x = 0 No solutions x = 0 x = 0 α =1 x = 0 x = 0 α = − x = 0 x x + x x = 0 No solutions x x + x x = 0 VIIIIX α = − X α =1 The connected parts of the subspaces denoted in the above diagram are the orbits of Aut all , c ( s ) on Y s . To obtainthe orbits of Aut all ( s ) on Y s , whereAut all ( s ) = T T T T T T T : T , T ∈ R \{ } , T , T , T , T ∈ R , we verify whether some of the connected components of the orbits of Aut all , c ( s ) on Y s are additionally connectedby an element of Aut all ( s ). To do so, we use, as previously, the lift to Λ s of one element of each connectedcomponent of Aut all ( s ), for instance T λ ,λ := λ λ λ
00 0 0 1 ⇒ Λ T λ ,λ = λ λ λ λ λ λ
00 0 0 0 0 λ , λ , λ ∈ {± } . (7.4)For α ∈ R \{ , } , one has that Aut( s α ) = Aut all ( s ), where s α stands for the Lie algebra s for a fixed value of α .Then, the classes of equivalent r -matrices (up to Lie algebra automorphisms of s ) on the Lie algebra s α can easilybe obtained and they are summarised in Table 2. Moreover, the classes of equivalent coboundary coproducts for α = − a ) II(zero class) , b ) I + , III + , c ) I − , III − , d ) IV , e ) VII α = − , f ) V , g ) VI + , h ) VI − , i ) VIII . For those s α with | α | 6 = 1, one has (Λ s α ) s α = 0 and the classes of equivalent coboundary coproducts are given byeach one of the following subsets a ) I − , b ) I + , c ) II , d ) III + , e ) III − , f ) IV , g ) V , h ) VI + , i ) VI − , j ) VII α = − , k ) VIII , l ) IX α = − − , m ) IX α = − . Note that, for values | α | 6 = 1, not all the above classes may be simultaneously available as they arise for particularvalues of α .Let us consider now the case of the Lie algebra s , i.e. the Lie algebra of the class s for α = 1. In this case,the group of Lie algebra automorphisms is slightly larger than for remaining admissible values of α . In particular,Aut( s ) = T T T T T T T T T : T , T ∈ R \{ } , T , T , T , T , T , T ∈ R and Aut( s ) is the group resulting of composing Aut all ( s ) with the Lie algebra automorphisms of s of the form T ( e ) = e , T ( e ) = e + λe , T ( e ) = e + µe , T ( e ) = e , ∀ λ, µ ∈ R . Hence, to obtain the orbits of the action Aut( s ) on Y s , it is enough to act Λ T A on the orbits of Aut all ( s ). Notethat Λ T ( e ) = e + λe , Λ T ( e ) = e , Λ T ( e ) = e , Λ T ( e ) = e − e µ − λµe , Λ T ( e ) = e + λe , Λ T = e + µe , for every λ, µ ∈ R . This will give the final orbits detailed in Table 2. Since (Λ s ) s = 0, the families of equivalentcoboundary coproducts are given by the ones induced by the families of equivalent r -matrices.21 .5 Lie algebra s Structure constants for Lie algebra s are given in Table 1. Using this information, one can compute the Schoutenbrackets between the elements of s , Λ s , and Λ s from the information contained in Tables 3–5. In particular,we obtain (Λ s ) s = h e i for β = 0 and (Λ s ) s = { } for β = 0. Moreover, (Λ s ) s = 0 for α + 2 β = 0 and(Λ s ) s = h e i otherwise.By Remark 2.1, we obtain that the derivations of s read der ( s ) := µ µ µ µ µ − µ µ µ : µ , µ , µ , µ , µ , µ ∈ R , which give rise to the basis of V s of the form X = x ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x + x ∂ x + 2 x ∂ x + x ∂ x + x ∂ x ,X = x ∂ x − x ∂ x + x ∂ x − x ∂ x , X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x . For an element r ∈ Λ s , we get[ r, r ] = 2[ x x + ( α + β ) x x − ( α + β ) x x + x x + 2 βx x ] e + 2[( β − α ) x x + x x ] e + 2[ − x x + ( β − α ) x x ] e − x + x ) e . If α + 2 β = 0, then (Λ s ) s = 0. Thus, the mCYBE and the CYBE are equal in this case and they read βx x = 0 , x = 0 , x = 0 . For α + 2 β = 0, we have (Λ s ) s = h e i and the mCYBE reads x = 0 , x = 0 . Since x is the only brick for s , we start our Darboux tree with the cases x = 0 and x = 0. The full Darbouxtree is presented below. x = 0 x = 0 No solutions x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x + x = 0 x + x = 0 x + x = 0 x + x = 0 0IIIIII x = 0 x = 0 No solutions x = 0 x = 0 No solutions x = 0 x = 0 β =0 ∨ α +2 β =0 x = 0 IVV α = − ββ =0 Orbits of Aut c ( s ) are given by the connected parts of the loci of the above Darboux tree. Solutions are describedin Table 2.The automorphism group of s readsAut( s ) = T T T T T − T T T : T ∈ R , ( T ) + ( T ) > , T , T , T , T , T ∈ R . GL (2 , R ) of matrices of the form (cid:18) T T − T T (cid:19) ∈ GL (2 , R ) , ( T ) + ( T ) > , can be parametrised via φ ∈ [0 , π [ and µ := [( T ) + ( T ) ] / ∈ R + by setting T = µ cos φ and T = µ sin φ ,there are two connected components of Aut( s ). A representative of each connected part and its lift to Λ s read T λ := λ = ⇒ Λ T λ := λ λ λ , λ ∈ {± } . (7.5)Using techniques from previous sections, we can easily verify whether the orbits of Aut c ( s ) are additionallyconnected by a Lie algebra automorphism of s via Λ T λ . Our results are summarised in Table 2. Moreover, for β = 0, the classes of coboundary coproducts are induced from the following families of r -matrices: a ) I , b ) II + , c ) II − , d ) III + , e ) III − , f ) IV , g ) V α = − β + , h ) V α = − β − . Meanwhile, for β = 0, the list of families of equivalents coboundary coproducts read a ) II ± , b ) I , III ± , c ) IV , V β =0 ± . s Structure constants of Lie algebra s are given in Table 1. Using this information, we can compute the Schoutenbrackets of elements of s , Λ s , and Λ s (see Tables 3–5). As in previous sections, Tables 1,3–5 contain thenecessary information to accomplish the following calculations. In particular, we have (Λ s ) s = h e i while(Λ s ) s = { } .By Remark 2.1, one obtains that der ( s ) := µ µ µ µ µ − µ µ − µ − µ : µ , µ , µ , µ , µ ∈ R , which gives rise to the basis of V s of the form X = x ∂ x + 2 x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = ( − x + x ) ∂ x + x ∂ x − x ∂ x ,X = ( − x − x ) ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x − x ∂ x + x ∂ x − x ∂ x . For an element r ∈ Λ s , we get[ r, r ] = 2( x x + x x + x ) e + 2( x + x ) x e + 2( x − x ) x e − x x e . Since (Λ s ) s = h e i , the mCYBE reads( x + x ) x = 0 ( x − x ) x = 0 , x x = 0 , whereas the CYBE is x x + x x + x = 0 , ( x + x ) x = 0 , ( x − x ) x = 0 , x x = 0 . Note that the CYBE obtained above is exactly the result obtained in [6, eq. (3.7)] under the substitution x = − α + , x = − α − , x = ξ and x = ϑ .Since x , x are the bricks of s , our Darboux tree starts with cases x i = 0 and x i = 0, for i ∈ { , } . The fullDarboux tree is presented below. 23 = 0 x = 0 x = 0 x = 0 x = 0 No solutions x + x = 0 x + x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x − x = 0 x + x = 0 x x + x = 0 x x + x = 0 k/ ∈{− , , } x − kx = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 0II ext IIIIIIVVVIV ext VI ext VII | k | / ∈{ , } VIIIIX x = 0 x = 0 No solutions x = 0 No solutions − x + x = 0 − x + x = 0 x x + x = 0 x x + x = 0 VIII ext IX ext The connected parts of the subsets denoted in the above Darboux tree are the orbits of Aut c ( s ).The Lie algebra automorphism group of s readsAut( s ) = − T T T T T T T T T T T − , T T − T T − T T T T T T T : T , T ∈ R \{ } ,T , T ∈ R \{ } ,T , T , T ∈ R . One element of each connected component of Aut( s ), which are eight, and their lifts to Λ s are given by ∓ λ λ θ ( ∓ λ θ ( ± λ θ ( ± λ θ ( ∓ λ
00 0 0 ∓ ⇒ θ ( ∓ λ − θ ( ± λ − θ ( ± λ θ ( ∓ λ λ λ ∓ λ λ θ ( ∓ λ − θ ( ± λ − θ ( ± λ θ ( ∓ λ , (7.6)where θ ( x ) stands for the Heaviside function and λ , λ ∈ { , − } . By using the previous information, we can verifywhether some of the orbits of the action of Aut c ( s ) on Y s can be connected by an element of Aut( s ), which gives24ise to the classes of equivalent r -matrices up to the action of Lie algebra automorphisms of s . Since (Λ s ) s = 0,each class of equivalent r -matrices gives rise to a separate class of coboundary Lie bialgebras on s . Hence, theclasses of equivalent coboundary Lie bialgebras are given by a ) I , I ext b ) II , c ) , III , d ) IV , e ) V , V ext f ) VI , VI ext , g ) VII | k |6 = { , } , h ) VIII , VIII ext , i ) IX , IX ext . s The structure constants of Lie algebra s are given in Table 1. Using this information, we can compute somerelevant Schouten brackets between the elements of the bases of s , Λ s , and Λ s (see Tables 3–5). Note that,from these tables, we get (Λ s ) s = 0 and (Λ s ) s = h e i .By Remark 2.1, one obtains that der ( s ) := µ µ µ µ µ µ µ − µ
23 12 µ µ : µ , µ , µ , µ , µ ∈ R , which give rise to the basis of V s of the form X = 32 x ∂ x + 32 x ∂ x + x ∂ x + x ∂ x + 12 x ∂ x + 12 x ∂ x , X = x ∂ x + x ∂ x + x ∂ x − x ∂ x ,X = − x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x − x ∂ x + x ∂ x − x ∂ x . Then, [ r, r ] = 2( x + x x + x x ) e + 2( x x + x x ) e + 2( x x − x x ) e − x + x ) e . Since (Λ s ) s = h e i , then the mCYBE reads x = x = 0 . Meanwhile, the CYBE takes the form x = x = x = 0 . The Darboux tree is presented below. x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x − kx = 0 x + x = 0 x + x = 0 0IIIIIIIV k =0 The connected parts of the subsets denoted in the above Darboux tree are the orbits of Aut c ( s ) in Y s . Byusing the extension of the automorphism group Aut( s ) to Λ s , we can verify whether some of these parts areadditionally connected by a Lie algebra automorphism of s . The results are summarised in Table 2. As before,each family of r -matrices induces a separate class of coboundary coproducts.The Lie algebra automorphism group of s readsAut( s ) = ± [( T ) + ( T ) ] ± T T − T T T T ± T T T T T T ∓ T ± T T ± : ( T ) + ( T ) ∈ R \{ } T , T , T , T , T ∈ R . GL (2 , R ) of matrices of the form (cid:18) T T ∓ T ± T (cid:19) ∈ GL (2 , R ) , ( T ) + ( T ) > , can be parametrised via φ ∈ [0 , π [ and µ := [( T ) + ( T ) ] / ∈ R + by setting T = µ cos φ and T = µ sin φ ,one gets that Aut( s ) has two connected components. As usual, we only need one element for every connectedcomponent of Aut( s ) and their lifts to Λ s , namely T ± := ± ± ± = ⇒ Λ T ± = ± ± ± . Taking this into account and since (Λ s ) s = 0, we obtain that the families or equivalent coboundary Liebialgebras are given by a) I , b) II + , c) II − , d) III , e) (IV + ) | k |∈ R + f) (IV − ) | k |∈ R + . s The structure constants for the Lie algebra s are given in Table 1. Recall that α ∈ ] − , \{ } . As in the previouscases, one can obtain some relevant Schouten brackets between the basis elements of s , Λ s , and Λ s (see Tables3–5). In view of Tables 1,3–5, one has that (Λ s ) s = h e i for α = − and (Λ s ) s = 0, otherwise. Moreover,(Λ s ) s = 0.By Remark 2.1, one obtains that the derivations of s α , for a fixed value α ∈ ] − , \{ } read der ( s α ) = µ µ µ µ µ − µ µ − µ αµ : µ , µ , µ , µ , µ ∈ R . (7.7)In case α = 1, one gets der ( s ) = µ µ µ µ µ µ − µ µ µ − µ µ : µ , µ , µ , µ , µ , µ , µ ∈ R . (7.8)For any value of α ∈ ] − , \{ } and an element r ∈ Λ s , one obtains[ r, r ] = 2[(2+ α ) x x − (1+2 α ) x x +(1+ α ) x x + x ] e +2( x − αx ) x e +2 x ( x − x ) e +2( α − x x e . Since (Λ s ) s = 0, the mCYBE reads(2 + α ) x x − (1 + 2 α ) x x + (1 + α ) x x + x = 0 , ( x − αx ) x = 0 , x ( x − x ) = 0 , ( α − x x = 0 . Let us now consider two cases given by α ∈ ] − , \{ } and α = 1. In the first case, the space of derivations givesrise to the basis of V s α of the form X = x ∂ x + 2 x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = ( x α + x ) ∂ x + x ∂ x + αx ∂ x ,X = − ( x + x ) ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x − x ∂ x + x ∂ x − x ∂ x . (7.9)The following diagram depicts the Darboux families that give us the orbits of Aut c ( s α ) on Y s α . Since x , x are thebricks for every s α , our Darboux tree starts with the cases x i = 0 and x i = 0, for i ∈ { , } . The full Darboux treeis presented below. 26 = 0 x = 0 x = 0 x = 0 x = 0 x − αx = 0 No solutions x − αx = 0 x = 0 x = 0 ( x =0) x + x = 0 No solutions x + x = 0 αx − x x = 0 ( α = − / αx − x x = 0 x = 0 x = 0(1 + α ) x + x = 0 No solutions (1 + α ) x + x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVVVIVII α = − / x = 0 x = 0 ( α =1) x = 0 x − x = 0 No solutions x − x = 0 x − x = 0 No solutions x − x = 0 No solutions x + x x = 0 x + x x = 0 x x − x x + x = 0 No solutions x x − x x + x = 0 VIIIIX α =1 The Lie algebra automorphism group of s α for each α ∈ ] − , \{ } , readsAut( s α ) = T T T T T T − T T T αT T : T , T ∈ R \{ } T , T , T ∈ R . In reality, we are only concerned with obtaining one element of Aut( s α ) and its lift to Λ s for each one of itsconnected components. For instance, we can choose T λ ,λ = λ λ λ λ
00 0 0 1 = ⇒ Λ T λ ,λ = λ λ λ λ λ λ λ
00 0 0 0 0 λ , λ , λ ∈ {± } . (7.10)As in previous sections, the maps Λ T λ ,λ allow us to identify the orbits of the action of Aut( s α ) on Y s α . Ourresults are presented in Table 2. Moreover, for α = − , each family of equivalent r -matrices give rise to a separateclass of equivalent coboundary Lie bialgebras. For α = − , we obtain eight families of equivalent coboundary Liebialgebras given by: a ) II (zero class) , b ) I , III , c ) IV , d ) V , e ) VI , VII α = − − , VII α = − + , f ) VIII . α = 1 using the previous results. Since (7.7) for α = 1 is a Lie subalgebra of the space ofderivations for s α given in (7.8), the Lie algebra spanned by the vector fields of (7.9) for α = 1 is a Lie subalgebra ofthe Lie algebra of fundamental vector fields of the action of Aut( s ) on Λ s . Hence, the loci of the above Darbouxfamilies allow us to characterise the strata of the distribution spanned by the vector fields (7.9) for α = 1, which inturn gives as the orbits of a Lie subgroup of Aut( s ). To easily follow our discussion, we detail thatAut( s α ) = T T − T T − T T + T T − T T + T T T T T T T T T : T T − T T = 0 , T , T , T ∈ R . To obtain the orbits of Aut( s ) on Y s , it is enough to write Aut( s ) as a composition of the previous subgroupwith certain Lie algebra automorphisms of Aut( s ), e.g. the Lie algebra automorphisms T A = (det A )Id ⊗ A ⊗ Idfor every A ∈ GL (2 , R ). Hence, Λ T A = (det A ) A ⊗ (det A )Id ⊗ (det A )Id. The action of the Λ T A on the lociof the Darboux families of (7.9) assuming α = 1 allows us to obtain the orbits of the action of Aut( s ) on Y s .In particular, one obtains the subsets given in Table 2. Since (Λ s ) s = 0, the classes of equivalent coboundarycoproducts for s are given by a ) I , b ) II , c ) III , d ) IV . s As previously, we use the structure constants for s in Table 1 to compute the Schouten brackets given in Tables 3–5.Moreover, (Λ s ) s = 0 and (Λ s ) s = 0 for every α >
0. These calculations are enough to verify the remainingresults of present subsection.By Remark 2.1, one obtains that der ( s α ) := µ µ µ µ µ µ µ − αµ − µ µ αµ + µ : µ , µ , µ , µ , µ ∈ R . The obtained derivations give rise to the basis of V s α of the form X = x ∂ x + ( αx + x ) ∂ x + x ∂ x + ( αx − x ) ∂ x , X = ( − αx − x ) ∂ x + x ∂ x + x ∂ x + ( x + αx ) ∂ x ,X = − x ∂ x − x ∂ x , X = 3 x ∂ x + 3 x ∂ x + 2 x ∂ x + 2 x ∂ x + x ∂ x + x ∂ x ,X = x ∂ x − x ∂ x + x ∂ x − x ∂ x . For an element r ∈ Λ s , we get[ r, r ] = 2( x x + 3 αx x − αx x + x x + 2 αx x + x ) e + 2( − αx x + x x + x x ) e + 2( x x − x x − αx x ) e − x + x ) e . And since (Λ s ) s = 0 for every value of α , the mCYBE and the CYBE are equal and read(2 αx + x ) x = 0 , x = 0 , x = 0 . The Darboux tree for the class s is presented below. x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x + x = 0 x + x = 0 x + 2 αx = 0 No solutions x + 2 αx = 0 0IIIIII
28f we define ∆ := ( T ) + ( T ) , the Lie algebra automorphisms group of each s α readsAut( s α ) = ∆ T ( T + αT )+ T ( αT − T )1+ α ( − αT + T ) T + T ( T + αT )1+ α T T T T − T T T : ∆ ∈ R + ,T , T , T , T , T ∈ R . Using ideas from Section 7.5, we obtain that each Aut( s α ) has one connected component. Consequently, theorbits of Aut( s α ) on Λ s α are the strata of E s α . Since (Λ s ) s = 0 for every α >
0, the strata of E s α within Y s α amount for the families of equivalent coboundary Lie bialgebras on s α . Our final results are summarised in Table2. s As in the previous cases, we hereafter use the structure constants for Lie algebra s in Table 1 and the in-duced Schouten brackets between elements of s , Λ s , and Λ s depicted in Tables 3–5. It is remarkable that(Λ s ) s = 0 and (Λ s ) s = 0.By Remark 2.1, the derivations of s read der ( s ) = µ µ µ µ µ µ µ − µ µ µ : µ , µ , µ , µ , µ ∈ R , which give rise to the basis of V s of the form X = 32 x ∂ x + 32 x ∂ x + x ∂ x + x ∂ x + 12 x ∂ x + 12 x ∂ x , X = x ∂ x + ( x + x ) ∂ x + x ∂ x + ( x − x ) ∂ x ,X = ( − x − x ) ∂ x + x ∂ x + x ∂ x , X = − x ∂ x − x ∂ x , X = x ∂ x + x ∂ x . For an element r ∈ Λ s , we get[ r, r ] = 2(3 x x − x x + x x + 2 x x + x ) e + 2( − x x + x x + x x ) e + 2( x − x ) x e − x e . Since (Λ s ) s = 0, the mCYBE and the CYBE are equal and they read − x x + 2 x x + x = 0 , ( x − x ) x = 0 , x = 0 . Since x is the only brick for s , our Darboux tree starts with the cases x = 0 and x = 0. The full Darbouxtree is presented below. x = 0 x = 0 x = 0 x = 0 x = 0 No solutions x − x = 0 x − x = 0 x = 0 x = 0 x + x = 0 No solutions x + x = 0 No solutions x − x x = 0 x − x x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVV c ( s ) within Y s . Toobtain the orbits of Aut( s ) within Y s , we proceed as in previous sections and we derive the automorphism groupof s , which takes the formAut( s ) = ( T ) T T T T + T T − T T T T T T T T : T ∈ R \{ } ,T , T , T , T ∈ R . In reality, we do not need the full group. It is enough to note that there are two connected components of Aut( s ),represented by two automorphisms that read, along with their extensions to Λ s , as follows T λ := λ λ
00 0 0 1 = ⇒ Λ T λ := λ λ λ
00 0 0 0 0 λ , λ ∈ {± } . (7.11)By applying Λ T λ to the orbits of Aut c ( s ) in Y s , depicted in Table 2, we obtain the orbits of Aut( s ) in Y s .Since (Λ s ) s = 0, such orbits amount to the classification of equivalent coboundary Lie bialgebras up to Liealgebra automorphisms of s . s As previously, the information in Tables 1, 3–5 allows us to accomplish the calculations of this section. In particular,one gets (Λ s ) s = 0 and (Λ s ) s = 0.By Remark 2.1, one obtains der ( s ) = µ µ µ µ µ − µ µ − µ
00 0 0 0 : µ , µ , µ , µ , µ ∈ R . Such derivations give rise to the basis of V s of the form X = x ∂ x + 2 x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x , X = ( − x − x ) ∂ x + x ∂ x + x ∂ x ,X = − x ∂ x − x ∂ x , X = x ∂ x − x ∂ x + x ∂ x − x ∂ x , For an element r ∈ Λ s , we get[ r, r ] = 2(2 x x − x x + x x + x ) e + 2 x x e + 2( x − x ) x e − x x e . Since (Λ s ) s = { } , the mCYBE and the CYBE are equal and read2 x x − x x + x x + x = 0 , x x = 0 , ( x − x ) x = 0 , x x = 0 . Since x , x are the bricks for s , our Darboux tree starts with the cases x i = 0 and x i = 0, for i ∈ { , } . TheDarboux tree is presented below. x = 0 x = 0 No solutions x = 0 x − x = 0 No solutions x − x = 0 x + x x = 0 No solutions x + x x = 0 IX = 0 x = 0 x = 0 x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x + x = 0 No solutions x + x = 0 x = 0 No solutions x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVVVIVIIVIII The above Darboux tree gives the orbits of Aut c ( s ) in Y s . To obtain the equivalence classes of r -matricesup to Lie algebra automorphisms of Aut( s ), we obtain thatAut( s ) = T T T − T T T T T T
00 0 0 1 : T , T ∈ R \{ } T , T , T ∈ R . In fact, only one representative for each connected component is needed for our purposes. There are four connectedcomponents of Aut( s ), each one represented by one of the following automorphisms, which are given togetherwith their lifts to Λ s , namely T λ ,λ := λ λ λ λ
00 0 0 1 = ⇒ Λ T λ ,λ = λ λ λ λ λ λ λ
00 0 0 0 0 λ , λ , λ ∈ {± } . (7.12)The Λ T λ ,λ allow us to check whether the orbits of Aut c ( s ) are connected by an element of Aut( s ) acting onΛ s . The equivalence classes of such r -matrices (up to the action of elements of Aut( s ), as standard) are givenin Table 2. Since (Λ s ) s = 0, each family of r -matrices gives rise to a separate class of equivalent coboundaryLie bialgebras on s . s The structure constants for Lie algebra s are given in Table 1. This along with the selected Schouten bracketsbetween basis elements of s , Λ s , and Λ s described in Tables 3–5, allow us to accomplish the calculations ofthis section. In particular, we obtain that (Λ s ) s = 0 and (Λ s ) s = 0.By Remark 2.1, one obtains that der ( s ) = µ µ µ µ − µ µ µ − µ : µ , µ , µ , µ ∈ R . V s given by X = 2 x ∂ x + x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = x ∂ x + x ∂ x − x ∂ x − x ∂ x ,X = ( − x − x ) ∂ x + x ∂ x + x ∂ x , X = ( x − x ) ∂ x − x ∂ x + x ∂ x . For an element r ∈ Λ s , we get[ r, r ] = − x x + x x ) e + 2( − x x + x x − x − x x − x ) e − x + x ) x e + 2( x − x ) x e . Since (Λ s ) s = 0, the mCYBE and CYBE are equal and read x x + x x = 0 , x x − x x + x + x x + x = 0 , ( x + x ) x = 0 , ( x − x ) x = 0 . Since x is the only brick for s , our Darboux tree starts with the cases x = 0 and x = 0. The full Darbouxtree is given next. x = 0 x − x = 0 x + x = 0 x + x = 0 x − x = 0 x + x = 0 No solutions x + x = 0 x + x = 0 x + x = 0 x = 0 x = 0 x x + x x = 0 No solutions x x + x x = 0 No solutions − x x + x + x x + x = 0 − x x + x + x x + x = 0 0IIIIII x − k = 0, ( k = 0) x + x = 0 x − x = 0 No solutions x + x = 0 ∨ x − x = 0 x + x − x x + x x + 2 x x = 0 No solutions x + x − x x + x x + 2 x x = 0 IV | k | > The connected parts of the loci of the Darboux families of the Darboux tree are the orbits of Aut c ( s ) within Y s .These are given by the connected parts of the subsets given in Table 2. Let us obtain the orbits of Aut( s ) on Y sl .The Lie algebra automorphism group of s readsAut( s ) = T T T ± T ∓ T ± T T ∓ T ± : ( T ) + ( T ) ∈ R \{ } T , T , T , T ∈ R . There are two connected components of Aut( s ). One such element for each connected component of Aut( s ) andtheir extensions to Λ s are given by T ± := ± ± = ⇒ Λ T ± = ± ± ± ± . By using Λ T ± on the loci of the above Darboux tree, we obtain the orbits of Aut( s ) that are given by the classesof equivalent r -matrices detailed in Table 2. Since (Λ s ) s = 0, each family of equivalent r -matrices gives rise toa separate class of equivalent coboundary Lie bialgebras.32 .13 Lie algebra n The structure constants for the Lie algebra n are given in Table 1, while relevant Schouten brackets between theelements of bases of n , Λ n , and Λ n to be used hereafter are displayed in Tables 3–5. From these calculations,one obtains that (Λ n ) n = h e i and (Λ n ) n = h e , e i . By Remark 2.1, one has that the derivations of n take the form der ( n ) = µ µ µ µ µ µ µ µ − µ µ µ − µ : µ , µ , µ , µ , µ , µ , µ ∈ R , which, by lifting them to Λ n , give rise to the basis of V n of the form X = x ∂ x + 2 x ∂ x − x ∂ x + x ∂ x , X = x ∂ x + x ∂ x + x ∂ x + x ∂ x , X = − x ∂ x + x ∂ x ,X = − x ∂ x − x ∂ x , X = x ∂ x + 2 x ∂ x − x ∂ x + 3 x ∂ x + x ∂ x ,X = x ∂ x − x ∂ x , X = x ∂ x + x ∂ x . For an element r ∈ Λ n , we get[ r, r ] = 2( x x − x x ) e + 2( x − x x ) e + 2 x x e + 2 x e . Thus, CYBE reads x = 0 , x = 0 . Since (Λ n ) n = h e , e i , the mCYBE differs from the CYBE and reads x = 0 . The Darboux tree for this Lie algebra reads x = 0 x = 0 x = 0 x = 0 x = 0 x x − x x = 0 x x − x x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 0IIIIIIIVVVIVII As in all previous sections, the orbits of Aut c ( n ) in Y n are given by the connected components of the loci of theDarboux families of the above Darboux tree. Results can be found in Table 2.Let us again obtain the orbits of Aut( n ) in Y n . The automorphism group of n takes the formAut( n ) = T ( T ) T T T T T T T T T T T : T , T ∈ R \{ } , T , T , T , T , T ∈ R . Note that the dimension of the space of derivation matches the result given in the WˇS classification
33t has four connected components represented by the following Lie algebra automorphisms, which are also lifted toΛ n : T λ ,λ := λ λ λ λ
00 0 0 λ = ⇒ Λ T λ ,λ = λ λ λ λ λ
00 0 0 0 0 λ λ , λ , λ ∈ {± } . The action of the Λ T λ ,λ on the loci of the above Darboux tree gives the families of equivalent r -matrices givenin Table 2. Since (Λ n ) n = h e i and using the information in Table 2, we see that the families of equivalentcoboundary Lie bialgebras on n are given by orbits of Aut( n ) with the exception that the class I and the zeroclass (given by the zero r -matrix) give the Lie bialgebra in n with a zero coproduct. This work has devised a generalisation of the theory of Darboux polynomials to determine and to classify up to Liealgebra automorphisms the coboundary real Lie bialgebras over indecomposable real four-dimensional Lie algebrasin a geometric manner. As a byproduct, a technique for the matrix representation of Lie algebras with non-trivialkernel has been developed. Such matrix representations are frequently useful in calculations.The procedures devised in this work are good enough to make affordable the classification, up to Lie algebraautomorphisms, of coboundary Lie bialgebras on real and complex, of at least dimension five, indecomposable Liealgebras via the ˇSnobl and Winternitz classification [50]. The case of two- and three-dimensional Lie bialgebras canalso be obtained. Moreover, a brief analysis of the classification of coboundary coproducts on real four-dimensionaldecomposable Lie algebras through our methods shows that their classification relies partially on the classificationof coboundary coproducts on three- and two-dimensional Lie algebras, while the complexity of the procedure issignificantly easier than in the present work [25, 35]. We expect to tackle this task in the future and to compareour results with previous works on the topic [11, 35, 30]. Our techniques are also expected to be applied to otherhigher-dimensional Lie algebras of particular types, e.g. five-dimensional nilpotent or semi-simple Lie algebras.In all previously mentioned cases, we aim to inspect new properties of the Darboux families used to study theseproblems.The determination of coboundary Lie bialgebras is relevant to the study of Poisson Lie groups. In particular, itrepresents an initial step in the study of Poisson Lie groups of dimension four, which would give rise to an extensionof the paper [3].Our work has focused on the classification of coboundary Lie bialgebras on indecomposable Lie algebras. Prob-ably, one laborious task in the application of our method is the determination of a representative of each connectedpart of the Lie group of Lie algebra automorphisms of a Lie algebra. In the case of semi-simple Lie algebras, thisis much easier, as algebraic techniques based on Dynkin diagrams and other results can be applied [13, 40]. It isleft for further works to study this problem and to search for coboundary Lie bialgebras in the Lie algebra so (3 , D. Wysocki acknowledges support from a grant financed by the University of Warsaw (UW) and the Kartezjuszprogram of the Jagiellonian University and UW. J. de Lucas acknowledges partial financial support from the NCNgrant HARMONIA 2016/22/M/ST1/00542.
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Orbit Dim x x x x x x Repr. element s I ± R ± ± e II 1 0 R \{ } e III ± R ± R \{ } ± e + e IV 2 0 R R \{ } e V ± R ± R R \{ } ± e + e VI 3
R R R \{ } e VII 3 0 R R R \{ } e VIII ± R ± R R R \{ } ± e + e s I ± R ± ± e II ± R R ± ± e III 3
R R R \{ } e IV ± R R R ± ± e I 1 R \{ } e II 1 0 R \{ } e III 2 R \{ } R \{ } e + e IV 1 0 0 0 R \{ } e V 2 R \{ } R \{ } e + e s α,β VI 2 0 R \{ } R \{ } e + e differ VII ± R \{ } R \{ } ± x x x > ± ( e + e + e ) α, β, R R R \{ } e IX α + β ∈{ , − ∗} R R R \{ } R \{ } e + e X 3 R R R \{ } e XI α + β = − R R \{ } R R \{ } e + e XIV 3 0 R R R \{ } e XV α = − α + β = − , ∗ R \{ } R R R \{ } e + e I R x + x = 0 0 0 0 0 e II R \{ } e s α,α III R x + x = 0 0 R \{ } e + e IV R R R \{ } e α =1 V ∗ α = − / R R R \{ } R \{ } e + e VI ∗ α = − / x x − x x = 0 0 R R x + x = 0 e + e VII x x − x x = 0 0 R R x + x = 0 e I R \{ } e I I R x + x = 0 0 0 e s ,β I I I R \{ } R x + x = 0 0 0 e + e I V R R x + x = 0 R x x − x x = 0 0 e β =1 V β = − R R x + x = 0 R x x − x x = 0 0 e + e V I R R R \{ } e i 3 R R x + x + x = 0 0 0 e s , ii 5 x x + x x = x x R x x + x x = x x R x + x + x = 0 e + e s α I ± R ± ± e II 1 0 R \{ } e α / ∈ { , } III ± R ± R \{ } ± e + e IV 3
R R R \{ } e V 2 0 R R \{ } e VI ± R ± R R \{ } ± e + e VII α ∈{− ∗ , − } R R R \{ } R \{ } e + e VIII 3 0 R R R \{ } e IX α = − , ∗± R ± R R R \{ } ± e + e X α =1 R R R \{ } − x x x R \{ } e + e s I ± R ± R ± e II R \{ } e III R R R \{ } e IV R R R \{ } e V R R x x − x x = 0 0 R \{ } e + e s I 1 R x + x = 0 0 0 0 0 e II ± R ± ± e III ± R x + x = 0 0 R ± e ± e IV 3
R R R \{ } e (V ± ) β =0 α = − β ∗ R R R \{ } R ± e ± e s I 1 R \{ } e I ext R \{ } R \{ } R \{ } e + e III 3
R R R \{ } e IV ∗ R R R \{ } e V ∗ R x R \{ } e + e V ext ∗ R − x R \{ } Orbit Dim. x x x x x x Repr. element s VI ∗ R R \{ } x R \{ } e + e + e VI ext ∗ R \{ } R − x R \{ } ∗| k | / ∈{ , } R R kx R \{ } ke + e VIII 4 R − x /x R − x R \{ } e VIII ext − x x R R x R \{ } IX ∗ R x + x /x = 0 R − x R \{ } e + e IX ext ∗ x + x x = 0 R R x R \{ } s I 2 R x + x = 0 0 0 0 0 e II ± R R R ± ± e III ∗ R R R \{ } e (IV ± ) ∗| k |∈ R + R R R ± kx ± e ± ke s I 1 R \{ } e II 1 0 R \{ } e III 2 R \{ } R \{ } e + e IV 3
R R R \{ } e V 3
R R R \{ } − (1 + α ) x e − (1 + α ) e VI 3 R αx x R αx R \{ } e VII α = − / ± R x x − αx ∈ R ± R αx R \{ } e ± e VIII 3 − x x R R x R \{ } e s I R x + x = 0 0 0 0 0 e II R R R \{ } e III R R R \{ } − x e − e IV x x − x x + x = 0 x R x + x = 0 e s I 2 R x + x = 0 0 0 0 0 e II ± R R R ± ± e III ± R R R ± − αx ± αe ∓ e s I 1 R \{ } e II 2
R R \{ } e III ± R R R ± ± e IV ± R R R ± − x ± e ∓ e V 3 R x x R x R \{ } e s I 1 R \{ } e II 1 0 R \{ } e III 2 R \{ } R \{ } e + e IV 2 R R \{ } e V 3
R R \{ } R \{ } e + e VI 2 0
R R \{ } − x e − e VII 3 R \{ } R R \{ } − x e + e − e VIII 3 R R R \{ } e IX 3 − x x R R x R \{ } e s I 1 R \{ } e II 3
R R x + x = 0 0 0 e III 3 R x R − x x + x = 0 0 e + e IV | k | > − x + x x R R x − x k ke n I 1 R \{ } e II ± R R ± ± e III 3
R R R \{ } e IV 3
R R R \{ } e V 4
R R R \{ } R \{ } e + e VI ∗ R x x x R R R \{ } e VII ∗± R x − x x x ∈ R ± R R R \{ } e ± e Table 2: Orbits of the action of Aut( g ) on Y g , their dimensions, elements, and representatives, for real four-dimensional indecomposable Lie algebras g . Each Lie algebra is divided into several subsets enumerated by romannumbers, which classify the orbits of Aut( g ) in Y g . The orbits of Aut c ( g ) are given by the (topologically) connectedcomponents of each orbit of Aut( g ). The trivial orbits given by 0 ∈ Λ g are not considered. Symbol A ± stands fortwo orbits, one with + and other with − . In these cases, R ± stands for R + for the first orbit and R − for the second.If an orbit of Aut( g ) in Λ g belongs to Y g only for a certain set of parameters, each family of possible values of theparameters is indicated in a subindex, first, or as a superindex, if a second family of parameters is available. A star( ∗ ) is used to denote r -matrices that are not solutions to the CYBE.38 e e e e e s e e e e e − e − e e e e + e e e s e e e e − e e e + e e − e − e − e e e e e + 2 e e e + 2 e e + e e + e s e e e e − αe αe e − βe − βe e (1 + α ) e (1 + β ) e e ( α + β ) e αe βe s e e e e − e e e + e e − αe − αe e e (1 + α ) e e e + (1 + α ) e e + e αe s e αe αe e e − βe e βe e − e − βe − βe e e ( α + β ) e − e e + ( α + β ) e αe βe βe − e e + βe s e e − e − e e + e e e − e − e + e e e − e e − e s e e e − e e e e − e − e − e e e − e e − e e s e α ) e (1 + α ) e e − e − e e + e e − αe − e − e − αe e (2 + α ) e (1 + 2 α ) e (1 + α ) e (1 + α ) e e αe s e αe αe e e − αe − e e e + αe e − e − αe − e − e − αe e e αe − e αe + e αe αe αe − e e + αe s e e e e − e − e e + e e − e − e − e − e − e e e e e + 3 e e e e e + e s e e e e − e − e e + e e − e − e e e e e e e s e e e − e − e e − e e − e + e e e e e e e e − e − e e e n e e − e − e e e − e e − e − e − e − e Table 3: Schouten brackets between basis elements of g and Λ g for real four-dimensional indecomposable Liealgebras. 39 e e e e e s e e − e e e e e − e e − e − e + e e s e e e − e e e e e e − e e − e − e e − e s e α ) e e − (1 + β ) e e α + β ) e ( α − e ( β − e e e β − α ) e e s e e e − (1 + α ) e e α ) e α − e e − e e − e ( α − e − e e s e e ( α + β ) e e − ( α + β ) e e e βe ( β − α ) e − e ( β − α ) e + e e e − e e − e s e e e e e e − e e e e e e − e e e e e e s e − e e e e e e e − e e − e s e α ) e e − (1 + 2 α ) e e α ) e − αe − e e e e e e α − e e s e e αe e − αe e e αe − αe − e e − αe e e e e e − e e − e s e e e − e e e e − e e − e e e e e e e − e s e e e − e e e − e e e e e e − e e s e − e e − e e − e e − e − e e e − e − e e − e − e e n e e − e e − e e e e e e e e Table 4: Schouten brackets between basis elements of Λ g for four-dimensional indecomposable Lie algebras. e e e s e e − e e − e e e e e + e s e − e e e − e e − e e e e e e + 2 e e + 2 e s e − e e αe e − βe e (1 + α + β ) e (1 + α ) e (1 + β ) e ( α + β ) e s e e e e − e e − αe e (2 + α ) e e (1 + α ) e e + (1 + α ) e s e − αe e e βe e − βe e e ( α + 2 β ) e ( α + β ) e − e e + ( α + β ) e e s e e e − e e e − e e e − e s e e e − e e e − e e − e e s e − (1 + α ) e e e − e e − αe − e e α ) e (2 + α ) e (1 + 2 α ) e (1 + α ) e s e − αe e e αe − e e − αe e − e e αe αe − e αe + e αe s e − e e e − e e − e e − e e e e e + 3 e e s e − e e e − e e − e e e e e e s e − e − e e − e − e e e e e e e − e e n e e e e − e e − e − e Table 5: Schouten brackets between basis elements of g and Λ gg