aa r X i v : . [ m a t h - ph ] D ec Wei-Norman equations for classical groups
Szymon Charzy´nskiFaculty of Mathematics and Natural Sciences,Cardinal Stefan Wyszy´nski University,ul. W´oycickiego 1/3, 01-938 Warszawa, Poland [email protected]
Marek Ku´sCenter for Theoretical Physics, Polish Academy of Sciences,Aleja Lotnik´ow 32/46, 02-668 Warszawa, Poland [email protected]
December 16, 2013
Abstract
We show that the non-linear autonomus Wei-Norman equations, expressing the solutionof a linear system of non-autonomous equations on a Lie algebra, can be reduced to thehierarchy of matrix Riccati equations in the case of all classical simple Lie algebras. Theresult generalizes our previous one concerning the complex Lie algebra of the special lineargroup. We show that it cannot be extended to all simple Lie algebras, in particular to theexceptional G algebra. For a non-autonomous system of N linear differential equations ddt x ( t ) = M ( t ) x ( t ) , (1)where x ( t ) is an N dimensional vector and the N × N coefficient matrix is time-dependent it is,in general, difficult to find a solution in a finite form (not invoking a series expansion). A possiblenon-commutativity of the coefficient matrices M ( t ) and M ( t ′ ) calculated in different times t , t ′ prevents, namely, an explicit calculation of the ordered time product of exponentials of the timedependent matrix variable.Wei and Norman [1, 2] proposed a method for solving such a system of equations by trans-forming it to an autonomous, albeit nonlinear system using Lie-algebras techniques. Roughlyspeaking, the method consists of writing M ( t ) as a linear combination with time dependent co-efficients of generators X i of a Lie algebra, M ( t ) = P i a i ( t ) X i , and looking for the solution interms of a product of exponentials of the generators (i.e. elements of the corresponding Lie group), x ( t ) = Q i exp( u i ( t ) X i ) x (0). The resulting system of nonlinear equations derived from the orig-inal, non-autonomous, linear one (1) connects the unknown functions u i ( t ) with the coefficients a k ( t ).In [3] we have shown that using the Wei-Norman technique [1, 2] in the unitary case, i.e. whenthe solution of the linear system is given by a unitary evolution operator, the nonlinear systemby an appropriate choice of ordering can be reduced to a hierarchy of matrix Riccati equations.To this end we have considered a general linear non-autonomous dynamical system on the speciallinear group SL ( N + 1 , C ). The algebra sl ( N + 1 , C ) is one of the classical Lie algebras, of the type1enoted by A N . In the present paper we generalize the method developed in [3] to all classicalLie algebras, namely the algebras of type A N , B N , C N and D N . In particular we show that inall classical cases the resulting nonlinear system can be always reduced to a hierarchy of matrixRiccati equations if we chose an appropriate ordered basis of the underlying Lie-algebra. Thus, weexpand the applicability of the Wei-Norman method from the unitary and special linear groupsto orthogonal and symplectic groups.To exhibit the Riccati structure of the Wei-Norman equations in the case of A N algebras consid-ered in [3] we proved several facts concerning their structural properties, among them two crucialones. The first establishing a decomposition of the sl ( N +1 , C ) Lie algebra into a semidirect sum ofAbelian subalgebras and the second concerning the nilpotency of the adjoint endomorphism. Herewe show that these two observations remain valid for all classical simple Lie algebras. Interestinglythey cannot be extended to all simple Lie algebras - we show this in the case of an exceptional Liealgebra G . We briefly recall our analysis of the Wei-Norman method presented in greater detail in [3]. Let G be an n -dimensional Lie group and g - its Lie algebra. We assume in the following that g is asimple complex classical Lie algebra. Let also R ∋ t M ( t ) ∈ g be a curve in g and K ( t ) - acurve in G given by the differential equation ddt K ( t ) = M ( t ) K ( t ) , K (0) = I. (2)In g we choose some basis X k , k = 1 , . . . , n in which M ( t ) takes the form M ( t ) = n X k =1 a k ( t ) X k . (3)We look for the solution K ( t ) in the form K ( t ) = n Y k =1 exp (cid:0) u k ( t ) X k (cid:1) , (4)involving n unknown functions u k and this is the original idea of Wei and Norman [1], to useproduct of exponentials instead of most commonly used exponential of a linear combination of Liealgebra generators.Substituting (3) and (4) into (2) we straightforwardly arrive at the following system of coupleddifferential equations for u k in terms of a l (for a detailed derivation see [3]): n X k =1 a k X k = n X l =1 u ′ l Y k
The subalgebra a k is a commutative ideal in b k and e a k is a commutative ideal in e b k , for k ∈ { , . . . , N } .Proof. Recall that root system Φ is a partially ordered set with respect to the relation ≺ definedas α ≺ β iff β − α is a positive root or β = α . Each root system has a unique maximal elementwith respect to this relation. These maximal elements have the following forms in terms of thepositive simple roots [4]: A N : β max = α + α + . . . + α N ,B N : β max = α + 2 α + . . . + 2 α N ,C N : β max = α + 2 α + . . . + 2 α N ,D N : β max = α + 2 α + . . . + 2 α N − + α N − + α N . (20)Numbering of simple roots is again as indicated in Figure 1. Since the elements (20) are uniqueand maximal, for any classical algebra and any positive root β = P Ni =1 n i α i we have n = 0 or n = 1 and, according to definition (16), a is generated by X β for which n = 1. It follows from(11) that all these generators commute, since the sum of such roots would have n = 2 and itwould not be a root. So a is a commutative subalgebra of g . Moreover if X β ∈ a and X γ ∈ b ,then [ X β , X γ ] ⊂ span ( X β + γ ) and β + γ is a root with the first coordinate n = 1 or [ X β , X γ ] = 0,so [ X β , X γ ] ∈ a and a is a commuting ideal in b .Consider now a k and b k for k >
1. In this case we deal with algebras generated by root vectors X β corresponding to the root system resulting from omission of first k − a k and b k are isomorphic to subalgebras a and b of smaller simple algebrasof one of the type A N − k +1 , B N − k +1 or D N − k +1 (omission of α in C N yields A N − root system).It follows, that a k is a commuting ideal in b k .The proof for subalgebras e a k and e b k is analogous.Observe, that the proof of the Lemma 4.1 can be summarized in the following way. If oneexpresses root β in the basis ∆ numbered according to Figure 1, β = P Ni =1 n i α i , then β can beidentified with the vector of its coefficients: β = ( n , n , . . . , n N ). In the Lemma 4.1 we haveshown that in this notation the functional β = (0 , . . . , | {z } k − , , n k +1 , . . . , n N ) is never a root for the5lassical algebras and the considered simple roots numbering. This fact together with (11) yieldsthe thesis of the lemma. Thus, it follows from (13) and the Lemma 4.1 that Φ a k consists of rootsof the following form in the basis ∆, β = (0 , . . . , | {z } k − , , n k +1 , . . . , n N ) (21)whereas Φ b k consists of roots with coordinates of the form: β = (0 , . . . , | {z } k − , δ, n k +1 , . . . , n N ) , (22)where δ = 0 or δ = 1 and n i stands for any nonnegative integer admissible in the given rootsystem.Since the sets Φ a k defined in (13) are mutually disjoint (see (15)) this induces an order in theset Φ + = N [ k =1 Φ a k , (23)where we first order elements by the ascending index k corresponding to Φ a k to which the givenelement belongs, and then by the order of Φ a k . In what follows we will use the following Definition 4.2.
The ordered basis Ξ of classical Lie algebra is a basis consisting of three sequencesof generators: first the sequence of root vectors { X β : β ∈ Φ + } ordered in the same order as theset (23), next the sequence of N generators of Cartan subalgebra h , and finally the sequence of rootvectors { X − β : β ∈ Φ + } ordered in the order reverse to the order of the set (23). The main result of this section can be summarized by the following:
Corollary 4.3.
Let g be a simple complex classical Lie algebra of rank N . The equation (19)defines the decomposition of g into a sum of N + 1 commutative subalgebras. The basis Ξ is consistent with this splitting of g . We now set about formulating and proving the second important ingredient of our approachconcerning the degree of nilpotency and invariance properties of the adjoint endomorphisms ofclassical simple Lie algebras.
Lemma 5.1.
Let X α ∈ g be a root vector corresponding to a positive root α ∈ Φ + . In the basis Ξ defined by Definition 4.2, the matrix of ad X α is strictly upper triangular and the matrix of ad X − α is strictly lower triangular.Proof. Let α ∈ Φ + and X α ∈ n + . We consider ad X α ( X ) for X being an element of the basis Ξ inthree cases:1. X ∈ n + , so X = X β for some β ∈ Φ + . Moreover X α ∈ a k and X β ∈ a l for some k and l .The roots have the following coordinates in terms of the basis ∆: α = (0 , . . . , | {z } k − , , n k +1 , . . . , n N ) , β = (0 , . . . , | {z } l − , , n l +1 , . . . , n N ) . (24)If k = l then ad X α ( X β ) = 0.For k < l , if α + β is a root, then α + β ∈ Φ a k and ad X α ( X β ) ∈ a k , so the matrix elementcorresponding to ad X α ( X β ) lies above the diagonal.For k > l , if α + β is a root, then α + β ∈ Φ a l and ad X α ( X β ) ∈ a l . Since both α and β arepositive roots, the heights fulfill ht( α + β ) > ht( β ) hence X α + β precedes X β in the basis Ξand the corresponding matrix element lies above the diagonal.6. X ∈ h . Since [ n + , h ] ⊂ n + , the only nonzero matrix elements of ad X α in the sector corre-sponding to h lie above the diagonal.3. X ∈ n − , so X = X β for β ∈ Φ − . X α ∈ a k and X β ∈ e a l for some k and l . In this case theroots have the following coordinates in the basis ∆: α = (0 , . . . , | {z } k − , , n k +1 , . . . , n N ) , β = (0 , . . . , | {z } l − , − , n l +1 , . . . , n N ) . (25)For k = l the first nonzero coordinates in (25) cancel, so if α + β is a root, then X α + β ∈ a l +1 or X α + β ∈ e a l +1 . The generators of both a l +1 and e a l +1 appear earlier in the basis Ξ thanthe generators of e a l , consequently the matrix element in question lies above the diagonal.For k < l , if α + β is a root, then α + β ∈ Φ a k and ad X α ( X β ) ∈ a k , so the matrix elementcorresponding to ad X α ( X β ) lies above the diagonal.For k > l , if α + β is a root, then α + β ∈ Φ − and ad X α ( X β ) ∈ e a l . Since α is a positive rootand β is a negative one, the heights fulfill ht( − ( α + β )) < ht( − β ). Thus X α + β precedes X β in the basis Ξ and the corresponding matrix element lies above the diagonal.The statement is also true for X α ∈ n − , since it can be obtained by substituting all the roots bytheir negatives β → − β and reversing the order of the basis Ξ. Lemma 5.2.
Let α be a root and X α ∈ g be the corresponding root vector where g is one of theclassical Lie algebras A n , B n , C n or D n , then (ad X α ) = 0 .Proof. First observe, that for any H ∈ h ,(ad X α ) ( H ) = [ X α , [ X α , H ]] = − α ( H )[ X α , X α ] = 0 , thus it suffices to proof that (ad X α ) ( X β ) = 0, ∀ β ∈ ∆, since the elements X β generate n + ∪ n − and g = n + ⊕ h ⊕ n − . Recall that [4],ad X α ( X β ) = [ X α , X β ] = H α , α + β = 0 ,N α,β X α + β , α + β ∈ ∆0 , otherwise , (26)where H α ∈ h corresponds to the root α and N α,β is some constant. We have(ad X α ) ( X β ) = [ X α , [ X α , X β ]] = − α ( H α ) X α , α + β = 0 ,N ′ α,β X α + β , α + β ∈ ∆ , , otherwise , (27)where N ′ α,β is some other constant. Finally, since [ X α , X α ] = 0 we obtain:(ad X α ) ( X β ) = [ X α , [ X α , [ X α , X β ]]] = ( N ′′ α,β X α + β , α + β ∈ ∆ , , otherwise. (28)We will show that 3 α + β is never a root for a classical algebra g . A pair of roots α, β generatesa 2-dimensional root system. According to [4] there are exactly four 2-dimensional root systems: A × A , A , B and G . The equations (26-28) imply that (ad X α ) ( X β ) = 0, provided thereexist a, so called, root string of length four consisting of roots β , α + β , 2 α + β and 3 α + β . Butthe only 2-dimensional root system containing a root string of length four is the G root systemand the angle between roots α and β is equal 150 ◦ in this case. On the other hand according toclassification of simple Lee algebras [4], the only simple algebra with a pair of roots connectedby such an angle is the G algebra itself. So no other simple Lie algebra can have root systemcontaining root string or length four. 7bserve that if α ∈ Φ a and β ∈ Φ a then ad X α and ad X β are commuting nilpotent operatorsof nilpotency order 3. In general, if we have two commuting matrices of a given nilpotency order r then the sum of the matrices is also nilpotent of order r . It follows from the Jordan theorem [4]- the matrices can be expressed in the Jordan form in the same basis and are block-diagonal withthe same blocks of maximal size r −
1. So the Lemmas 4.1 and 5.2 yield:
Corollary 5.3. If X ∈ a k or X ∈ e a k then (ad X ) = 0 . Next we proof the crucial invariance property of the decomposition (19).
Lemma 5.4.
Let X α ∈ a k ⊂ g or X α ∈ e a k ⊂ g , where α ∈ Φ a is the corresponding root. Thesubalgebras a l , e a l for l < k and the subalgebra b k ⊕ h ⊕ e b k are invariant subspaces of ad X α .Proof. Let X α ∈ a k . We consider three cases:1. X β ∈ a l , l < k . It follows from (16) and (18) that a k ⊂ b k ⊂ b l . In this case X α ∈ b l and X β ∈ b l . Thus, ad X α ( X β ) = [ X α , X β ] ∈ b l . On the other hand X β ∈ a l and by Lemma 4.1, a l is an ideal in b l , so ad X α ( X β ) ∈ a l .2. Y ∈ b k ⊕ h ⊕ e b k . We have also X α ∈ b k ⊕ h ⊕ e b k and the property ad X α ( Y ) ∈ b k ⊕ h ⊕ e b k follows from the fact that b k ⊕ h ⊕ e b k is subalgebra of g and this a direct consequence of thedefinition of b k and e b k (see (17)).3. X β ∈ e a l , l < k . Since α ∈ Φ a k and − β ∈ Φ b l , in this case (see definitions (13-14)) thecoordinates of roots α and β in the basis ∆ have the following form (see (21-22)): α = (0 , . . . , | {z } k − , , n k +1 , . . . , n N ) , β = (0 , . . . , | {z } l − , − , n l +1 , . . . , n N ) , (29)Since l < k , the sum α + β has the form: α + β = (0 , . . . , | {z } l − , − , n l +1 , . . . , n N ) , (30)so if α + β is not a root then ad X α ( X β ) = 0 and if α + β is a root then since it has the form(30), we have α + β ∈ Φ a l and ad X α ( X β ) = X α + β ∈ a l .The analogous reasoning holds for X α ∈ e a k . The lemmas proved in the preceding sections allow now to exhibit the structure of the exponentialof the adjoint endomorphism. Its quadrating dependence on the exponent results in the Wei-Norman equations in the Riccati form, whereas its triangular and block-diagonal structure ordersthe resulting Riccati equations in a specific hierarchy.Lemma 5.4 implies that ad X for X ∈ a k or X ∈ e a k is a block diagonal operator with respectto the following decomposition: g = a ⊕ . . . ⊕ a k − ⊕ (cid:16) b k ⊕ h ⊕ e b k (cid:17)| {z } one block ⊕ e a k − ⊕ . . . ⊕ e a . (31)From Corollary 5.3 we know that it is also nilpotent, ad X = 0. Thus Corollary 5.3 and Lemmas5.4 and 5.1 yield: Corollary 6.1.
For X ∈ a k or X ∈ e a k the matrix of exp(ad X ) is a quadratic polynomial in ad X and it is block diagonal with respect to the decomposition (31). Moreover in the basis Ξ (Definition4.2) the matrix of exp(ad X ) is upper triangular for X ∈ a k and lower triangular for X ∈ e a k . X i and define: U k := Y X i ∈ a k exp( u i ad X i ) , e U k := Y X i ∈ e a k exp( u i ad X i ) . (32)Since a k and e a k are commutative subalgebras, the order of factors in (32) does not matter. For l < dim a k we also define: V kl := Y i ∈ I kl exp( u i ad X i ) , e V kl := Y i ∈ ˜ I kl exp( u i ad X i ) , (33)where for a given k the index i runs over the first l − a k and e a k respectively. So I kl is the string of the first (with respect to the ordering of Ξ) l indices i , such that X i ∈ a k , and X i ∈ Ξ and analogously, I kl is the sequence of the first l indices i , such that X i ∈ e a k , and X i ∈ Ξ.We set U = e U N = V k = e V k = . Observe that commutativity of the subalgebras a k implies that: U k = Y X i ∈ e a k exp( u i ad X i ) = exp X X i ∈ e a k u i ad X i = exp (cid:18) ad (cid:16)P Xi ∈ e a k u i X i (cid:17) (cid:19) , hence U k equals exp(ad X ) for some X ∈ a k (analogously e U k equals exp(ad X ) for some X ∈ e a k ).Thus by Corollary 6.1 we have: Corollary 6.2.
The matrix elements of the operators U k , V kl , e U k and e V kl are polynomial functionsof the parameters u i of degree at most 2. The matrices U k , V kl , e U k and e V kl are block diagonalwith respect to the decomposition (31) and the matrices U k and V kl are upper triangular, whereasthe matrices e U k and e V kl are lower triangular. We also define exponentials corresponding to Cartan subalgebra h : H := Y X i ∈ h exp( u i ad X i ) , H l := Y i ∈ I h l exp( u i ad X i ) , (34)where I h l is the index range numbering the first (with respect to the ordering of Ξ) l generators of h . H and H l are diagonal matrices and we set H = . Since the operators defined in (32), (33) and (34) have exactly the same properties as corre-sponding operators in Section 5 of [3], the method of separation of equation (5) into subsystemscorresponding to the elements of the decomposition (19) described in Section 6 of [3] works with-out any modification. We will not repeat the proof given in [3] here, but we will show a fewexample results for low values of N . It should be stressed that a crucial ingredient for realizingthe described in [3] algorithm in practice is the order of the generators in the basis Ξ describedin Definition 4.2. Once this basis is used for computations and the inverse in (9) is successfullycomputed, the separation of the system of equations comes up automatically.In what follows we present example results. The results for A N , N = 1 , , A , B , B , B , C , C and D . (There are the so called9 ccidental isomorphisms in low dimensions: A ≃ B ≃ C , D ≃ A × A , B ≃ C and A ≃ D .)In what follows we use matrix representation of classical algebras as traceless matrices preservinggiven bilinear form. There are many equivalent conventions in defining these preserved bilinearforms. We use the same convention as in [4]. In all cases we provide explicit parametrization of theconsidered matrix algebra. The results for B case are presented in great detail. For all exampleswe give the basis Ξ fulfilling the Definition 4.2, which is crucial for the calculations. Once thebasis is known, the equations for u i can be easily computed, so we do not present them for N = 4and for N = 3 we give only the Riccati equations corresponding to n + . B algebra (Lie algebra of O (5 , C ) group and Sp (4 , C ) ) The ordered basis of B algebra fulfilling the Definition 4.2 is encoded in the matrix: X i =1 a i X i = − a − a a a − a a a a − a a a − a − a a − a − a − a a a − a a − a . (35)According to (19) the algebra of type B splits in the following way: g = a ⊕ a ⊕ h ⊕ e a ⊕ e a . (36)The order of the basis defined in (35) is consistent with the splitting (36) in the sense that a =span { X , X , X } , a = span { X } , h = span { X , X } , e a = span { X } and e a = span { X , X , X } .The decomposition (36) yields the separation of system (9) into five subsystems. First we get the3-dimensional matrix Riccati equation corresponding to a : u ′ = − a u + 12 a u − a u u − a u + 2 a u + a ,u ′ = − a u − a u u − a u u + a u u + a u + a u − a u + a ,u ′ = 12 a u − a u − a u u + a u + (2 a − a ) u + a . for u , u and u . Once we solve it, we can solve the Riccati equation for u : u ′ = 12 ( a u − a u − a ) u + ( a u − a u − a + 2 a ) u − a u + a u + a . After the solution of this equation is found the rest of the solutions for u i can be found by simpleconsecutive integrations. The equations corresponding to h are: u ′ = − a u − a u − a u + a ,u ′ = 12 a u u − a u u − a u − a u − a u + a . The equation corresponding to the sector e a reads u ′ = − ( a u − a u − a ) e − u +2 u . And finally, the equations for e a : u ′ = (cid:18) − a u + a u + a (cid:19) e u − u ,u ′ = 12 (cid:0) − a u + 2 a u + 2 a (cid:1) u e u − u + ( − a u + a ) e u ,u ′ = − (cid:0) − a u + 2 a u + 2 a (cid:1) u e u − u + a e u − ( − a u + a ) u e u . .2 B algebra (Lie algebra of O (7 , C ) group and Sp (6 , C ) ) In this case the algebra decomposes into g = a ⊕ a ⊕ a ⊕ h ⊕ e a ⊕ e a ⊕ e a and the ordered basisfulfilling the Definition 4.2 yields the following parametrization: X i =1 a i X i = − a − a − a a a a − a a a a a a − a a a − a a − a a − a a a a − a − a − a a − a − a − a − a − a a a − a − a − a + a − a a a a − a − a − a + 2 a , where a = span { X , X , X , X , X } , a = span { X , X , X } , a = span { X } , h = span { X , X , X } , e a = span { X } , e a = span { X , X , X } and e a = span { X , X , X , X , X } .The matrix Riccati equation corresponding to a reads: u ′ = a u u + 12 a u − a u u − a u u − a u u − a u + − a u − a u − a u + a u + a ,u ′ = − a u u + a u u + 12 a u − a u u − a u − a u u ++ a u − a u − a u + ( a − a + 2 a ) u + a ,u ′ = − a u u − a u u + a u u + a u u − a u − a u u − a u u ++ a u + a u + a u − a u − a u + a ,u ′ = 12 u a − a u u − a u + a u u − a u u − a u u ++ u a − a u + a u − (2 a − a − a ) u + a ,u ′ = − a u − a u u − a u u − a u u + a u u + 12 a u ++ a u + a u + a u + (2 a − a ) u + a . The system of equation corresponding to a is u ′ = ( a u − a u − a ) u + ( a u − a u − a ) u u ++ 12 ( − a u + a u + a ) u + ( a u − a u − a ) u ++ ( a u + a u − a u − a u − a + 2 a ) u − a u + a u + a ,u ′ = ( a u − a u − a ) u u + ( u a − a u + a ) u u + −
12 ( u a − a u + a ) u − ( u a − a u + a ) u u + − ( u a − a u + a ) u − ( a u − a u + a − a ) u + − ( u a − u a − a ) u − u a + a u + a ,u ′ = −
12 ( a u − a u − a ) u − ( u a − a u + a ) u u + − ( u a − a u + a ) u + ( u a − a u + a ) u ++ ( u a − a u − a u + a u − a − a + 2 a ) u + a u − a u + a . a reads u ′ = − (cid:16) − u a u + a u u − a u u + a u u ++ a u − a u − a u + a u + a (cid:17) u + − (cid:16) a u u − a u u − a u u + a u u ++ u a − a u − a u + a u − a + a (cid:17) u ++ a u u − a u u + a u u − a u u − a u + a u − a u + a u + a . The remaining 12 equations which we do not present here can be solved by consecutive integrations. C algebra (Lie algebra of Sp (6 , C ) group) In this case the ordered basis fulfilling the Definition 4.2 yields the following parametrization: X i =1 a i X i = a − a a a a a a a a − a a a a a a a a a a a a a a − a + a − a − a a a a − a a − a − a a a a − a − a − a , which is consistent with the following decomposition: g = a ⊕ a ⊕ a ⊕ h ⊕ e a ⊕ e a ⊕ e a == span { X , . . . , X } ⊕ span { X , X } ⊕ span { X } ⊕⊕ span { X , X , X } ⊕⊕ span { X } ⊕ span { X , X } ⊕ span { X , . . . , X } . The equations corresponding to a , a and a are: u ′ = − a u − a u u − a u − a u u − a u u − a u ++ 2 a u + 2 a u + 2 a u + a ,u ′ = − a u u − a u u − a u u − a u u − a u u − a u u + − a u u − a u − a u u + a u + a u + a u + a u + a u + a ,u ′ = − a u u − a u u − a u u − a u u − a u u − a u + − a u u − a u u − a u u ++ a u + a u + a u + a u + ( a − a + a ) u + a ,u ′ = − a u − a u u − a u − a u u − a u u − a u ++ 2 a u + 2 a u + (2 a − a ) u + a ,u ′ = − a u u − a u u − a u − a u u − a u u − a u u − a u u + − a u u − a u u + a u + a u + a u + a u + ( a − a ) u + a ,u ′ = − a u − a u u − a u − a u u − a u u − a u ++ 2 a u + 2 a u + (2 a − a ) u + a ,u ′ = + ( a u + a u + a u − a ) u + ( a u + a u + a u − a ) u u ++ ( a u + a u − a u − a u − a + a + a ) u ++ ( − a u − a u − a u + a ) u − a u − a u − a u + a , ′ = + ( a u + a u + a u − a ) u u + ( a u + a u + a u − a ) u ++ ( a u − a u + a u − a u − a + 2 a − a ) u ++ ( − a u − a u − a u + a ) u − a u − a u − a u + a ,u ′ = ( − a u u − a u u − a u u + a u + a u + a u + a u − a ) u ++ (cid:16) − a u u − a u u + a u u − a u u + a u u + a u u ++ a u − a u + a u + a u − a u − a u − a + 2 a (cid:17) u ++ a u u + a u u + a u u − a u − a u − a u − a u + a . The remaining 12 equations which we do not present here can be again solved by consecutiveintegrations. A algebra (Lie algebra of SL (5 , C ) group) In this case the decomposition (19) of g has the following form a ⊕ a ⊕ a ⊕ a ⊕ h ⊕ e a ⊕ e a ⊕ e a ⊕ e a == span { X , . . . , X } ⊕ span { X , X , X } ⊕ span { X , X } ⊕ span { X } ⊕⊕ span { X , . . . , X } ⊕⊕ span { X } ⊕ span { X , X } ⊕ span { X , X , X } ⊕ span { X , . . . , X } , where the generators X i form an ordered basis fulfilling the Definition 4.2. The explicit parametriza-tion of the algebra by these generators reads: X i =1 a i X i = a a a a a a − a + a a a a a a − a + a a a a a a − a + a a a a a a − a . B algebra (Lie algebra of O (9 , C ) group) In this case the decomposition (19) of g has the following form a ⊕ a ⊕ a ⊕ a ⊕ h ⊕ e a ⊕ e a ⊕ e a ⊕ e a == span { X , . . . , X } ⊕ span { X , . . . , X } ⊕ span { X , X , X } ⊕ span { X } ⊕⊕ span { X , . . . , X } ⊕⊕ span { X } ⊕ span { X , X , X } ⊕ span { X , . . . , X } ⊕ span { X , . . . , X } , where the generators X i form an ordered basis fulfilling the Definition 4.2. The explicit parametriza-tion of the algebra by these generators P i =1 a i X i reads: − a − a − a − a a a a a − a a a a a a a a − a a a − a a a − a a a − a a a a − a a − a − a a − a a a a a − a − a − a − a a − a − a − a − a − a − a − a a a − a − a − a − a + a − a − a a a a − a − a − a − a + a − a a a a a − a − a − a − a + 2 a . .6 C (Lie algebra of Sp (8 , C ) group) In this case the decomposition (19) of g has the following form a ⊕ a ⊕ a ⊕ a ⊕ h ⊕ e a ⊕ e a ⊕ e a ⊕ e a == span { X , . . . , X } ⊕ span { X , X , X } ⊕ span { X , X } ⊕ span { X } ⊕⊕ span { X , . . . , X } ⊕⊕ span { X } ⊕ span { X , X } ⊕ span { X , X , X } ⊕ span { X , . . . , X } , where the generators X i form an ordered basis fulfilling the Definition 4.2. The explicit parametriza-tion of the algebra by these generators P i =1 a i X i reads: a − a a a a a a a a a − a + a a a a a a a a a a − a a a a a a a a a a a a a a a a a a − a + a − a − a − a a a a a − a a − a − a − a a a a a − a − a − a + a − a a a a a − a − a − a − a . D algebra (Lie algebra of O (8 , C ) group) In this case the decomposition (19) of g has the following form a ⊕ a ⊕ a ⊕ a ⊕ h ⊕ e a ⊕ e a ⊕ e a ⊕ e a == span { X , . . . , X } ⊕ span { X , . . . , X } ⊕ span { X } ⊕ span { X } ⊕⊕ span { X , . . . , X } ⊕⊕ span { X } ⊕ span { X } ⊕ span { X , . . . , X } ⊕ span { X , . . . , X } , where the generators X i form an ordered basis fulfilling the Definition 4.2. The explicit parametriza-tion of the algebra by these generators P i =1 a i X i reads: a − a a a a a a a a a − a + a a a − a a a a a a − a a − a − a a a a a a − a − a − a − a − a − a − a + a − a − a − a a − a − a − a − a + a − a − a − a a a − a − a − a − a + a − a a a a − a − a − a − a . We have presented fully applicable algorithm for reducing the highly non linear system of equations(5) for the parameters u k to a hierarchy of Riccati matrix equations and integrals. The resultspresented in Section 7 were obtained by computer program implementing this algorithm writtenunder Maple . The existence of working computer program proves the usability of the algorithm.The computation complexity of this algorithm grows fast with the rank of the algebra N . Thecrucial point is computation or the inverse (9), which has to be realized by Cramer’s rule of acomputational complexity scaling as d ! with the dimension of the inverted matrix d . Since theproblem splits into subsystems corresponding to decomposition (19) the computation of the inverse(9) is substantially simplified, since it can be calculated by inverting the matrices corresponding14o each a k separately. Thus, the crucial ingredient for the computation complexity turns out to bethe inversion of the matrix corresponding to a which has the largest dimension. The dimensionof a depends on the algebra type in the following way: A N B N C N D N dim a N N − N ( N + 1) 2 N − a for C N differs from the others since in growsquadratically with N while for A N , B N and D N there is a linear growth of dim a with N , sothe computational complexity grows much faster with N for algebras C N than for the other ones.On a standard PC the algorithm is executed in a reasonable time for dim a
10 and the lastalgebras in each series fulfilling this condition are A , B , C and D . G G algebra. We discus how the fact that the G algebradoes not fulfill some of the lemmas proved for classical groups affects the usability of Wei-Normanmethod in this case and to what extend some of the results presented in this paper apply to G .The lowest dimensional matrix representation of G is 7-dimensional [4]. We use the followingexplicit matrix representation of G : X i =1 a i X i = − a √ a √ a √ − a √ a √ − a √ a √ a a a − a a − a √ − a − a + a a a a a √ a a − a − a − a a √ a a − a a − a − a √ − a a − a a − a − a − a √ − a − a − a − a a , (38)where X i are the root vectors, n + = span { X , . . . , X } , h = span { X , X } , n − = span { X , . . . , X } and the root vectors corresponding to positive simple roots are X (long) and X (short).First obstacle was already mentioned in the proof of Lemma 5.2. The G root system containsroot strings or length 4, so the nilpotency order of adjoint operators ad X may be equal to 4 forsome root vectors X and exp( u ad X ) for such vectors is a polynomial of order 3 in u i . For examplein the parametrization (38): exp( u ad X ) = − u − u − u
00 1 0 0 0 0 0 0 3 u − u u − u − u u u
00 0 0 0 0 1 0 0 0 0 0 0 u
00 0 0 0 0 0 1 0 0 0 0 u u − u u − u . As a consequence, the equations obtained are no longer Riccati equations.15he second obstacle is that for G group the decomposition (19) having the properties describedby Lemma 4.1 and Corollary 4.3 does not exist. The rank of G is 2, so one would expectthe decomposition of n + into two commuting subalgebras generated by some disjoint subsets ofroot vectors, but it turns out that this is not the case. Since there are finitely many potentialdecompositions of this type it is easy to check all of them. It turns out that the only nontrivialdecomposition of n + into two subalgebras is n + = c ⊕ c := span { X , X , X } ⊕ span { X , X , X } , (39)where X i are defined in (38). It easy to check that c is commutative, but c is not. Moreover c isnot an ideal in n + . Thus, both algebras c and c do not fulfill the Lemma 4.1. As a consequencethe invariance properties described by Lemma 5.4 are lacking and the system (8) will not separateinto blocks smaller than those corresponding to the decomposition g = n − ⊕ h ⊕ n + . This is theonly decomposition that survives.The equations corresponding to sector n + are: u ′ =2 a u u + 3 a u u + 6 a u u + 2 a u + 3 a u u + a u + − a u + 3 a u − a u u − a u u − a u + a u + 3 a u ++ ( a + a ) u + a ,u ′ = − a u − a u − a u u − a u − a u u − a u − a u ++ ( − a + 2 a ) u + a ,u ′ = a u u + 2 a u u + 4 a u u + a u + 2 a u u − a u − a u u + − a u u + a u + a u + 2 a u − a u + a ,u ′ = a u u + 3 a u u u − a u + 3 a u u + a u − a u + a u u ++ 3 a u u − a u u − a u − a u + (2 a − a ) u + a ,u ′ = a u u u + a u u + a u u + 2 a u u + a u + a u u + a u u + − a u u + a u − a u u − a u + a u + a u − a u + a ,u ′ = a u u u + a u u + 2 a u u + a u u + a u u + a u u u ++ a u u u + a u + a u u + a u u + a u u + a u u − a u u ++ a u − a u + 2 a u − a u + ( a − a ) u + a , The polynomials on the right hand side of above system are of order 4 and not only 3. The maximalpower of single variable is equal to 3, but the lack of decomposition into commuting subalgebrascauses the appearance of higher order terms, because the product of two noncommuting operatorsof nilpotency order 4 does not have to be of the same nilpotency order.The remaining eight functions u i corresponding to sectors h and n − can be computed byconsecutive integration, provided the solutions of the above six equations are found. This propertyof the decomposition g = n − ⊕ h ⊕ n + holds also in this case.The question of applicability of our method to other exceptional Lie groups is also interestingand definitely worth answering. The root systems of other exceptional Lie groups do not containroot strings of order 4, but the problem of existence of the decompositions into sum of commut-ing subalgebras with relevant invariance properties deserves deeper study and will be providedelsewhere.
10 Discussion and remarks
The general method for solving the matrix Riccati equation is not known, so the method presentedin this paper does not provide the general solution of the system (2-3). Nevertheless there aremany methods to study matrix Riccati equations (see for example [6] and references therein), soour method yields a major reduction of complexity of the original problem (2-3).16calar and matrix Riccati equations and system of the form (2-3) are examples of the so called
Lie systems . The theory of such systems is still being developed, see for example [5] and the mostrecent review paper [6] with exhaustive list of references therein. It is known that every systemof Riccati equations is related to a Lie group action and solution of this system is equivalent tosolution of the system of the form (2-3), but not every system of the form (2-3) is equivalent toRiccati equation system. In [3] we have shown that for SU ( N + 1) and SL ( N + 1 , C ) the system(2-3) is equivalent to hierarchy of Riccati equations and linear equations. Here we have shownthat the same statements holds for all classical Lie groups, but can not be generalized to all Liegroups, presenting G as a counterexample. Acknowledgments
We gratefully acknowledge fruitful discussions with Javier de Lucas. The presented results areobtained in frames of the the Polish National Science Center project MAESTRO DEC-2011/02/A/ST1/00208 support of which is gratefully acknowledged by both authors.
ReferencesReferences [1] J. Wei and E. Norman. Lie algebraic solution of linear differential equations.
J. Math. Phys. ,4(4):575–581, 1963.[2] J. Wei and E. Norman. On global representations of the solutions of linear differential equa-tions as a product of exponentials.
Proc. Am. Math. Soc. , 15(2):327–334, 1964.[3] S. Charzy´nski and M. Ku´s . Wei-Norman equations for a unitary evolution.
J. Phys. A:Math. Theor. , 46, 265208, 2013.[4] J. E. Humphreys. Introduction to Lie Algebras and Representation Theory. Springer-Verlag1980.[5] P. Winternitz, Lie groups and solutions of nonlinear differential equations (in Nonlinearphenomena, Oaxtepec, 1982),
Lecture Notes in Phys. , vol. 189, p.263-331, Spinger 1983.[6] J. F. Cari˜nena, and J. de Lucas, Lie systems: theory, generalisations, and applications.