aa r X i v : . [ m a t h . R A ] J a n ABOUT APPROXIMATE AND DUAL LIE ALGEBRAS
V.V. GorbatsevichIntroduction o
In this article the concept of an approximate Lie algebra, depending on smallparameter, and which is naturally arising when studying approximate symmetry(about which see, for example, fundamental article [1]) of the differential equations,is considered. At the same time it is natural to pass to more general concept ofdual or D -Lie algebras, for which many results are represented in more convenientform. The main purpose of this article — to clear some algebraic concepts, whichare used when studying approximate solutions of the equations and their approx-imate symmetry. The author hopes that the results, which are contained in thisarticle, will be useful when studying, in particular, of Lie algebras of infinitesimalapproximate symmetry of differential equations.In Introduction the way, which led to use of the concept of dual or D -Liealgebras, will be described.The concept of the approximate Lie algebra, which we will study, was enteredfor a long time. In the beginning it was used only for Lie algebras of vector fields(or — it was very frequent — for the corresponding differential operators) and asa separate object of the research is allocated recently (see, for example, [2,3,4])and further underwent quite detailed studying. At the same time the basic ideaof consideration of functions and vector fields was to approximate them — upto o ( ǫ ), what is equivalent to their linearization. For example, smooth (or evenanalytical) function F ( x, ǫ ) is replaced with the function of the type of f ( x ) + ǫf ( x ), where f , f — some smooth (or analytical) functions. Such replacementwill be designated further by symbol ≈ . Similarly, for smooth (or analytical) vectorfield X ( x, ǫ ), depending on parameter ǫ , we have approximate equality X ( x, ǫ ) ≈ X ( x ) + ǫX ( x ) , where X ( x ) , X ( x ) — vector fields. The commutation of suchapproximate vector fields can be calculated by such formulas:[ X, Y ] ≈ [ X + ǫX , Y + ǫY ] ≈ [ X , Y ] + ǫ ([ X , Y ] + [ X , Y ])Here the bilinearity of operation of such commutation and skew-symmetry areremain, Jacobi’s identity is carried out too. So we receive some Lie algebra. Butif to consider the standard commutation of the linearized vector fields, then theresult will not be, generally speaking, linear on ǫ any more. Therefore the givenabove commutation is not standard operation of commutation of vector fields — itis made in a different way. We come to the new concept, in which in the uniformform all Lie algebras, depending on parameter ǫ , are combined. This object is calledas approximate Lie algebra and now we will consider several different approachesto consideration of such objects. Typeset by
AMS -TEX x ∈ R n . Let’s consider the set of vector fields of the form X + ǫY , where X, Y ∈ Φ, and ǫ is some parameter( ǫ = 0). This set we will designate by Φ( ǫ ) orwe will present it in the form Φ+ ǫ Φ. Generally speaking, this set of vector fields (orof jets) does not form a Lie algebra, as operation of commutation can lead to thevector fields which are not lying in Φ( ǫ ). But if we define on Φ( ǫ ) a commutationoperation in a different way (as it is made above for approximate vector fields),then we will get, as it is easy to be verified, the Lie algebra (but it will not be thea Lie algebra of vector fields any more). Such Lie algebras are called approximateone’s in the articles stated above.Except the approximate Lie algebras, which are already constructed by such away, it is necessary to consider also some of their subalgebras — those, which arestable under multiplication on ǫ . Such Lie subalgebras correspond to Lie groups ofapproximate symmetry of differential equations, which studying initiated introduc-tion of the concept of the approximate Lie algebra.Let’s note also, that consideration of expressions of the similar type ¯ a + ω ¯ a ,where ¯ a, ¯ a are three-dimensional vectors, and ω be the symbol, for which ω = 0,is used in mechanics as subject to screw calculation.Construction 2Let Φ be any Lie algebra. Let’s consider two-dimensional algebra D (algebraof dual numbers of the form a + ǫb ; sometimes it is called Studi’s algebra and itselements—Studi’s numbers) over R with generators 1 and ǫ , where ǫ = 0. Let’snote that this algebra D (which is commutative, associative and with a unit) hasthe exact matrix representation, which matrices are (cid:16) a b a (cid:17) , where a, b ∈ R . It is alsointeresting to note, that unlike the field R . which group of automorphisms is trivial,and field of complex numbers C , which group of automorphisms is isomorphic Z (with complex conjugation as the only nontrivial automorphism), the algebra D has one-dimensional group of automorphisms—it consists of transformations of theform a + ǫb → a + ǫαb for any nonzero α ∈ R .Let’s denote ˜Φ = Φ ⊗ D - it is the tensor product of algebras Φ and D . Itis possible to write down elements of ˜Φ in the form of a + ǫb , where a, b ∈ Φ.Operation of commutation in the algebra ˜Φ is set naturally (with commutationon the first and multiplication on the second tensor factors), so we get on ˜Φ thestructure of the Lie algebra. Up to isomorphism of Lie algebras ˜Φ can be consideredas the semidirect sum Φ + ad Φ of the subalgebra, which is isomorphic to Φ, and anAbelian ideal, corresponding to the adjoint action ad of Lie algebra Φ on itself ason the vector space. It is clear, that if Φ be a Lie algebra of vector fields, then bysuch construction we get the approximate Lie algebra from the Construction 1.This Construction 2 can be generalized. For example, we may consider thealgebra of the form D p = R [ ǫ ] / < ǫ p > , where R ( ǫ ) is the algebra of polynomialsand < ǫ p > is the ideal, generated by element ǫ p . We get an associative andcommutative algebra of dimension p with unit, sometimes it is called as algebra ofplural numbers. For p = 2 this algebra was already entered above. Let’s considerthe tensor product Φ ⊗ D p with natural operation of commutation— we get here theLie algebra, which can be considered as the approximate Lie algebra when usingapproximations of order p by means of o ( ǫ p ). It is presented in the form of thesemidirect sum of subalgebra, isomorphic to Φ, and the nilpotent ideal of the form2 Φ + ǫ · Φ · · · + ǫ p − · Φ.The Lie algebras, constructed above by us , are graduated. It is possible to studyfor the bigger expansion of the concept of the approximate Lie algebra also moregeneral graduated Lie algebras—for example, any 2-graduated Lie algebras of thetype of L = L + L (which can be written down, by analogy with the aforesaid,in the form of L + ǫL ), where L — some Lie subalgebra, and L is the Abelianideal in L . It is possible to consider also some longer graduation (they arise abovefor p > L ⊗ D — tensor products of ordinary Lie algebras L and two-dimensionalalgebra D = < , ǫ > (the algebra of dual numbers). Let’s note that such Liealgebras can be considered as D -modules. Moreover, approximate Lie algebrasof symmetry of approximate differential equations always have a structure of D -modules too. It follows from such simple fact: if the vector field X is an infinitesimalsymmetry of a differential equation, that and vector field ǫX — too. Thereby theLie algebra of infinitesimal symmetry is invariant under multiplication on elementsof algebra D .Except Lie algebras of the type of L ⊗ D , it is possible (and it is necessary)to consider — for the purpose of their further applications — and their subalge-bras too. Generally speaking, any Lie subalgebra in L ⊗ D is not closed underthe multiplication on ǫ (and therefore is not a D -Lie subalgebra ). For example,for any element X ∈ L spanned on it one-dimensional abelian Lie subalgebra in L ⊗ D do not D -invariant. But if we are interested in Lie algebras of approxi-mate symmetries, then it is natural to consider only such Lie subalgebras, whichare D -modules. Also it is possible to call them as approximate Lie algebras (inthat sense, what arises when we study approximate differential equations and theirsymmetries). Thereby we come to need to consider some class of the Lie algebras,defined over D . Namely, as the next approach to our final definition of approxi-mate Lie algebras, it make sense to call as approximate subalgebras any invariantunder multiplication on ǫ Lie subalgebras in Lie algebras of the type of L ⊗ D forvarious Lie algebras L (in the theory of approximate symmetry of differential equa-tions it is used as L the Lie algebras of smooth or even analytical vector fields). Butto separate this general concept from that, which is connected with vector fields,we will call such Lie algebras— which are D -modules - as D -Lie algebras (alsothe name ”dual Lie algebras” is admissible, though it is a little ambiguous).So, our main object, which we will also study further, is any D -Lie algebra.The linear operator of multiplication on ǫ for such Lie algebras we will denote by E (actually it determines D -structure). At the same time, it is natural to study alsothe corresponding Lie subalgebras, ideals, homomorphisms, etc. Let’s note that as D -module the approximate Lie algebra can be trivial (i.e. operator E for it can bezero). Thereby ordinary Lie algebras, for which E = 0, can be considered as D -Liealgebras. However, further we will be interested generally in cases when E 6 = 0.Moreover, sometimes it is useful to consider only ”nondegenerate” D -structures,when the rank of operator E is equal to the half of the dimension of the Lie algebra(which, at the same time, is supposed, naturally, to be even). It is clear, that it3ill be in only case when the Lie algebra L is a free D -module. However theclass of such nondegenerate Lie algebras is not closed concerning transition to Liesubalgebras (or even only to ideals), or under transitions to quotient algebras byideals. Generally, upon such transitions the linear operator E can be ”degenerate”,in particular—be zero (when we have the trivial D -module and the corresponding D -Lie algebra is the ordinary real Lie algebra without additional nontrivial D -structure). Further in article some main properties of any D -Lie algebras will bestudied in detail. In particular, it gives a lot of information about approximate Liealgebras, that can be useful for researches in which these concepts arose.As algebra D is not a field (the division by any numbers of type ǫb is impossi-ble), the normal concept of the basis for its modules is inapplicable. By essentialbasis (or D -basis) for D -Lie subalgebra we understand the minimal set of the vec-tors, generating this subalgebra as D -module. Speaking in other words, it is suchminimal set of vectors, which together with their multiplications on ǫ generates theLie algebra as the vector space over R . Such bases were entered in ([5], p. 41; therethey are called essential parameters). Let’s note that only for Lie algebras over D (and for D -Lie subalgebras) the concept of essential basis makes sense. Thenumber of elements in it (the number of essential parameters) is some analog of thedimension for the considered Lie algebra L over D (here it is possible to use theterm ” D -dimension” and to designate this number as d ( L )). For example, D -Liesubalgebra in the Lie algebra of the type L ⊗ D , generated by the element of type ǫX (where X ∈ L ) is one-dimensional over R , but Lie subalgebra, generated by theelement X , is two-dimensional (here X = X + ǫ · V is some finite-dimensional free D -the module, then V isisomorphic to D m for some natural m . In this case it is natural to consider in VD -bases (systems, free generating the module) and linear operators are defined bymatrices (which elements belong to D ).Let’s note also, that the concept of the characteristic polynomial is badly adaptedfor use in D -the situation. First, the notion of the basis is absent (in normalsense of this word), and therefore it is difficult to use the concept of the matrix.As for polynomials and their roots, then in general there are a lot of surprises.For example, the polynomial can have infinitely many roots—for example, for thepolynomial ( z − a ) for the fixed a ∈ R any dual number of the form a + ǫb withan arbitrary value of the real number b , will be the root of this polynomial.Here the Introduction, having generally methodological focus, comes to an endand we pass to the studying of the object, entered above— the category of D -Liealgebras. We will consider generally real Lie algebras, though absolutely similarconsiderations can be carried out for complex D -Lie algebras (note that algebra D can be considered also over R and over C ). § D -Lie algebras Here some properties of finite-dimensional D -Lie algebras (i.e. such real Liealgebras, which are D -modules) will be considered. Let’s remind that D is the al-gebra of dual numbers (which can be considered as result of Cayley-Dixon doublingfor the algebra of real numbers) and therefore it is possible to call the correspond-ing Lie algebras as dual Lie algebras. It is possible to consider also algebra of dualnumbers over C (several times it will be stated below). We will also be limited hereto consideration of finite-dimensional Lie algebras (though some statements, given4elow, are true also without this restriction).On any real Lie algebra L the structure of D -Lie algebra is set by the linearoperator E (which we will often consider as the operator of multiplication by somespecial element ǫ ∈ D ), for which there are two following properties:1. [ E X, Y ] = [ X, E Y ] = E [ X, Y ]On another way this condition can be expressed so: the operator of the D -structure commutes with operators of the adjoint representation of the Lie algebra L . 2. E = 0In suitable (Jordan) basis the matrix of the linear operator E is the direct sum ⊕ J (0) ⊕ J (0) or order 2, corresponding to the zeroeigenvalue, and some zero square matrix (designated above as 0; it can be absentin this decomposition). The number of Jordan cells J (0) in this decomposition isequal to the rank of the matrix E , and also to the dimension of the image of thislinear operator.Trivial examples of D -(or dual) Lie algebras are ordinary real Lie algebras, forwhich the corresponding linear operator E equals to zero.The simplest nonstandard examples of D -Lie algebras are Lie algebras of thetype of L ⊗ D — tensor products of any real Lie algebra L and the algebra D (commutation operation is defined naturally here: [ X ⊗ α, Y ⊗ β ] = [ X, Y ] ⊗ ( αβ )). D -Lie algebras, which we get by such construction, are even dimensional, if toconsider them as real Lie algebras. However there are also odd-dimensional D -Lie algebras—for example, such is any one-dimensional Lie subalgebra in any D -Lie algebra, spanned by the element ǫX . The specified Lie algebras L ⊗ D are,obviously, free D -modules. The corresponding linear operator E has the greatestpossible rank (which equals to dimension of the initial Lie algebra L ).Odd-dimensional Lie algebras can not be obtained by means of the specifiedoperation of the tensor product (which it is possible to call as operation of thedualization of the initial Lie algebra). However there are also even dimensional D -Lie algebras which are not result of the dualization of any Lie algebra. For example,such is the Abelian Lie algebra R (with trivial D -structure). But there are suchexamples and in the case of nontrivial D -structures. Here we have an elementary(having the minimal possible dimension) example - it is the four-dimensional Liealgebra L , having basis X, Y, ǫX, ǫY and the defining bracket [
X, Y ] = ǫX (othernontrivial brackets are zero). It is easy to understand that, as the real Lie algebra,this L is isomorphic to the Lie algebra n (3 , R ) ⊕ R , where n (3 , R ) is the three-dimensional nilpotent Lie algebra (which in suitable basis U, V, W is defined by thebracket [
U, V ] = W ). If there is such Lie algebra P , that L is isomorphic to P ⊗ D ,then the Lie algebra P would have to be two-dimensional. But as is well-known,that there are only two—up to isomorphism, as it is natural—real two-dimensionalLie algebras: they are one Abelian and one solvable (which is not nilpotent). Sothe Lie algebra P ⊗ D has to be or Abelian or not nilpotent. But ours Lie algebra L is nilpotent and therefore the assumption of existence of the Lie algebra P leadsto the contradiction. The same conclusion can be obtained more simply (and notbased on classification). The matter is that D -structure on the considered Liealgebra is degenerated (the rank of operator E is equal to 1). But Lie algebrasof the type L ⊗ D always have due to their construction a nondegenerate D -structure. Therefore the considered Lie algebra L is not the dualization of any Liealgebra. If for D -Lie algebra the structure is nondegenerate, then examples of this5ort can be constructed too, but they will be more bulky (some arguments in thisdirectiont will be given below).For any D -Lie algebra L by L R we will denote the realification of the Lie algebra L , i.e. it is Lie algebra L considered over field R (ignoring its D - structure). As itwill be shown below, many properties of D -Lie algebras it is convenient to prove,using the known results, applied to real Lie algebras L R . However there are alsoresults specific to D -Lie algebras.Let L be some finite-dimensional D -Lie algebra. Let’s consider the linear sub-space P = Im ( E )—the image of the linear operator E , and by Q we will denoteits kernel Ker ( E ). The condition E = 0 is equivalent to the inclusion P ⊂ Q . Itis obvious that the subspace P is zero in only case, when the operator E is zero.Let’s note that number of essential parameters d ( L ) for D -Lie algebra (mentionedabove) equals to the real dimension of the vector space L/P . We pass to detailedstudying of some properties of D -Lie algebras. Proposition 1.
Let L be some D -Lie algebra and U be any linear subspace in L .Then E ( U ) is the Abelian subalgebra in L .Proof. We have [ E ( U ) , E ( U )] = E ([ U, U ]) = { } .For any linear subspace U ⊂ L we will put ˆ U = U + E U . Actually ˆ U equals to D · U — it is minimal D -subspace containing U (or D - saturation of the subspace U ). Proposition 2.
Let U be some ideal in D -Lie algebra. Then(i) E ( U ) is the ideal in L (ii) ˆ U is the ideal in L (and [ ˆ U , L ] ⊂ U )Proof.
1. We have [ E ( U ) , L ] = E ([ U, L ]). But U is the ideal in L and therefore[ U, L ] ⊂ U , so we get [ E ( U ) , L ] ⊂ E ( U ), i.e. E ( U ) is the ideal in L .2. We have [ ˆ U , L ] = [ U + E U, L ] ⊂ [ U, L ] + [ E U, L ]. But [
U, L ] ⊂ U (because U is the ideal), and also [ E U, L ] = [ U, E L ] ⊂ U . We get [ ˆ U , L ] ⊂ ˆ U , i.e. ˆ U is the idealin L . Also [ ˆ U , L ] ⊂ U In particular, it follows from Propositions 1 and 2, that subspace P , introducedabove, is the Abelian ideal in L . This statement, as well as the Corollary 1, givenbelow, was noted for the first time in [3]. Let’s note that in the specified articlesome designation is not standard for the theory of Lie algebras — it is used thenotation of decomposition L = L ⊕ L , using designations for the direct sum,though actually the speech there implicitly goes about decomposition to disjointunion of two subsets, from which one— L —is not the subspace. Corollary 1. If L is some D -Lie algebra and E 6 =
0, than L as the Lie algebraover R will never be semi-simple.Proof. . If E 6 = 0, then the subspace P is an nontrivial Abelian ideal, that makessemi-simplicity of the Lie algebra L impossible.In addition to the statement (ii) of the Proposition 2 we will prove Proposition 3.
Let U be some linear subspace in D -Lie algebra L . Then ˆ U isthe D -Lie subalgebra in L . roof. We have [ ˆ
U , ˆ U ] = [ U + E U, U + E U ] ⊂ [ U, U ] + [ E U, U ] + [ E U, E U ]. But[ E U, U ] = E [ U, U ] ⊂ ˆ U and [ E U, E U ] = E [ U, U ] = 0, and therefore [ ˆ
U , ˆ U ] ⊂ ˆ U , i.e.ˆ U is the Lie subalgebra.Concepts of the solvable and nilpotent Lie algebras are defined for the class of D -Lie algebras naturally (by the sequences of the corresponding commutants). Itis easy to be convinced that all members of the central series (upper and lower)and the members of commutants series for D -Lie algebra are D -Lie subalgebrastoo. For example, we will show it for the center Z ( L ): let X ∈ Z ( L ), then for anyelement Y ∈ L we have [ E X, Y ] = E [ X, Y ] = 0, therefore E X ∈ Z ( L ).More general statement can be similarly proved Proposition 4.
Let L be some D -Lie algebra, and U is some D -Lie subalgebrain it. Then the centralizer Z L ( U ) and the normalizer N L ( U ) of this Lie subalgebraare D -Lie subalgebras in L . Further, the concept of the solvable radical is naturally defined (it is the maximalsolvable D -ideal). As it appears, D -radical in D -Lie algebra equals to the ordi-nary radical (in the Lie algebra L R ). The situation for the nilradical (the maximalnilpotent ideal of the Lie algebra of L R is similar. Lemma 1.
Let L be some D -Lie algebra, and R is the radical (the maximalsolvable ideal) in the Lie algebra L R . Then R is D -radical (the maximal solvable D -ideal) in the Lie algebra L . If N is the nilradical in L R , then it is also D -nilradical.Proof. We will consider Lie algebra ˆ R = R + E R . The ideal R will be an D -idealin only case, when ˆ R = R . But E ( R ) contains, due to the Proposition 1, in theAbelian ideal P of the Lie algebra L . We have P ⊂ R and therefore ˆ R ⊂ R .Therefore ˆ R = R .For the case of the nilradical the proof is much simpler. It, as we know, con-sists of the radical’s elements, for which the operator of the adjoint representation,if restricted on the radical, has only zero eigenvalues. But by multiplication byelements from D nonzero eigenvalues cannot appear. Therefore N = D · N . Lemma 2.
Let L be some D -Lie algebra and N — its nilradical. Then N containsthe Abelian ideal P = Im E . In particular, dim N ≥ dim P .Proof. Due to the Proposition 1, subspace P is the Abelian ideal in L and thereforecontains in the radical of the Lie algebra L . Let X ∈ P be any element. Then ad X ( L ) ⊂ P and therefore ad X ( L ) = { } , i.e. operator ad x is nilpotent. Buttherefore X contains in N .The statement of the Lemma 2 can be deduced by a different way: any nilpotentideal of the Lie algebra (which is contained always in its radical), contains also inits nilradical. It follows from the known description of the nilradical in the solvableLie algebra, that it is maximal among all nilpotent ideals of the radical.There are some difficulties with the concept of semisimple D -Lie algebras andsemisimple D -Lie subalgebras. For them E = 0 (see the Proposition 1). Butfurther for theory of D -Lie algebras a lot of things go to as usual—for example,Levi’s decomposition L = S + R takes place. At the same time, the radical R willbe D -subalgebra, and the semi-simple part S —not always (because Lie subalgebra E ( S )—if it is nonzero—never contains in S ). As the such example we will consider7he Lie algebra S ⊗ D , where S — some semi-simple real Lie algebra. Then followsfrom the construction of the tensor product, that E S = { } , but E S ∩ S = { } (itis the typical case, see the Proposition 5 below).Let’s study in more detail the situation, connected with semi-simple subalgebrasin D -Lie algebra L . Proposition 5.
Let S be some semi-simple Lie subalgebra in the finite-dimensional D -Lie algebra L (more precisely, in L R ). Then1. E S ∩ S = { }
2. Lie algebra ˆ S has Levi’s decomposition S + A —it is the semidirect sum of theLie subalgebra S and some Abelian ideal A ⊂ ˆ S .Proof.
1. We have [ E ( S ) , S ] = E ([ S, S ]) = E ( S ) (because [ S, S ] = S for any semi-simpleLie algebra S ). From this it follows, that subspace E ( S ) is invariant under actionof the Lie algebra S , induced by the adjoint representation of the Lie algebra L .From another point of view, E ( S ) contains in the Abelian ideal P = E ( L ) in the Liealgebra L (see the Proposition 1 above). As the semi-simple Lie algebra S has nonontrivial Abelian ideals, we see that E S ∩ S = { } , that proves our first statement.2. Let’s consider the Lie subalgebra ˆ L ⊂ L . By the definition it is the sumof two subspaces S and A = E ( S ) having, as soon as it was proved above, trivialintersection. At the same time the subspace A is the Abelian Lie algebra (as shownabove), and it is the Lie subalgebra, which is invariant under the action of S .All this together means that Lie subalgebra ˆ S is the semidirect sum of the Liesubalgebra S and the Abelian ideal A (being also its radical).It is possible to suppose, that if D - structure on Lie algebra ˆ S has the maximumrank, then dim A ≥ dim S . Equality takes place, for example, if L = S ⊗ D , where S —any semi-simple Lie algebra.Let’s consider now Levi’s decomposition L R = S + R for realified D -Lie algebras.Due to the Proposition 5 E ( S ) always has the trivial intersection with S , i.e. the Liesubalgebra S is far from being invariant under multiplication by elements from D .This S is isomorphic to quotient D -Lie algebra (including nondegenerate one’s)for the D -ideal (radical), perhaps degenerate. For nilpotent D -Lie algebra itscenter Z ( L ) is always nontrivial. Therefore for nilpotent D -Lie algebras, by usingstandard induction on dimension, we come to the D -analog of the Engel theorem: Proposition 6.
Let L ⊂ gl ( V ) be some linear D -Lie algebra, consisting of nilpo-tent elements (here V is some free D -module). Then there is such D -basis in V ,in which all matrices of the Lie algebra L have the nilpotent triangular form.Proof. Due to the classical Engel theorem, in V exists vector X , which is nullifiedby all elements from L . As V is a free D -module, the element X can be extendedto a free basis of V . Now we take quotient space V / < X > and apply inductionon dimension.Further, for solvable D -Lie algebras some analog of the theorem of Lie takesplace (which is formulated usually for complex Lie algebras, but also for real Liealgebras some analog takes place). The classical theorem of Lie for complex linear(or with the fixed linear representation) solvable Lie algebras L claims, that there isone-dimensional invariant subspace in L . From this fact by induction it is deducedthat all matrices of this linear Lie algebra L in some basis have the triangular8orm (it is the statement of Lie’s theorem). For real solvable Lie algebras the one-dimensional invariant subspace not always exists, but always there is no more thantwo-dimensional invariant subspace. Further by induction it is proved, that suchLie algebra has in suitable basis a quasitriangular form (over C — triangular) —on diagonal there are cells of orders 1 or 2, and below—zero elements). It appearsthat for D -Lie algebras a similar statement is valid, because it is possible to provethat the specified invariant subspace can be chosen as D -invariant. For simplicityof the statement we will prove this statement only in case of the adjoint linearrepresentation. Proposition 7.
Let L be a real solvable finite-dimensional D -Lie algebra. Thenin L there is an Abelian D -ideal of dimension 1 or 2. If L is a complex D -Liealgebra, then exists D -ideal of dimension 1.Proof. We will consider the realification L R of the Lie algebra L . For the beginningwe will assume, that the linear operator E is nontrivial. Then P = Im E is theAbelian ideal of positive dimension. Due to the real analog of the theorem of Lie,stated above, in P there is some L -invariant subspace of W of dimension (real) 1 or2. As P is the Abelian ideal, this subspace of W also is the Abelian ideal. At thesame time E ( P ) = { } (as E = 0). Therefore E ( W ) = { } and W is the D -idealof dimension (real) 1 or 2.If operator E is trivial, then the statement, necessary to us, is the real analog ofthe Theorem of Lie, mentioned above.For the case of the complex Lie algebra the reasoning is similar, only the initialsubspace W is one-dimensional.Thereby the Proposition 7 is completely proved.Let’s show, that for real D -Lie algebras the one-dimensional ideal not alwaysexists. For this purpose we will consider the Lie algebra E (2) ⊗ D , where E (2)is the Lie algebra of motions of the Euclidean plane (it is isomorphic to the Liealgebra so (2) + R ). In the Lie algebra E (2) there are no one-dimensional ideals.But then it is easy to check, that there are no one-dimensional ideals also in D -Liealgebra E (2) ⊗ D .Let’s consider now for D -Lie algebras the analog of the theorem of Ado aboutexistence of faithful finite-dimensional linear representation for any real finite-dimensional Lie algebra.Let V be some vector space over R . By V ( ǫ ) we will denote the vector space V ⊗ D over algebra D . Above such construction was used for introduction of theconcept of approximate Lie algebras.For Lie algebra gl n ( R ) of linear transformation of the vector space R n we willconsider corresponding D -Lie algebra gl n ( D ) = gl n ( R )( ǫ ) — it is the Lie algebraof linear transformations of D -vector space D n (free D -module). Let’s note thatit is nondegenerate D -Lie algebra. It can be presented as consisting of cells of order2, each of which has the form (cid:0) x y x (cid:1) . Further, similarly it is possible to introduceother analogs of classical matrix Lie algebras. By N ( n, D ) we will denote thenilpotent Lie algebra, consisting of all nilpotent upper triangular matrixes withelements from D (it is isomorphic to N ( n, R ) ⊗ D ) and it can be written down inthe form of the set of block nilpotent matrixes which nonzero blocks are matricesof the form (cid:0) x y x (cid:1) . Similarly, by T ( n, D ) we will denote the set of upper triangularmatrices with elements from D —it is solvable (and even triangular) Lie algebra,9hich matrix representation consists of block matrices of order 2. Let’s remindthat the real Lie algebra is called triangular (sometimes—completely solvable), ifall characteristic roots of operators of its adjoint representation are real. Proposition 8.
Let L be any finite-dimensional Lie algebra over D . Then it isisomorphic to some D -Lie subalgebra in gl n ( D ) . In other words, the Lie algebra L has an exact finite-dimensional linear representation over D .If L is nilpotent, then it is isomorphic to some D -Lie subalgebra in N ( n, D ) .And when L is triangular (if we consider it as the real Lie algebra), then it isisomorphic to the D -Lie subalgebra in T ( n, D ) .Proof. We will consider L as the Lie algebra L R over R . Due to the classicaltheorem of Ado there is an embedding of this Lie algebra L R into Lie algebra gl ( n, R ) for some n ∈ N . It is clear, that there is the natural embedding L ⊂ L R ( ǫ ).But then we get the D -Lie algebra L embedding into gl ( n, R )( ǫ ).For cases, when L is nilpotent or triangular, the corresponding statements fromthe Proposition 8 are proved by similar method, proceeding from the correspondingwell-known statements in the theory of real Lie algebras (specifying Ado’s theoremin these cases).Let’s review briefly the situation with triangular D -Lie algebras. Unlike anycomplex solvable Lie algebras, the famous fixed point theorem in the real casetakes place only for triangular Lie algebras. For any solvable Lie algebras it is nottrue (even for one-dimensional Abelian Lie algebras). Due to the concept of thetriangular Lie algebra we will consider one question, which has a little more generalcharacter—it concerns also the general theory of Lie algebras.When studying solvable complex groups and Lie algebras, an important role isplayed by the concept of Borel subgroup and the corresponding Lie subalgebra—themaximum connected solvable Lie subgroup and the Lie subalgebra. It is connected,in particular, with the fact that for complex solvable groups and Lie algebras thefixed point theorem takes place. In the real case for any solvable groups and Liealgebras such statement, generally speaking, is incorrect. However it is true fortriangular groups and Lie algebras. There is the natural question— how propertiesof triangular Lie algebras and any solvable Lie algebras in the complex case areconnected. For example, whether it is true that, if R is some solvable complex Liealgebras, and T —its maximal triangular subgroup, then the complexification of thisLie subalgebra equals to R . It turns out that it is incorrect and therefore it turnsout, that in case of complex solvable Lie algebras we have stronger statements thatwhen we consider them as real Lie algebras.Here we give some elementary example. Let’s consider the complex Lie algebraof the type K = C + C n —the semidirect sum, which is defined by a homomorphism φ : C → gl ( n, C ). Let’s consider matrix A = φ (1) and we will assume, that itseigenvalues are linearly independent over R (for example, when n = 2, it means thatthe quotient of two eigenvalues of the corresponding matrix A is not real number).Let T be the maximum triangular subgroup in such R . Let’s prove that T = C n ,what will means that the complexification of the Lie algebra of T is not equal to R . It is clear, that C n ⊂ T . Let X ∈ K \ C n . Let’s prove that X does not belongs to T . For this purpose we will consider eigenvalues of the operator ad X for its actionon C n . It is clear, that they have to be proportional to the matrix A eigenvalues.10ut then, as appears from our choice of matrix A , eigenvalues of ad X cannot bereal. So it is proved that X does not belongs to T and therefore we have T = C n .Similar questions arise also for D -Lie algebras. If T is some maximal triangularLie subalgebra in the complex Lie algebra R , whether its complexification will equalsto R and whether it is possible to tell something similar about its dualization?Answers to both of these questions are similar to stated above— the are negative.For this purpose it is enough to consider the Lie algebra R ⊗ D , where R —thecomplex Lie algebra of the form C + C n , constructed above. Here operator E is nontrivial. There is a simpler example (though degenerated)—to take the Liealgebra R with trivial E . § A classification of Lie algebras over D is in many respects differs from a classi-fication of real or complex Lie algebras. Let’s begin with the fact, that there are no”purely” semi-simple Lie algebras here. Further, here can be several non isomor-phic D -structures on one real Lie algebra, i.e. there can be non isomorphic D -Liealgebras, which realifications are isomorphic. It strongly expands the classificationlist of D -Lie algebras.It is possible to carry out a classification, for example, by number d ( L ) of es-sential parameters. It is obvious that for d ( L ) =1 there are only two (up to D -isomorphism) D -Lie algebras, they are Abelian: one-dimensional R (with trivial E ) and two-dimensional D (isomorphic to R as the real Lie algebra).There are known classifications of D -Lie algebras for cases of 2 and 3 essentialparameters (see [3,4]). There are reason to believe, that actually D -Lie algebrasin sense of our definition, at which the number of essential parameters is equal 2 or3, are classified there (though concepts of D -Lie algebras are not used there). Wewill proceed from such understanding of articles [3,4].It turns out that only since dimension 6 (and with three essential parameters)there are unsolvable D -Lie algebras. These are two such Lie algebras su (2) ⊗ D and sl (2 , R ) ⊗ D (both are of the type S ⊗ D , where S — one of two real simplethree-dimensional Lie algebras). There are no classifications for bigger numberof essential parameters and for bigger dimensions, because for this purpose, inparticular, it is necessary to have classifications of real Lie algebras of dimensions8 and more, that looks now as a unavailable task. The top of today’s achievementsin this direction is the classification of all Lie algebras of dimensions ≤ D -Lie algebras, but the advantage of such work is not clear).Let’s provide the list of nilpotent D -Lie algebras of real dimensions ≤ D -isomorphism)—they are Abelian R , R (with trivial E ), D andnon-Abelian N (3 , R ) , D ⊕ R . For cases of dimensions of 4,5 and 6 see [3].In classifications of D -Lie algebras, which can be useful for applications, itis useful to distinguish different classes of D -Lie algebras, because Lie algebras,which are of interest, belongs sometimes only to one of this classes. Below aredescribed some classes of D -Lie algebras (all of them are supposed real and finite-dimensional):1. Any D -Lie algebras2. D -Lie algebras of the form L ⊗ D for various ”ordinary” Lie algebras L
11. The Lie subalgebras in Lie algebras of the type L ⊗ D , which are D -modules4. D -free Lie algebras (i.e. D -Lie algebras, being free as D -modules)5. Nondegenerate D -Lie algebras.Let’s specify those relations, which take place between the listed five classes of D -Lie algebras.First, classes 1 and 3 are coincide. It follows from the Proposition 8 (Ado’stheorem). It is clear also, that the class 1 is the most general, it contains all others.It is necessary to find out interrelations of classes 2, 4 and 5. It is clear, that theclass 2 contains in classes 4 and 5. The classes 4 and 5 are coincide — it follows fromthe analysis of their definitions. But these two classes do not coincide with class 2.In other words, not any nondegenerate D -Lie algebra can be presented in the formof the dualization of some Lie algebra. The matter is that dualization gives to Liealgebras very special structure — they are presented in the form of the semidirectsum of the initial Lie algebra and some Abelian ideal (and both summands haveidentical dimensions). For any nondegenerate D -Lie algebras such decompositionnot always exists.In conclusion of article—several remarks about D -Lie algebras L of vector fieldson the D -line (which is not the ordinary two-dimensional real plane R , but sup-plied standard D -structure).For the line D there is one dimensional invariant distribution, generated by thesubspace of purely imaginary elements ǫb . It is integrable and therefore all D -Liealgebras of vector fields on D are imprimitive (i.e. they keep some one-dimensionalfoliations). There are only four primitive Lie algebras of vector fields on the plane R , their dimensions are ≤ D (including those,which have a D -structure), may have arbitrarily large dimension.Let’s note that classical classification of Lie of vector fields on R uses the conceptof similarity of Lie algebras of vector fields, and their similarity as D -subalgebrasdoes not follow from this classification. It is necessary to carry out additionalconsiderations, that greatly increases problems of classification. However there isone useful simplification — such Lie algebras of vector fields on the plane are surelyimprimitive.Also let’s note what full classification of Lie algebras of vector fields on the dualplane D would demand to have classification of Lie algebras of vector fields on R ,that gives us now an unattainable task. Classifications of approximate Lie algebrasof dimensions ≤ D -Lie algebras) on the plane see in [4]. List of references
1. Baykov V. A., Gazizov R. K., Ibragimov N. H.,
Approximate symmetry , Mat. Sbornik (1988), 436 – 450.2. Baykov V. A., Gazizov R. K., Ibragimov N. H.,
Approximate transformation groups , Differ-ential equations (1993), 1712 – 1732.3. Gazizov R. K., Lukashchuk V. O., Classification of approximate Lie algebras with three es-sential vectors , Izv. VUZov, Mathematics (2010), no. 10, 3 – 17.4. Gazizov R. K., Lukashchuk V. O.,
Classification of nonsimilar approximate Lie algebras withtwo essential symmetries on plane , Works of the fifth All-Russian scientific conference with theinternational participation (on May 29-31, 2008). Part 3, Differential equations and boundary-value problems, Matem. modeling and boundary problems, SAMGTU, Samara, 2008, pp. 62– 64.5. Ibragimov N.H.,
CRC handbook of Lie group analysis of differential equations.Vol.3:Newtrends in theoretical development and computational methods , CRC Press, FL, 1996. . Chebotaryov N.G., Lie group theory , Moscow, Editorial URSS, 2011., Moscow, Editorial URSS, 2011.