Integrable systems associated to the filtrations of Lie algebras

In tegrable systems assoc iated to the filtrations of Lie algebras Bo ˇ zidar Jo v ano vi´ c, Tijana ˇ Sukilo vi ´ c, Sr djan V ukmiro vi ´ c Abstract. In 1983 Bogoy avlenski conjectured that if the Euler equa tions on a Lie al- gebra g 0 are in tegrable, then their certain extensions to semisimple lie alge bras g related to the filtrations of Lie algebras g 0 ⊂ g 1 ⊂ g 2 · · · ⊂ g n − 1 ⊂ g n = g are i n tegrable as we ll. In particular, b y t aking g 0 = { 0 } and natural filtrations of so ( n ) and u ( n ), we hav e Gel’fand-Cetlin in tegrable systems. W e pro ved the conjecture for filtrations of compact Lie algebras g : t he system are in tegrable in a nonco mmutat ive sense b y means of p oly- nomial in tegrals. V arious constructions of complete comm utativ e p olynomial in tegrals for t he system are also gi ven. 1. In tro duction Let G be a compac t connected Lie gr oup. Consider a chain of connected compact Lie subgroups G 0 ⊂ G 1 ⊂ G 2 ⊂ · · · ⊂ G n − 1 ⊂ G n = G and the corr esp onding filtra tion o f the Lie algebra g = Li e ( G ) g 0 ⊂ g 1 ⊂ g 2 · · · ⊂ g n − 1 ⊂ g n = g . (1) W e study in tegrable E ule r equations related to the filtration (1). One can consider non compact Lie algebras a s well. In fa c t, one of the first contribution is g iven by T r ofimov, who constructed integrable systems on Borel subgr o ups of complex semisimple Lie algebr as (see [ 36 ]). Later, Bogoya v lenski [ 2 ] considered filtration of semisimple Lie algebr as, s uch that the restrictio ns o f the Killing form to g i , i = 0 , . . . , n are non-degenera te. W e restrict ourself to the compact case and a genera lization of Gel’fand-Cetlin s ystems on L ie algebr as so ( n ) and u ( n ) given by filtrations (9) and (1 0) below in order to insur e compact inv ariant manifolds of the flows. Similar statemen ts ca n be formulated for reductiv e g roups a s well. Fix a n inv ariant sca lar pro duct h · , · i on g and deno te the restr ictions of h · , · i to g i also by h · , · i . By the use of h · , · i , w e identify g ∼ = g ∗ and g i ∼ = g ∗ i , i = 0 , . . . , n . Let p i be the orthogo nal complement o f g i − 1 in g i , p 0 = g 0 and pr p i and pr g i the orthogona l pro jections onto p i and g i , r e s pe c tively . F o r x ∈ g , we denote y i = pr p i ( x ) , x i = y 0 + y 1 + · · · + y i = pr g i ( x ) , i = 0 , . . . , n. The Euler eq ua tions ˙ x = [ x, ω ] , ω = A ( x ) (2) asso ciated with a symmetric positive op er ator o f the form A ( x ) = A 0 ( y 0 ) + n X i =1 s i y i , A 0 : g 0 → g 0 , s i ∈ R , i = 1 , . . . , n (3) 2010 Mathematics Subje ct Classific ation. 37J35, 17B63, 17B80, 53D20 . Key wor ds and phr ases. Noncomm utativ e inte grability , inv ariant p olynomials, Gel’fand-Cetlin systems. 1 2 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C were s tudied by Bogoya v le nski [ 2 ]. The equa tio ns are Ha milto nia n with resp ect to the Lie–Poisson brack et 1 { f , g } | x = − h x, [ ∇ f ( x ) , ∇ g ( x )] i (4) and the Hamiltonian function H ( x ) = 1 2 h x, ω i = 1 2 h A ( x ) , x i . Due to the relations [ p i , p j ] ⊂ p j , 0 ≤ i < j ≤ n, the Euler equa tions (2 ) can be rewritten into the form ˙ y 0 =[ y 0 , A 0 ( y 0 )] , (5) ˙ y i =[ y i , A 0 ( y 0 ) − s i y 0 + ( s 1 − s i ) y 1 + · · · + ( s i − 1 − s i ) y i − 1 ] , i = 1 , . . . , n. (6) Spec ia lly , if g 0 = { 0 } is a trivial Lie alge br a, we hav e y 1 = const and the comp onents of y 2 are elemen tary functions of the time t . The system (5) , (6) has ob vious fa mily of po lynomial first integrals I = I 1 + I 2 + · · · + I n , (7) where I i are inv ariants R [ g i ] G i lifted to g along the pro jection to g i : I i = pr ∗ g i R [ g i ] G i , i = 1 , . . . , n . According to the following (quite s imple, but imp or ta nt ) lemma, it is clear that I is a c o mmu tative algebra with resp ect to the Lie-Poisson bra ck et (4 ). Lemma 1 ( [ 2 , 35 , 36 ]) . If f and g Li e–Poisson c ommute on g i , then their lifts ˜ f = pr ∗ g i f and ˜ g = pr ∗ g i g Lie–Poisson c ommute on g . Bogoya vlens k i found also a lar ge clas s of additional first in tegr als obtained b y the trans- lations of inv ariants of g i along the suba lgebra g i − 1 p α,β ( x ) = p ( αx i + β y i ) , p ∈ R [ g i ] G i , i = 1 , . . . , n, α, β ∈ R (8) and conjectured that the equations (5), (6) a re completely integrable if this is true for the Euler equatio ns (5) (see [ 2 ]). In [ 25 ], Mikityuk prov ed that Bogoy avlenski’s integrals (8) imply complete comm utative in teg rability in the case when ( g i , g i − 1 ) a re symmetric pair s, that is when [ p i , p i ] ⊂ g i − 1 , i = 1 , . . . , n. On the other hand, Thimm used c ha ins of suba lgebras (1) in studying the in tegrability of geo desic flows on homogeneo us space s (see [ 35 ]). He prov ed that integrals (7 ) form a complete commu tative alg ebras o n the Lie a lgebras so ( n ) and u ( n ), with res pe c t to the natural filtrations so (2) ⊂ so (3) ⊂ · · · ⊂ so ( n − 1) ⊂ so ( n ) (9) and u (1) ⊂ u (2) ⊂ · · · ⊂ u ( n − 1) ⊂ u ( n ) , (10) resp ectively . After [ 15 ], the corresp onding integrable systems are refereed a s Gel’fand-Cetlin systems on so ( n ) a nd u ( n ). Namely , Gel’fand and Cetlin constructed ca nonical bases fo r a finite- dimensional repr e sentation of the orthog onal and unitary gr oups b y the decomp osition o f the repres ent ation b y a chain o f subgro ups [ 12 , 13 ]. The co rresp onding integrable systems on the adjoint or bits with in tegrals I can be seen as a symplectic geometric version of the Gelfand-Cetlin construction [ 15 ]. Also, Thimm’s examples motiv ated Guillemin and Sternberg to intro duce an imp ortant notion of multiplicit y free Hamilto nain actions [ 14 , 16 ]. The construction is also used in the study of integrabilit y of geo de s ic flows on homogeneous 1 The gradient i s determined by an i n v ari an t metric: d f ( ξ ) = h∇ f ( x ) , ξ i . Also, to simplify notation, t he Lie brack ets, th e Lie-Poisson brack ets and the gradien ts of the functions on g i will b e denote d by the same symbols as on g , i = 1 , . . . , n . INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 3 spaces and bi-q uo tient s of Lie gro ups (see [ 1 , 5 , 7 , 35 ]). The nonholonimic systems on compact Lie g roups G with left inv ariant metrics defined by the Hamiltonians o f the form H = 1 2 h A ( x ) , x i , wher e A is given b y (3), and left-in v aria nt constraints are studied in [ 19 ]. In this paper w e prove complete integrability in a noncommutativ e sense of the sy s- tem (5) , (6) and calcula te the dimension of in v ariant isotropic tori δ (Theorem 2, Section 3): Main resul t 1 . As s u me that the Euler e quations (5) ar e int e gr able by p olynomial inte gr als. Then t he Euler e quations (5 ) , (6) ar e c ompletely inte gr able in a nonc ommut ative sense by me ans of p olynomial inte gr als as wel l. Concerning dynamics, nonco mm utative (or sup erintegrability) is a str onger character- istic then the usual commutativ e (or Liouville) integrability . The sys tem is solv able by quadratures , regular compact in v aria nt manifolds a re isotropic tori, and there exis t an ap- propriate a ction–angle co o rdinates in which the dyna mics is linearized [ 29 , 30 ]. It implies the Lio uville integrabilit y , at least by means of smo o th functions: inv ar iant isotr opic tori can be alwa ys organized into re s onant Lagrangia n tori [ 6 ]. According to the Mischenko-F omenko conjecture, a natural algebra ic problem is a con- struction of a c omplete commutativ e set of p olyno mia l in tegrals . The problem can be formulated in terms of pairs ( G i − 1 , G i ): we need to construct b ( g i , g i − 1 ) = 1 2 (dim p i + rank g i − rank g i − 1 ) (11) m utually indep endent commuting Ad G i − 1 –inv a riants on g i that are indep endent from the po lynomials o n g i − 1 (Corollar y 4, Theorem 5, Section 4). The basic examples ar e pairs ( G i − 1 , G i ) where G i − 1 is a multiplicit y free o r an a lmost multiplicit y free s ubgroup o f G i (see Examples 1 and 2, Section 4). In Section 5 we adopt v arious constr uctions of commutativ e po lynomials on g i to pro- vide complete commutativ e po lynomial int egra ls for the system (5), (6). When G i − 1 is an isotropy subgro up of a i ∈ g i , one ca n us e the Mishchenk o -F o menko shifting of argument metho d in b oth cases: when a i is regular (see [ 3 , 28 ]), but a lso a singular element of g i (see Theor em 8). F urther, as a n example of a v ariation o f Mikityuk’s construction and a complement to (9) and (10), a co mplete commutative sets of p olynomials for the filtra tio ns sp ( k 0 ) ⊂ sp ( k 1 ) ⊂ · · · ⊂ sp ( k n ) , k 0 < k 1 < · · · < k n are giv en (Pr op osition 10, see also [ 17 ]). Also, in subsection 5.3 we estimate the num b er of indep endent Bogoya v lenski’s in te- grals (8) (Theor em 12). Main resul t 2 . Un der c ertain generic assumptions, amo ng Bo goyavlenski’s int e gr als (8 ) ther e ar e at le ast b ( g i , g i − 1 ) mutu al ly indep endent p olynomials (se e (1 1 ) ) that ar e indep en- dent fr om the p olynomials on g i − 1 . Although the p olynomia ls (8) do not commute in general, we pr ovide exa mples of com- plete co mmu tative sets of integrals obtained b y B o goy avlenski’s metho d (Example 5). Fi- nally , in Theor ems 13, 15, 16 w e reca ll the results obtained in [ 3 , 9 , 20 , 3 3 ], which so lve the problem in several interesting cases. F or the completeness of the exp os ition, we b egin the presentation by briefly reca lling on the concept of no ncommutativ e in tegrability . 2. Complete algebras of functions on P o isson manifolds Let ( M , Λ) b e a Poisson manifold. The Poisson bra ck et of t wo smo o th functions is defined by the use of the Poisson tensor Λ as usual { f , g }| x = Λ x ( d f ( x ) , dg ( x )), giv ing the Lie algebra structur e to C ∞ ( M ). L et x = ( x 1 , . . . , x m ) b e lo cal co ordinates on M . Let f and g b e the first integrals of the Hamiltonian equatio ns with the Hamiltonian H : ˙ x = X H = X i { x i , H } ∂ ∂ x i = X i,j Λ ij ∂ H ∂ x j ∂ ∂ x i , (12) 4 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C that is { f , H } = { g , H } = 0 . Then, due to the Ja cobi iden tity , their Poisson brack et { f , g } is a lso the first in tegral: {{ f , g } , H } = 0 . Therefore we can consider the Lie algebra F of first integrals. Let F x = { d f | f ∈ F } be a s ubspace of T ∗ x M spanned b y diff erentials of the functions from F at x ∈ M . Assume that the dimensio ns of F x and dim ker Λ x | F x are consta nt o n an op en dense set U o f M . The corres p o nding dimensions are denoted by ddim F ( differ en tial dimension of F ) and dind F ( differ ential index o f F ), resp ectively . Remark 1 . The num b ers l = ddim F and k = dind F ha ve a clea r g eometrical meaning. Let y 1 , y 2 , . . . , y l ∈ F b e indep endent functions within a domain V ⊂ U , suc h that F : V → W ⊂ R l , F ( x ) = ( y 1 ( x ) , . . . , y l ( x )) is a submers ion and { y i , y j } = a ij ( y 1 , . . . , y l ) , (13) where a ij are smoo th functions on W . Then (13) defines a Poisson str ucture on W of a constant corank k . F is a c omplete algebr a (or a c omplete algebr a at x , x ∈ U ) if ddim F + dind F = dim M + corank Λ , or, equiv alent ly , if F Λ x = { ξ ∈ T ∗ x M | Λ x ( F x , ξ ) = 0 } ⊂ F x , x ∈ U. (14) Similarly , if F is an ar bitrary Lie subalgebra of C ∞ ( M ) and F 0 ⊂ F its subalgebra, w e say that F 0 is a c omplete sub algebr a of F if ddim F 0 + dind F 0 = ddim F + dind F . In particula r, F is a complete a lgebra o n M if and only if it is a complete subalgebra of C ∞ ( M ). If F is a complete algebra o n M then F | S (the r estrictions of the functions to S ) will be a complete algebra on a gener ic symplectic le a f S ⊂ M . Sp ecially , we may b e int erested in the comp etent ness of F not on M but on a particular, regular or singula r, symplectic le af S 0 (see Le mma 3 below). Then the co nditio n (14) is sligh tly c ha nged: F | S 0 is co mplete ddim ( F | S 0 ) + dind ( F | S 0 ) = dim S 0 , if F Λ x = { ξ ∈ T ∗ x M | Λ x ( F x , ξ ) = 0 } ⊂ F x + k er Λ x , for a generic x ∈ S 0 . The Hamiltonian system (12) is called c ompletely inte gr able in a nonc ommutative sense (or sup erinte gr able ), if it has a co mplete algebra F of first integrals. The in tegrable s ystem is solv able by qua dr atures (at least lo cally), the regular compact connected comp onents o f the level s e ts determined by functions in F are δ –dimens ional (isotropic considered on the symplecic lea ves of Λ) tori ( δ = dim M − ddim F = dind F − cora nk Λ), and there exist an appropria te action–a ngle co or dinates in which the dynamics is linearized [ 29 , 30 ]. When F is a complete commut ative algebr a, ddim F = dind F = a ( M ) = 1 2  dim M + cor a nk Λ  , we hav e the usual Liouville in tegr ability – regula r compact inv ar iant level sets o f the func- tions fro m F are δ 0 = dim M − a ( M ) dimensio na l (Lagr angian on the s ymplectic leav es) tori. Remark 2 . Note that a ( M ) is the maximal num b er o f Poisson co mm uting indep endent functions on M . F or a n ar bitrary F we have the inequalit y ddim F + dind F ≤ 2 a ( M ) , (15) (or, conce rning Remar k 1, a ( W ) ≤ a ( M )) and equalit y holds if and o nly if F is a complete algebra. INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 5 Mishchenk o and F o menko stated the conjecture that noncommutativ e integrability im- plies the Liouville in tegr ability b y means of an algebr a of integrals that belong to the same functional class as the original one [ 29 ]. Note tha t in the cas e o f nonc o mm utative integra- bilit y tra jecto ries o f (12) b elong to the tori of dimensio n δ < δ 0 = dim M − a ( M ), that is, δ 0 –dimensional Lagr angian inv ariant tori are reso na nt : they are filled with ( δ 0 − δ )– parametric family of δ - dimensional in v a riant tor i. In a smooth categ ory the pro blem is easy to solve: we can alwa ys semi–lo cally reor ganize isotropic toric foliatio n into the Lagr angian toric foliation (see [ 6 ] 2 ). The polynomial Mishch enko-F omenko conjecture sa ys tha t one can find independent commuting functions p 1 , . . . , p a , that ar e po ly nomials in functions from F . The co njecture is solved for finite dimensional Lie a lgebras F (see [ 4 , 34 ]). F r om a point of view of the dynamics, nonco mmu tative in teg rability is stro nger then the Liouville one since Lagra ng ian tori are r e sonant and not an in trins ic pr o p erty of the system. 3. P o lynomial noncommutat ive integrabilit y Suppo se that the E uler equations (5) a re completely integrable b y means of a complete algebra A 0 of polyno mia l integrals, ddim A 0 + dind A 0 = dim g 0 + rank g 0 , and let δ 0 = dind A 0 − rank g 0 be the dimension o f generic inv a riant tori. Let A i be the alge bra R [ g i ] G i − 1 of Ad G i − 1 –inv a riant p olynomials on g i . Consider the lift o f algebras A i to the Lie algebra g : A i = pr ∗ g i A i , i = 0 , . . . , n. In particular, since the in v ariants on g i belo ng to A i , w e hav e I i ⊂ A i . Theorem 2 . The system (5) , (6) is c ompletely inte gr able with a c omplete set of p oly- nomial inte gr als A = A 0 + A 1 + · · · + A n . A generic motion is a quasi-p erio dic winding over δ = δ 0 + rank g 0 − rank g + n X i =1 dim pr p i ( g i ( x i )) dimensional invariant tori determine d by the inte gr als A . Her e we take generic elements x i ∈ g i , i = 1 , . . . , n . 3 Recall that for a generic x i ∈ g i , g i ( x i ) is a Ca rtan subalgebra in g that is spa nned b y the gradien ts of rank g i basic in v a riant p oly nomials in R [ g i ] G i , whic h also c oincides with the kernel of the Lie-Poisson structur e of g i at x i . Proof. First, no te that p oly nomials A i ( i > 0) ar e indeed integrals of the equations. The polyno mia l p belo ngs to A i , i = 1 , . . . , n , if and only if h∇ p ( x i ) , [ ξ , x i ] i = 0 , for all ξ ∈ g i − 1 , x i ∈ g i , which is equiv alent to the Poisson comm uting of p with the lifting o f a ll po lynomials from R [ g i − 1 ] to R [ g i ]. Let ˜ p = pr ∗ g i p . Then d dt ˜ p ( x ) = h∇ p ( x i ) , ˙ x i i = h∇ p ( x i ) , [ x i , A ( x i )] i = h∇ p ( x i ) , [ x i , A ( x i − 1 ) − s i x i − 1 ] i = 0 . 2 In [ 6 ] the symplectic case is considered, but the pro of can b e ea sily modi fied to the Poisson case. 3 g l ( x k ) de notes the is otrop y algeb ra of x k ∈ g k within g l : g l ( x k ) = { ξ ∈ g l | [ ξ , x k ] = 0 } , l ≤ k. Generic m eans that t he dimensions of the i sotrop y algebras g i ( x i ) a nd g i − 1 ( x i ) a re minimal. 6 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C Let O i ⊂ g i be a gener ic G i -adjoint or bit. With a sign c o nv ention (4) , the moment um mapping of G i adjoint a ction is simply the inclusion mapping ı : O i ֒ → g i m ultiplied by − 1, while the momentum ma pping of the adjoint ac tio n of G i − 1 is Φ i − 1 : O i → g i − 1 , Φ i − 1 = − pr g i − 1 ◦ ı. According to the Lemma 3 on integrability related to Hamiltonian a ctions (see below), the a lgebra C ∞ G i − 1 ( O i ) + Φ ∗ i − 1 ( R [ g i − 1 ]) is a complete algebra on O i , where C ∞ G i − 1 ( O i ) is the algebra of smooth G i − 1 -inv a riant functions. Since generic orbit of Ad G i − 1 -action on g i are separated b y in v ar ia nt polynomia ls, we hav e that A i | O i + Φ ∗ i − 1 ( R [ g i − 1 ]) is a co mplete po lynomial algebra o n O i . Therefore A i + pr ∗ i − 1 ( R [ g i − 1 ]) (16) is a complete algebra o n g i (recall that the inv ar iants R [ g i ] G i are contained in A i = R [ g i ] G i − 1 ). Nex t, b y induction using the item (ii) of Lemma 3, w e g et that A is a complete algebra of integrals on g . In order to determine the dimension of in v aria nt to ri, note that the dimension of in v ari- ant tori determined by the functions A i | O i + Φ ∗ i − 1 ( R [ g i − 1 ]) on O i equals (Lemma 3): δ i =dind ( A i | O i + Φ ∗ i − 1 ( R [ g i − 1 ])) = dind ( A i | O i ) = dind (Φ ∗ i − 1 ( R [ g i − 1 ]))) = dim g i − 1 ( x i − 1 ) − dim g i − 1 ( x i ) = rank g i − 1 − dim( g i − 1 ∩ g i ( x i )) = rank g i − 1 − rank g i + dim pr p i ( g i ( x i )) . Here x i ∈ O i is a gener ic element and x i − 1 = pr g i − 1 ( x i ). Again, by induction using the item (ii) of Lemma 3, w e get the dimension of in v ariant tori: δ = δ 0 + δ 1 + · · · + δ n = δ 0 + rank g 0 − rank g 1 + dim pr p 1 ( g 1 ( x 1 )) + rank g 1 − rank g 2 + dim pr p 2 ( g 2 ( x 2 )) + · · · + r ank g n − 1 − rank g n + dim pr p n ( g n ( x n )) = δ 0 + rank g 0 − rank g n + dim pr p 1 ( g 1 ( x 1 )) + · · · + dim pr p n ( g n ( x n )) , for a generic x i ∈ g i , i = 1 , . . . , n .  Here w e recall o n the construction o f collectiv e in teg rable sy stems. Co ns ider th e Hamil- tonian action of a c o mpact c onnected Lie group K on the sy mplectic manifold M with the equiv aria nt momentum mapping Φ : M → k ∗ ∼ = k . Let C ∞ K ( M ) b e the algebra of K –inv ariant functions. According to the No ether theorem, { f , ˜ p } M = 0 for all f ∈ C ∞ K ( M ), ˜ p = Φ ◦ p , p ∈ R [ k ]. Also, if p ∈ R [ k ] K is an inv ariant po lynomial, then ˜ p is K –inv ariant. In [ 6 ] we proved the following quite simple but imp or tant statement. Lemma 3 ( [ 6 ]) . (i) The algebr a of functions C ∞ K ( M ) + Φ ∗ ( R [ k ]) is c omplete on M : ddim ( C ∞ K ( M ) + Φ ∗ ( R [ k ])) + dind ( C ∞ K ( M ) + Φ ∗ ( R [ k ])) = dim M and the dimension of invariant r e gular isotr opic tori is δ 0 = dind ( C ∞ K ( M ) + Φ ∗ ( R [ k ])) = dind C ∞ K ( M ) = dind Φ ∗ ( R [ k ]) = dim K µ − dim K x , wher e K x and K µ ar e isotr opic sub gr oups of a generic x ∈ M and µ = Φ( x ) ∈ k . (ii) L et F ⊂ R [ k ] b e c omplete on a generic adjoi nt orbit in the image of M , ddim ( F | O ) + dind ( F | O ) = dim O , O ⊂ Φ( M ) , and let F = Φ ∗ F . Then C ∞ K ( M ) + F is c omplete on M : ddim ( C ∞ K ( M ) + F ) + dind ( C ∞ K ( M ) + F ) = dim M INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 7 and the dimension of invariant isotr opic tori is δ 1 = dind ( C ∞ K ( M ) + F ) = δ 0 + dind ( F | O ) . Note that all adjoin t orbits in Φ( M ) co uld be singular and then the completeness of F on k do es not imply directly the co mpleteness of the restriction F | O , O ⊂ Φ( M ). 4. The problem of a p olynomi al commutativ e integrabilit y As above, w e suppose that the Euler equations (5) a re completely integrable by means of a complete comm utative algebr a A 0 of polyno mia l integrals a nd set A 0 = pr ∗ g 0 A 0 . According to Lemma 1 and Theorem 2, we ha ve Corollar y 4 . Supp ose that for every i = 1 , . . . , n ther e exist a c ommu tative sub algebr a B i of the algebr a of Ad G i − 1 –invariants A i = R [ g i ] G i − 1 , su ch that B i + pr ∗ g i − 1 ( R [ g i − 1 ]) is a c omplete algebr a on g i . Then B = A 0 + B 1 + · · · + B n , B i = pr ∗ g i ( B i ) , i = 1 , . . . , n (17) is a c omplete c ommutative set on g : ddim B = a ( g ) = 1 2  dim g + r ank g  . Therefore, the polyno mial comm utative in teg rability o f the sys tem (5), (6), r e duce s to a c onstruction of co mm utative subalgebras B i of A i , i = 1 , . . . , n . Mikityuk pro ved that Bog oy avlenski’s integrals (8) solves the pro blem in the case when ( g i , g i − 1 ) are symmetr ic pairs [ 25 ] (see also examples in [ 17 , 31 ]). T o simplify notation, let us denote K = G i − 1 , G = G i , k = g i − 1 , g = g i , p = p i , x = x i , y 0 = x i − 1 , y 1 = y i , A = R [ g ] K . The a lgebra of p olynomials A + pr ∗ k ( R [ k ]) (18) is co mplete with res pec t to the Lie Poisson brack et on g (see (16)). Let B be a commutativ e subset of the algebra of Ad K –inv a riants A . W e alwa y s a ssume that B contains Ad G –inv a riant po lynomials R [ g ] G and the lift of Ad K –inv a riants pr ∗ k ( R [ k ] K ) that belong to the center of A . Definition 1 . W e shall say that B is a c omplete c ommutative set of Ad K –invariants (or ad k – invariants ) if the set of p olynomials B + pr ∗ k ( R [ k ]) is complete on g with resp ect to the Lie-P oisson brac ket, or , equiv alently , if B is a complete commutativ e subset of A : ddim B = dind B = 1 2 (ddim A + dind A ) . Theorem 5 . A c ommutative set of Ad K –invariants B is c omplete if it has b ( g , k ) = 1 2 (dim p + r ank g − r ank k ) p olynomials indep endent fr om p olynomials pr ∗ k ( R [ k ]) . In other wor ds, B is c omplete if dim pr p span {∇ p ( x ) | p ∈ B } = 1 2 (dim p + r ank g − r ank k ) , for a generic x ∈ g . Proof. Assume that B is a commutativ e set of Ad K –inv a riants and let κ = dim pr p span {∇ p ( x ) | p ∈ B } . Then, due to inequality (15) and the completeness of (18 ), we ha ve ( κ + dim k ) + ( κ + rank k ) = ddim ( B + pr ∗ k ( R [ k ])) + ddim ( B + pr ∗ k ( R [ k ])) 8 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C ≤ ddim ( A + pr ∗ k ( R [ k ])) + ddim ( A + pr ∗ k ( R [ k ])) = 2 a ( g ) = dim g + rank g . Therefore, B + pr ∗ k ( R [ k ]) is complete if a nd only if κ = b ( g , k ).  Note that the construction of B is closely related to the construction of complete G – inv ar iant algebras o f functions on the cota ng ent bundle o f the homog eneous spac e G/K (see [ 5 , 7 , 9 , 10 , 20 , 21 , 23 , 26 , 27 ]). The simplest situation is the cas e when the algebra of Ad K –inv a riants A is already commutativ e – the center of A co incides w ith A . Then w e s ay that K ⊂ G is a multiplici ty fr e e sub gr oup of G . F or example, S O ( n − 1 ) a nd U ( n − 1) are m ultiplicit y free subgr oups of S O ( n ) and U ( n ), respectively . This is the reason that the inv ar iants (7) a re sufficient for the in teg rability of the Gel’fand-Cetlin systems on so ( n ) and u ( n ) [ 14 , 16 ]. The next natura l step is to cons ider a subgroup K ⊂ G whe n apart from Ad G –inv a riants, for a complete comm utative s e t o f Ad K –inv a riants w e ca n take arbitrar y Ad K –inv a riant po lynomial, w hich is not in the center of A . Then we say that K is a n almost multiplicity fr e e sub gr oup of G . The cla ssification of multiplicit y fre e subgroups K of compact Lie gr oups G is giv en b y Kr¨ amer [ 22 ] (see a lso Heckman [ 18 ]). If G is a simple group, the pa ir of corre s po nding Lie algebras ( g , k ) is ( B n , D n ) , ( D n , B n − 1 ) , or ( A n , A n − 1 ⊕ u (1)) . Example 1 . Multiplicity free pairs are: ( S U ( n ) , S ( U (1) × U ( n − 1 ))) , ( S U ( n ) , U ( n − 1)) , ( S U (4) , S p (2)) , ( S O ( n ) , S O ( n − 1)) , ( S O (4) , U (2)) , ( S O (4 ) , S U (2)) , ( S O (6 ) , U (3 )) , ( S O (8) , S pin (7)) , ( S pin (7 ) , S U (4)) . In the next statement w e obtained the class ifications o f almost mult iplicity free sub- groups of compa c t simple Lie groups: Theorem 6 . The p air of Lie algebr as ( g , k ) c orr esp onding t o the almost multiplicity fr e e sub gr oups K ⊂ G b elongs to the fol lowing list: ( A n , A n − 1 ) , ( A 3 , A 1 ⊕ A 1 ⊕ u (1)) , ( B 2 , u (2)) , ( B 2 , B 1 ⊕ u (1)) , ( B 3 , g 2 ) , ( g 2 , A 2 ) . The proo f will b e giv en in a separate pa pe r . Example 2 . Almost mult iplicity free pairs: ( S U ( n ) , S U ( n − 1)) , ( S U (4 ) , S ( U (2) × U (2 )) , ( S U (3) , S O (3)) , ( S O (5 ) , S O (3) × S O (2)) , ( S p (2) , S p (1) × U (1)) , ( S O (5) , U (2 )) , ( S p (2) , U (2)) , ( S O (6) , S O (4) × S O (2)) , ( S O (6) , S U (3)) , ( S pin (7) , G 2 ) , ( G 2 , S U (3)) , ( S O (3) × S O (4) , S O (3)) . 5. Commuta tiv e p ol ynomial in teg rals As in Sec tio n 4, let us deno te K = G i − 1 , G = G i , k = g i − 1 , g = g i , p = p i , x = x i , y 0 = x i − 1 , y 1 = y i , that is x = y 0 + y 1 . F urther, in this sectio n by p 1 , . . . , p r we denote the base of homo geneous inv ariant po lynomials o n g , r = rank g . INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 9 5.1. Isotropy s ubgroups and the Mi shcenk o-F omenko shifting of argumen t metho d. Mishchenk o and F omenko show ed that the set of po ly nomials induced from the inv ar iants b y shifting the argument B = { p j,a,k ( x ) | k = 0 , . . . , deg p j , j = 1 , . . . , r } , (19) p j,a,λ = p j ( x + λa ) = deg p j X k =0 p j,a,k ( x ) λ k is a complete comm utative set on g , for a ge neric a ∈ g (see [ 3 , 28 ]). The following Bols inov’s sta temen t will be useful to us frequently , hence w e are stating it for the sake of completeness. The pro of can b e found in [ 3 , Prop ositio n 2.5 ]. Lemma 7 ( [ 3 ]) . Consider a p encil of skew-symmetric line ar forms Π = { λ 1 Λ 1 + λ 2 Λ 2 | λ 1 , λ 2 ∈ R , | λ 1 | + | λ 2 | 6 = 0 } in R n and set R 0 = max Λ ∈ Π rank Λ . L et Λ 1 , . . . , Λ q ∈ Π C = { λ 1 Λ 1 + λ 2 Λ 2 | λ 1 , λ 2 ∈ C , | λ 1 | + | λ 2 | 6 = 0 } b e line arly indep endent skew-symmetric forms in C n with r ank less then R 0 . L et L ⊂ C n b e the u nion of kernels of al l forms in Π C , L 0 ⊂ C n b e the union of kernels of forms with the maximal r ank, and let L Λ 0 = { ξ ∈ C n | Λ( ξ , η ) = 0 , η ∈ L 0 } b e the Λ –ortho gonal sp ac e of L 0 . Then. (i) Λ –ortho gonal sp ac e of kernels of forms with the maximal r ank L Λ 0 do es not dep end on Λ ∈ Π C . It is an isotr opic subsp ac e and c ontains t he kernels of al l forms in Π C : L ⊂ L Λ 0 , L ⊂ L Λ 0 . (ii) The e quality L Λ 0 = L holds if and only if dim C ker( Λ | ker Λ i ) = dim { ξ ∈ k er Λ i | Λ( ξ , η ) = 0 , η ∈ ker Λ i } = n − R 0 , i = 1 , . . . , q , wher e Λ ∈ Π C is a skew-symmetric form of r ank R 0 at x . Theorem 8 . L et k b e a Lie algebr a e qual to the isotr opy algebr a of a element a ∈ g k = g ( a ) = { x ∈ g | [ x, a ] = 0 } and let K b e its c orr esp onding Lie gr oup. The set (19) is a c omplete c ommutative s et of Ad K -invariant p olynomials on g . Proof. It is clear that p olyno mia ls B are Ad K -inv a riant. Based on Lemma 7, Bolsino v prov ed that for a given x 0 ∈ g (regular or singular), o ne can find a r egular element a ∈ g , such that B is a complete commutativ e a lgebra on the adjoint o rbit through x 0 (see [ 8 ] and [ 37 , Theorem 5, page 230], the sta temen t is firstly pro ved by Mishchenko and F omenko by using a differen t appr oach [ 28 , 37 ]). One can reverse the roll of x 0 and a and s lig htly change the given theorem to prov e the required sta temen t. W e will present the detailed pro of since some of the arg ument s will b e used in the proof of Theorem 12 given below. The set (19 ) is the union of Ca simir p oly no mials of the Poisson brackets of maximal rank R 0 = dim g − rank g within the p encil Π o f compatible Poisson brack ets given b y Lie Poisson structure Λ and the a -brack et Λ a given b y { f , g } a | x = Λ a ( x )( ∇ f ( x ) , ∇ g ( x )) = −h a, [ ∇ f ( x ) , ∇ g ( x )] i . (20) It is sufficient to consider the case when a is a singular elemen t of g . Since g is compac t, there e xist x = y 0 + y 1 ∈ g , such that the complex plane ℓ ( x, a ) = { λ 0 y 0 + λ 1 y 1 | λ 0 , λ 1 ∈ C } int ersect the set of sing ular points in the complexified Lie algebra g C only at the line C · a and tha t y 0 is r egular in k C . Consider the co mplexified p encil of skew-symmetric forms Π C x = { λ 0 Λ( x ) + λ 1 Λ a ( x ) | λ 0 , λ 1 ∈ C , | λ 0 | + | λ 1 | 6 = 0 } . 10 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C The k ernel of λ 0 Λ( x ) + λ 1 Λ a ( x ) in g C is the isotro p y alg ebra g C ( λ 0 x + λ 1 a ) = { ξ ∈ g C | [ ξ , λ 0 x + λ 1 λa ] = 0 } . (21) Thu s, all forms Π C x are regular exc ept Λ a ( x ) with the kernel equals to k C . According to Lemma 7 , w e have L Λ 0 = L 0 + k C (22) if and only if 4 dim C { ξ ∈ k C | h x, [ ξ , η ] i = 0 , η ∈ k C } = dim C { ξ ∈ k C | [ ξ , y 0 ] = 0 } = rank g . Since rank k = ra nk g and y 0 is reg ular, the ab ove identit y is sa tisfied. The real pa rt of L 0 is spanned by the g radients o f the polyno mials in B at x . Therefo r e, in the real domain, (22) implies that B + pr ∗ k ( R [ k ]) is a complete set of functions at x , and therefore on a n open dense subset on g .  5.2. Mikityuk’s construction wi th symmetric pairs and its v ariation. In the ab ov e notation, Bog oy avlenski’s in tegrals (8) ca n b e defined as co efficients in λ in the ex- pansion of p j ( y 0 + λy 1 ): B = { p j,k ( x ) | k = 0 , . . . , deg p j , j = 1 , . . . , r } , (23) p j,λ ( x ) = p j ( y 0 + λy 1 ) = X k λ k p j,k ( x ) , j = 1 , . . . , r . Mikityuk has prov e d the following completeness statement for p olyno mia ls (23 ) (se e [ 25 , Prop ositio n 3, Theorems 1 and 2]). Theorem 9 ( [ 25 ]) . Assume that ( g , k ) is a symmetric p air. Then the set of p olyno- mials (23) is a c omplete c ommutative set of ad k –invariant p olynomials on g . Ther efor e, if al l p airs ( g i , g i − 1 ) , i = 1 , . . . , n ar e symmetric and A 0 is a c omplete c ommutative set on g 0 , then the asso ciate d set (17 ) is a c omplete c ommut ative set on g . There is a small v a riation o f Mikityuk’s construction that allows us to significa nt ly extend the class of examples if no t all of the pairs ( g i , g i − 1 ), i = 1 , . . . , n are symmetric. Simply , we can extend the or ig inal filtration and us e differen t metho ds at every step. As an illustration, consider an arbitra ry c hain of compact symplectic gro ups with standa rd inclusions S p ( k 0 ) ⊂ S p ( k 1 ) ⊂ · · · ⊂ S p ( k n ) , k 0 < k 1 < · · · < k n . (24) Then, w e extend (24) to the filtration (also using natural inclusions): S p ( k 0 ) ⊂ S p ( k 0 ) × S p ( k 1 − k 0 ) ⊂ S p ( k 1 ) ⊂ · · · ⊂ S p ( k n − 1 ) × S p ( k n − k n − 1 ) ⊂ S p ( k n ) . Now, the construction of functions in in volution is clear. If in the step ( g i , g i − 1 ) we hav e a symmetric pa ir ( sp ( k j ) , sp ( k j − 1 ) × sp ( k j − k j − 1 )), then B i is given by (23). On the other hand, if ( g i , g i − 1 ) is ( sp ( k j − 1 ) × sp ( k j − k j − 1 ) , sp ( k j − 1 )), then for B i we tak e an arbitrar y co mplete commutativ e set on sp ( k j − k j − 1 ) (for example using the argument s hift metho d [ 28 ]). Proposition 1 0 . A ssume that the Euler e qu ations (5) on sp ( k 0 ) ar e inte gr able by p oly- nomial inte gr als. T hen the Euler e quations (5 ) , (6) asso ciate d to t he filt r ation (24) ar e c ompletely inte gr able in a c ommutative sen s e by me ans of p olynomial inte gr als as wel l. Of cour se, the ab ove v ariatio n ca n be applied for other cons tructions of commutativ e po lynomials. If for a given Lie suba lgebra k ⊂ g one can find a Lie s ubalgebra h , k ⊂ h ⊂ g , having a complete commutativ e set o f ad k –inv a riant po lynomials on h a nd a co mplete commutativ e set o f ad h –inv a riant po ly nomials on g , then the problem is solved for a pair ( g , k ) as well. 4 By h· , · i w e also denote an inv ariant quadratic form on g C , the extension of the inv ariant scalar pro duct from g to g C . INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 11 5.3. Bogoy a vlenski’s integrals. First, we note the follo wing lemma. Lemma 11 . The p olynomials (23) ar e Ad K –invariant. Proof. As we already ment ioned, the p olynomials (23) commute with the lifting o f po lynomials fro m R [ k ] to R [ g ] if ( g , k ) is a symmetric pair (Theor em 9). The pro of g iven there c an be slightly mo dified, and ado pted to a general case . Let p ∈ R [ g ] G be an in v ariant p olynomia l and let p λ ( x ) = p ( y 0 + λy 1 ). The gra dient of p λ at x is given b y P λ = ∇ p λ ( x ) = P λ 0 + λP λ 1 , where P λ 0 = pr k ∇ p | y 0 + λy 1 , P λ 1 = pr p ∇ p | y 0 + λy 1 . Since p is an inv ariant po lynomial, w e ha ve [ P λ 0 + P λ 1 , y 0 + λy 1 ] = 0 . (25) Let ξ b e an arbitrary element in k . Then h∇ p λ ( x ) , [ ξ , x ] i = h P λ 0 + λP λ 1 , [ ξ , y 0 + y 1 ] i = h P λ 0 , [ ξ , y 0 ] i + λ h P λ 1 , [ ξ , y 1 ] i . (26) On the other hand, from (25) we get 0 = h [ P λ 0 + P λ 1 , y 0 + λy 1 ] , ξ i = h [ P λ 0 , y 0 ] , ξ i + h [ P λ 1 , λy 1 ] , ξ i . (27) Therefore, according to (26) and (27), p λ is a n Ad K –inv a riant polyno mial. W e can see this also directly b y us ing the ident ities: p (Ad g ( x )) = p ( x ), Ad g ( x ) = Ad g ( y 0 ) + Ad g ( y 1 ), g ∈ G , and Ad g ( y 0 ) ∈ k , Ad g ( y 1 ) ∈ p , g ∈ K .  Now, we w o uld lik e to use a relationship b etw e e n the a rgument shift metho d and trans- lations a long the subalgebr as presented ab ov e to estimate the num b er o f indep endent B o - goy avlenski’s in tegr als. Theorem 12 . L et S ing ( g C ) b e the s et of singular p oint in g C and let ℓ ( x ) = { λ 0 y 0 + λ 1 y 1 | λ 0 , λ 1 ∈ C } . Assume that t her e exist x ∈ g such that ℓ ( x ) ∩ S ing ( g C ) = { 0 } , or, ℓ ( x ) ∩ S ing ( g C ) = C · y 1 . Then, for a given x , for p olynomials (23) we have dim B ( x ) ≥ b ( g , k ) = 1 2 (dim p + rank g − rank k ) , wher e B ( x ) = pr p span {∇ p j,λ ( x ) | j = 1 , . . . , r, λ ∈ R } . Proof. Step 1. W e ca n obtain p olynomials (23 ) b y the transla tio n of in v ariants in the direction of k instead of p : p λ ( x ) = p ( λy 0 + y 1 ), p ∈ R [ g ] G . Since ∇ p λ ( x ) = λ pr k ∇ p | λy 0 + y 1 + pr p ∇ p | λy 0 + y 1 = λ pr k ∇ p | λy 0 + y 1 + pr p ∇ p | x + µy 0 , λ = µ + 1 , we need to estimate the dimension of the linear space B ( x ) = spa n { pr p ∇ p j | x + µy 0 | j = 1 , . . . , r , µ ∈ R } . The space B ( x ) is equal to the pro jection to p of the space C ( x ) spanned b y gradients of the p o ly nomials p a,µ ( x ) = p ( x + µa ) obtained by shifting of ar gument in the direction a = y 0 = pr k ( x ): C ( x ) = span {∇ p a,µ ( x ) | p ∈ R [ g ] G } Note that the sets o f p olynomials (19) a nd (23) are differ e nt, but the pro jections to p of their gradients at the given p oint x are the same. 12 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C Step 2. Le t us assume that the complex plane ℓ ( x ) intersects the set of singular p oints in g C only in 0 and consider the p encil o f c ompatible Poisson structures spanned by Lie-Poisson brack et (4) and the a -bracket (20), where a = y 0 . The kernel of the skew-symmetric form λ 0 Λ( x ) + λ 1 Λ a ( x ) in g C is the isotropy alge- bra (21). Thus all for ms in Π C x hav e the maximal ra nk, and, according to Lemma 7 , L Λ 0 = L 0 where L 0 ⊂ g C is the union of all kernels in in Π C x . Since C ( x ) = L 0 ∩ g , the set { p a,µ | µ ∈ R } is co mplete at x : dim C ( x ) = a ( g ) = 1 2 (dim g + rank g ) . Therefore dim B ( x ) = dim pr p C ( x ) = 1 2 (dim g + rank g ) − dim ( C ( x ) ∩ k ) . It remains to note that dim( C ( x ) ∩ k ) ≤ 1 2 (dim k + rank k ) , implying that dim B ( x ) ≥ b ( g , k ). Indeed, w e have { p a,µ 1 , f a,µ 2 } ( x ) = h x, [ ∇ p | x + µ 1 a , ∇ f | x + µ 2 a ] i = 0 , p, f ∈ R [ g ] G , and, if ∇ p | x + µ 1 a , ∇ f | x + µ 2 a ∈ k , then also h y 0 , [ ∇ p | x + µ 1 a , ∇ f | x + µ 2 a ] i = 0 . Thu s, C ( x ) ∩ k is an isotro pic subspace of k at y 0 with respect to the Lie-Poisson br ack e t on k . On the o ther hand, the maxima l isotropic subspace at y 0 (it is regular in k ) has the dimension 1 2 (dim k + rank k ). Step 3. Now assume that the complex plane ℓ ( x ) in ter s ects the set of s ingular po in ts in C · y 1 and in addition that pr g C ( y 1 ) x is regula r in g C ( y 1 ), where g C ( y 1 ) is the isotropy Lie algebra of y 1 within g C . Then all skew-symmetric forms in Π C x , except Λ 1 = Λ( x ) − Λ a ( x ), hav e the maxima l rank ( a = y 0 ). Since rank g C ( y 1 ) = rank g , we ge t dim C ker( Λ | ker Λ 1 ) = dim C { ξ ∈ g C ( y 1 ) | h x, [ ξ , η ] i = 0 , η ∈ g C ( y 1 ) } = dim C { ξ ∈ g C ( y 1 ) | [ ξ , pr g C ( y 1 ) x ] = 0 } = rank g . Thu s, from Lemma 7, w e ha ve L Λ 0 = L 0 + g C ( y 1 ), that in the real doma in implies C ( x ) Λ = C ( x ) + g ( y 1 ) . (28) F or y 1 that b e long an ope n dense set of p with minimal dimensions of the isotropy algebras g ( y 1 ) a nd k ( y 1 ), w e hav e that the semisimple part of g ( y 1 ) b elong s to k ( y 1 ) (e.g ., see [ 26 ]): [ g ( y 1 ) , g ( y 1 )] = [ k ( y 1 ) , k ( y 1 )] . Then g ( y 1 ) = k ( y 1 ) + z ( g ( y 1 )) , (29) where z ( g ( y 1 )) is the center of g ( y 1 ). F urther, w e ha ve (e.g., s ee [ 1 , Lemma 4]) z ( g ( y 1 )) = span {∇ p j ( y 1 ) | j = 1 , . . . , r } ⊂ C ( x ) . (30) By com bining (28), (29), and (30) we obtain C ( x ) Λ = C ( x ) + k ( y 1 ) . Thu s, there exist a subspa ce D ⊂ k ( y 1 ) suc h that C 1 ( x ) = C ( x ) + D is a ma ximal isotr opic subspace in g with r esp ect to Λ( x ) and dim C 1 ( x ) = 1 2 (dim g + rank g ) . INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 13 Using the identit y B ( x ) = pr p C ( x ) = pr p C 1 ( x ), the re s t of the pro of is the same a s in Step 2 with C ( x ) replaced b y C 1 ( x ).  Thu s, if B is comm utative and the conditions of Theor e m 12 a re satisfied, acco r ding to Theorem 5 and Lemma 11, (23) is a complete commutativ e set of ad k –inv a riant p oly nomials on g . Let f , p be inv ariant p olynomia ls, f λ ( x ) = f ( y 0 + λy 1 ), and p µ ( x ) = p ( y 0 + µy 1 ), µ 6 = λ , λ 2 + µ 2 6 = 0 . With the ab ov e notation w e hav e [ F λ 0 + F λ 1 , y 0 + λy 1 ] = 0 , [ P µ 0 + P µ 1 , y 0 + µy 1 ] = 0 , (31) which implies (see (27 )): h y 0 , [ F λ 0 , ξ ] i = λ h y 1 , [ ξ , F λ 1 ] i , h y 0 , [ P µ 0 , ξ ] i = µ h y 1 , [ ξ , P µ 1 ] i , ξ ∈ k . In particular, h y 0 , [ F λ 0 , P µ 0 ] i = λ h y 1 , [ P µ 0 , F λ 1 ] i , h y 0 , [ P µ 0 , F λ 0 ] i = µ h y 1 , [ F λ 0 , P µ 1 ] i . (32) Since x = y 0 + y 1 , y 0 , y 1 can be e xpressed a s line a r combinations of y 0 + λy 1 and y 0 + µy 1 , from (31) and (32) we g e t 0 = h y 0 , [ P µ 0 + P µ 1 , F λ 0 + F λ 1 ] i = h y 0 , [ P µ 0 , F λ 0 ] i + h y 0 , [ P µ 1 , F λ 1 ] i , 0 = h y 1 , [ P µ 0 + P µ 1 , F λ 0 + F λ 1 ] i = h y 1 , [ P µ 0 , F λ 1 ] i + h y 1 , [ P µ 1 , F λ 0 ] i + h y 1 , [ P µ 1 , F λ 1 ] i =  1 λ + 1 µ  h y 0 , [ F µ 0 , P λ 0 ] i + h y 1 , [ P µ 1 , F λ 1 ] i . Now, b y the use of the ab ov e iden tities, the Poisson brack et { f λ , p µ } r eads { f λ , p µ }| x = h y 0 + y 1 , [ P µ 0 + µP µ 1 , F λ 0 + λF λ 1 ] i = h y 0 , [ P µ 0 , F λ 0 ] i + λµ h y 0 , [ P µ 1 , F λ 1 ] i + λ h y 1 , [ P µ 0 , F λ 1 ] i + µ h y 1 , [ P µ 1 , F λ 0 ] i + λµ h y 1 , [ P µ 1 , F λ 1 ] i = h y 0 , [ P µ 0 , F λ 0 ] i + λµ h y 0 , [ F µ 0 , P λ 0 ] i + h y 0 , [ F µ 0 , P λ 0 ] i + h y 0 , [ F µ 0 , F λ P ] i − ( λ + µ ) h y 0 , [ F µ 0 , P λ 0 ] i =(1 − λ )(1 − µ ) h y 0 , [ F µ 0 , P λ 0 ] i . Example 3 . If k is a Ca rtan s ubalgebra, then { f λ , p µ }| x = (1 − λ )(1 − µ ) h y 0 , [ F µ 0 , P λ 0 ] i = 0 and in tegra ls (23) pr ovides a complete commutativ e set on g . The pair ( g , k ) is alr eady describ ed in Theor em 8: k = g ( a ), where a ∈ k is reg ular. It is also obvious that if ( g , k ) is a symmetric pair that B is commutativ e. In Bogoy- avlenski [ 2 , Theo rem 1] it is claimed that the set B is a lways c ommut ative, how ever the presented pro of co nt ains a small ga p. Recently , P anyushev and Y a kimov a gav e an example of the ca s e where B is not comm utative [ 32 , Example 2.3 ]. Example 4 . F or symmetric pairs ( so ( n ) , so ( p ) × so ( n − p )), Theorem 12 provides another pro of of Theo r em 9. Example 5 . In the following example w e verified the commut ativity of B by direct computations. The exa mples ar e a ls o co nv enient in the discussion of the co nditions in Theorem 12. Consider the ca se g = so ( 5). The Lie subalg e br as so (2) ⊕ so (2) , so (2) ⊕ so (3 ) , so (4) satisfy conditions of Theorem 1 2. Moreov er, S O (4) and S O (2) × S O (3) are multiplicit y free and almost mult iplicity fre e subgroups of S O (5). On the other side, if k = so (3) or k = so (2), a generic y 0 ∈ k is not regula r in so (5) and we can not apply Theor e m 12. W e hav e b ( so (5) , so (3)) = 4 , b ( so (5) , so (2)) = 5 . (33) 14 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C Let x = y 0 + y 1 ∈ so (5 ), with y 0 =  0 0 0 Q  , y 1 =  P 2 P 1 − P T 1 0  , where P 1 ∈ R 3 × 2 , P 2 ∈ so (2), Q ∈ so (3) in the case k = so (3) and P 1 ∈ R 2 × 3 , P 2 ∈ so (3), Q ∈ so (2) in the case k = so (2) . Bo goy avlenski’s in tegr als ar e given by: p 1 ,λ ( x ) = p 1 ,λ ( y 0 + λy 1 ) = tr ( y 0 + λy 1 ) 2 = 2 X k =0 λ k p j,k ( x ) , p 1 , 0 ( x ) = tr( y 2 0 ) , p 1 , 1 ( x ) = tr( y 0 y 1 + y 1 y 0 ) ≡ 0 , p 1 , 2 ( x ) = tr( y 2 1 ) , p 2 ,λ ( x ) = p 2 ,λ ( y 0 + λy 1 ) = tr ( y 0 + λy 1 ) 4 = 4 X k =0 λ k p j,k ( x ) , p 2 , 0 ( x ) = tr( y 4 0 ) , p 2 , 1 ( x ) = tr( y 0 y 1 y 2 0 + y 1 y 3 0 + y 2 0 y 1 y 0 + y 3 0 y 1 ) ≡ 0 , p 2 , 2 ( x ) = tr( y 2 0 y 2 1 + y 0 y 1 y 0 y 1 + y 0 y 2 1 y 0 + y 1 y 2 0 y 1 + y 1 y 0 y 1 y 0 + y 2 1 y 2 0 ) , p 2 , 3 ( x ) = tr( y 0 y 3 1 + y 1 y 0 y 2 1 + y 2 1 y 0 y 1 + y 3 1 y 0 ) , p 2 , 4 ( x ) = tr( y 4 1 ) . The polyno mials p j,k commute and we need to estimate the num b er of indep endent gradients after pro jections on to p , at a general point x ∈ so (5). F or this reason, w e consider the gradients: ∇ p 1 , 2 ( x ) = 2 y 1 , ∇ p 2 , 2 ( x ) = 4( y 2 0 y 1 + y 0 y 1 y 0 + y 1 y 2 0 + y 2 1 y 0 + y 1 y 0 y 1 + y 0 y 2 1 ) , ∇ p 2 , 3 ( x ) = 4( y 3 1 + y 0 y 2 1 + y 1 y 0 y 1 + y 2 1 y 0 ) , ∇ p 2 , 4 ( x ) = 4 y 3 1 . Let k = so (3) and let e ij be the standard basis of so (5). The equation µ 1 pr p ∇ p 1 , 2 ( x ) + µ 2 pr p ∇ p 2 , 2 ( x ) + µ 3 pr p ∇ p 2 , 3 ( x ) + µ 4 pr p ∇ p 2 , 4 ( x ) = 0 at the p oint x = e 45 + e 13 + e 23 + e 24 + e 15 + e 25 reduces to the system of eq uations µ 2 + µ 3 =0 , µ 1 − 8 µ 3 − 8 µ 4 =0 , µ 2 =0 , µ 3 − µ 4 =0 , µ 1 − 10 µ 3 − 10 µ 4 =0 , which has only the tr iv ial solution. Therefore, B is a complete set of Ad S O (3) –inv a riant po lynomials. Obviously , accor ding to (33), Bogoy avlenski’s integrals are not sufficient for k = so (2). How ever, we can consider the v ar iation of the method, by taking the chain of subalgebra s so (2) ⊂ so (2) ⊕ so (2) ⊂ so (5) or so (2) ⊂ so (2) ⊕ so (3) ⊂ so (5) . 5.4. Diagonal subgroups. Let G 0 be a co mpact connected Lie gro up and g 0 = Lie ( G 0 ) its Lie a lgebra. Consider the case when the group G is the pro duct G = G m 0 and the subgro up K ⊂ G is G 0 diagonally em b edded into the pro duct: K = diag ( G 0 ) = { ( g , . . . , g ) | g ∈ G 0 } ⊂ G. F or a purp ose of a construction o f integrable geo desic flows on m –symmetric spaces Q = G/K , the following c o nstruction related to the filtration g 0 ⊂ g 1 = g 0 ⊕ g 0 ⊂ · · · ⊂ g m − 1 = ( g 0 ) m = g INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 15 is given in [ 20 ]. Let f 1 , . . . , f r 0 be the base of homog eneous inv ar iant p olynomials o n g 0 , r 0 = ra nk g 0 , and let B = B 1 + B 2 + · · · + B m − 1 , B i = { f i α,k ( x ) | k = 0 , . . . , deg f j , j = 1 , . . . , r 0 } , where the p olynomia ls f i j,k ( x ) are defined b y: f j ( y 0 + y 1 + · · · + y i − 1 + λy i ) = deg f j X k =0 f i j,k ( y 0 , y 1 , . . . , y m − 1 ) λ k . Theorem 13 ( [ 2 0 ]) . (i) The set B is a c ommu tative set of Ad K -invariant p olynomials on g . (ii) The set B + µ ∗ ( R [ g 0 ]) is a c omplete set of p olynomials on g , wher e µ ( y 0 , y 1 , . . . , y m − 1 ) = y 0 + y 1 + · · · + y m − 1 . Therefore, the s et B solves our problem for the pair ( G, K ) = ( G m 0 , diag( G 0 )). 5.5. Reyman’s construction: the shi fting of argume n t and symm etric pairs. Here we present Reyman’s co nstruction of c ommut ative p olynomials r e lated to symmetric pairs [ 33 ]. Supp ose that g is a Lie subalgebra of a s emisimple Lie alg ebra h , suc h that ( h , g ) is a symmetric pair: [ g , m ] ⊂ m , [ m , m ] ⊂ g , where m is the or thogonal c o mplement of g with resp ect to a n inv ariant sc a lar pro duct h· , ·i . 5 F urther, supp ose there exist a ∈ m , such that k equals to the isotro p y alg ebra of a within g , k = g ( a ) = { ξ ∈ g | [ ξ , a ] = 0 } . Let h 1 , . . . , h s be the basic homogeneous inv a riant polynomials on h , s = r ank h and let K be the set of linear functions o n k , considered as line a r functions on h . On h we have a p encil Π of compatible Poisson bivectors spanned b y Λ 1 ( ξ 1 + η 1 , ξ 2 + η 2 ) | z = −h z , [ ξ 1 , ξ 2 ] + [ ξ 1 , η 2 ] + [ η 1 , ξ 2 ] i , Λ 2 ( ξ 1 + η 1 , ξ 2 + η 2 ) | z = −h z + a, [ ξ 1 + η 1 , ξ 2 + η 2 ] i , where z ∈ h , ξ 1 , ξ 2 ∈ g , η 1 , η 2 ∈ m (see Reyman [ 33 ]). The Poisson bivectors Λ λ 1 ,λ 2 , for λ 1 + λ 2 6 = 0 and λ 2 6 = 0, are isomor phic to the canonica l Lie-Poisson biv ec to r on h . Thus, the union of their Casimir functions B = { h λ,k ( z ) = h k ( λx + t + λ 2 a ) | k = 1 , 2 , . . . , s, λ ∈ R } , (34) where z = x + t , x ∈ g , t ∈ m , is a comm utative set with resp ect to the a ll brac kets from the penc il Π (se e [ 4 , 33 ]). Moreover, for a generic a ∈ m , the set of functions B + K is a complete non-commutativ e set on h with resp ect to Λ 1 (see [ 4 , Theorem 1 .5], for the detail pr o ofs of the ab ov e statement s, g iven for a n a rbitrary semi-simple sy mmetric pair, see [ 37 , pages 234-2 37]). The symplectic leaf within h (and the corresp o nding symplectic str uctur e) o f the bra ck et Λ 1 at a point x ∈ g coincide with the s ymplectic leaf through x of the Lie-Posson brack et (and the corr esp onding symplectic str ucture) on g . Therefore, the following statement holds. Proposition 14 ( [ 3 , 28 , 33 ]) . The re strictions of the p olynomial (34) B = { h j,a,k ( x ) | k = 1 , . . . , deg f j , j = 1 , . . . , s } , 5 Again, we use the same symbol for different ob jects. The r estriction of h· , · i to g is a p ositive definite inv arian t sca lar product w e are dealing with. Als o, as ab ov e, we identify h and h ∗ b y h· , ·i . 16 BO ˇ ZID AR JO V ANOVI ´ C, TIJANA ˇ SUKILOVI ´ C, SRDJAN VUKMIRO VI ´ C h j ( x + λa ) = deg h j X k =0 h j,a,k ( x ) λ k is a c ommu tative set of K -invariant p olynomials on g . Moreov er, the following is true (see the discussion in [ 3 , befor e Theo rem 1 .6]). Theorem 15 ( [ 3 ]) . F or a generic a ∈ m , t he the set B + pr ∗ k ( R [ k ]) is c omplete on g . Here, a ∈ m is generic if the dimension of the is otropy a lgebras g ( a ) and h ( a ) a re minimal. It would be interesting to prov e the ab ove statement in the s ing ular case a s well. In par ticular, w e ha ve the f ollowing statement (see [ 37 , pages 241–244] and [ 9 , Theor em 1]). Theorem 16 ( [ 9 , 37 ]) . Consider the symmetric p air ( h , g ) = ( sl ( n ) , so ( n )) and let a = diag( n 1 z }| { a 1 , . . . , a 1 , . . . , n k z }| { a k , . . . , a k ) , k = so ( n )( a ) = so ( n 1 ) ⊕ · · · ⊕ so ( n k ) . The set B is a c omplete c ommutative set of ad k –invariant p olynomials on so ( n ) Int egra ls B are referred as Ma na ko v integrals [ 24 ]. Theor e m 16 implies co mplete in- tegrability of a motion of a symmetric rigid b o dy abo ut a fixed point in R n (see [ 11 , 24 ]), having the o p erator of iner tia I = A − 1 of the form x = I ( ω ) = J ω + ω J, ω ∈ so ( n ) , where a mass tensor is J = diag ( b 1 , . . . , b n ), a i = b 2 i (see [ 9 , Subsection 1.6]). Ac knowledgmen ts. W e ar e g r ateful to the referees for thor ough review and construc- tive comments that g r eatly improved quality o f the pa per . The resea rch is supp orted by the Pro ject 774 4 592 ME GIC ”Int egra bilit y and E xtremal Pro blems in Mechanics, Geometr y and Co mbin atoric s” of the Science F und of Serbia. References [1] Bazaikin, Y a. V., Double quotients of Lie groups with an integ rable geodesic flow, Sibirsk. Mat. Zh. 2000, v ol. 41, pp. 513–530 (Russian); Engli sh translation: Sib erian Math. J. 2000, vol. 41, pp. 419–432. [2] Bogoy avlenski, O.I., In tegrable Euler equations asso ciated with filtrations of Li e algebras, Mat. Sb. 1983, vol. 121( 163), pp. 2 33–242 (Russian). E ngli sh translation: Sb ornik: Mathematics , 1984, v ol. 49, no. 1, pp. 229– 238. [3] Bolsinov, A. V., Compatible Po isson brack ets on Lie algebras and the completeness of f amilies of functions in i n volution, Izv. A c ad. Nauk SSSR, Ser. matem. 1991, v ol. 55, no. 1, pp. 68 –92 (Russian); English transl ation: Math. USSR-Izv. 1 992, v ol. 38, no . 1, pp. 69–90. [4] Bolsinov, A. V., Complete comm utativ e subalgebras in p olynomial P oisson algebras: a pro of of the Mischenk o–F omenk o conjecture, The or e tic al and Applie d Me chanics , 2016 , v ol. 43, pp. 145–16 8. [5] Bolsinov, A. V. a nd Jo v ano vi´ c, B., In tegrable geodesic flows on homoge neous spaces. Matem. Sb ornik , 2001, vol. 192, no. 7, pp. 21–40 (Russian). English translation: Sb. M at. 2001, v ol. 192, no. 7-8, 951–96 9. [6] Bolsinov, A. V. and Jo v anov i´ c, B., Non-commut ative int egrability , m omen t map and geodesic flows. Anna ls of Glob al Ana lysis and Ge ometry , 20 03, vo l. 23, pp. 305–3 22, arXiv: math-ph/010903 1. [7] Bolsinov, A. V. and Jo v ano vi´ c, B., Complete in vo lutive algebras of functions on cotangen t bundles of homogeneou s spaces, Mathematische Zeitschrift , 200 4, v ol. 2 46, no. 1-2, pp. 21 3–236. [8] Brailov, A.V., Construction of complet e in tegrable geodesic flo ws on compact symmetric spaces. Izv. A c ad. Nauk SSSR, Ser.matem. 1986, v ol. 50 no. 2, pp. 661–674 (Russi an); English translation: Izvestiya: Mathematics , 1987 , v ol. 29 no. 1, 19–31. [9] Dragovi ´ c, V., Ga ji´ c B. and Jov ano vi´ c, B., Singular Manak ov Flows and Geodesic Fl o ws of Homogeneous Spaces of S O ( n ), T r ansfomation Gr oups , 2009, v ol. 14, no . 3, 513–530, [10] Dragovi ´ c, V., Ga ji´ c B. and Jov ano vi´ c, B., On the completeness of the Manako v int egrals, F undam. Prikl. Mat. , 2015, vol. 20, no. 2, pp 35–49 (Russian). English translation: J. Math. Sci. , 2017, vol. 223, no. 6, pp. 675– 685, [11] F edoro v, Y u. N. and Kozlov, V. V ., V arious aspects of n -dimensional rigid b o dy dynamics, Amer. Math. So c. T r ansl. Series 2 , 1995, vol. 168, pp. 141–171. [12] Gel’fand, I and Ts etlin M., Finite-dimensional representation of the group of unimo dular matrices, Dokl. Akad. Nauk SSSR , 1950, v ol. 71, pp. 825–828. [13] Gel’fand, I and Tsetlin M. , Finite-dimensional represen tation of the group of orthogonal matrices, Dokl. Akad . Nauk SSSR , 1950, v ol. 71, pp. 1017–1 020. INTEGRABLE SYSTEMS ASSOCIA TED TO THE FIL TRA TIONS OF LIE ALGEBRAS 17 [14] Guillemin, V and Sternberg, S. , On collectiv e complete inte grability according to the method of Thimm, Er go d. Th. & Dynam. Sys. 19 83, v ol . 3, pp. 219–230. [15] Guillemin, V and Sternberg, S., The Gel’fand-Cetlin system and quantiza tion of the complex flag manifolds, Jou rnal of F unction Analysis , 1983, v ol. 52, 106–128. [16] Guillemin, V and Stern b erg, S., M ultiplicity-free spaces, J. Diff. Ge ometry , 19 84, v ol. 19, pp. 31–5 6. [17] Harada, M ., The symplectic geometry of the Gel’fand-Cetlin-M olev bases for representa tion of S p (2 n, C ), J. Sympl. Ge om. 2006, v ol. 4, pp. 1–41, arXiv:math/04 04485. [18] Heck man, G. J., Pro jections of orbits and asymptotic behav ior of multiplicities for co mpact co nnected Lie groups. Invent. Math. 1982 , v ol. 67, no. 2, pp. 333–356 . [19] Jov anovi ´ c, B., Geometry and integrabilit y of Euler–Poinca r´ e–Suslo v equations, Nonline arity , 2001, V ol. 14, 15 55–1567, arXiv:math-ph/01070 24. [20] Jov anovi ´ c, B. , Integrabilit y of Inv ariant Geodesic Flows on n - Symmetric Spaces, Annal s of Glob al Ana lysis and Geo metry , 2010, vol. 38, pp. 30 5–316, arXiv:1006.3693 [math.DG]. [21] Jov anovi ´ c, B., Geo desi c flo ws on Ri emannian g.o. spaces, R egula r and Chaotic Dy namics 2011, , v ol. 16, pp. 504–513, a rXiv: 1105.3651 [math.DG]. [22] Kr¨ amer, M., Multipl icity free subgroups of compact conne cted Lie groups, A r ch. Math. , 1976, v ol. 27, pp. 28– 36. [23] Lomp ert, K. and Panasyuk, A., Inv ariant Ni j enh uis T ensors and In tegrable Geodesic Flo ws, Symme- try, Integr ability and Ge ometry: Metho ds and Applic ations (SIGMA) , 2019, vo l. 15, 056, 30 pages, [24] Manako v, S. V., Note on the integrabilit y of the Euler equations of n - dimensional rigid bo dy dynamics, F unkc. A nal. Pril. , 1976, vol. 1 0, no. 4, pp. 93–94 (Russian). [25] Mikityuk, I. V., Integrabilit y of the Euler equations associated with filtrations of semisimple Lie alge- bras, Matem. Sb ornik , 1984 , v ol. 125(167), no.4 (Russian); English translation: Math. USSR Sb ornik 1986 v ol. 53, no. 2, pp . 541 –549. [26] Mikityuk, I.V., I nte grability of geo desic flo ws for m etrics on suborbits of the adjoin t orbits of c ompact groups, T r ansform. Gr oups , 2016, v ol. 21, pp. 531–553. [27] Mykyt yuk, I. V. and Panasyuk A., Bi-Poisson structures and integrabilit y of ge o desic flo ws on homo- geneous spa ces. T r ansformation Gr oups , 2004, v ol . 9, no. 3 , pp. 289–308. [28] Mishchenk o, A. S. and F omenk o, A. T., Eul er equations on finite -dimensional Lie gr oups, Izv. A c ad. Nauk SSSR, Ser. matem. , 1978, vol. 42, no. 2, pp. 396–415 (Russian); English translation: Math. USSR-Izv. 1 978, v ol. 12, no. 2, pp. 371– 389. [29] Mishchenk o, A. S. and F omenk o, A. T., Generalized Li ouville method of i n tegration of Hamiltonian systems. F unkts. A nal. Prilozh . 1978 , v ol. 12, no. 2, pp. 46–56 (Russian); English translation: F unct . Ana l. Appl. , 1978, vol. 1 2, pp. 113–121. [30] Nekhoroshev, N. N., A ction-angle v ariables and t heir generalization. T r. Mosk. Mat. O.- va , 1972, vol. 26, pp. 181–198 (Russian); English translation: T r ans. Mosc. Math. So c. , 1972, v ol. 26, 18 0–198. [31] Pan yushev, D. I. and Y akimo v a, O. S., P oisson-commutat ive subalgebras of S ( g ) asso ciated with in volu- tions, In ternational Mathematics Research Notices, 2021, vol. 2021, pp. 18367–18406, [32] Pan yushev, D. I. an d Y akimov a, O. S., Reduct ive subalgebras of semisimple Lie algebras and P oisson comm utativit y , Journal of Symplectic Geometry , 20 ( 2022), pp. 911–926, arXiv:2012.04014 . [33] Reyman, A. G., In tegrable Hamiltonian s ystems connected wi th graded Lie algebras, Zap. Nauchn. Semin. L O MI AN SSSR , 1980, vol. 95 , pp. 3–54 (Russi an); English translat ion: J. Sov. Math. , 1982, v ol. 19, pp. 1507–154 5. [34] Sadeto v, S. T., A pro of of the Mishchenk o-F omenko conjec ture (1981). Dokl. Akad. Nauk 2004, v ol. 397, no . 6, 751–754 (Russian). [35] Thimm A. , Inte grable geodesic flo ws on homogeneou s spaces, Er go d. Th. & Dynam. Sys. , 1981, v ol. 1, pp. 495 –517. [36] T rofimo v, V. V. , Euler equations on Borel subalgebras of semisimple Lie groups, Izv. A ca d. Nauk SSSR, Ser. matem. , 1979 , v ol. 43 , no. 3, pp. 714 –732 (Russian). [37] T rofimo v, V . V. and F omenko, A. T., Algebr a and ge ometry of inte g r able Hamiltonian differ ential e quations , Moskv a, F aktorial, 1995 ( Russian). B.J.: Mathema tical Institute, Serb ian Academy of Sciences and Ar ts, Kneza Miha ila 36, 11000 Belgrade, Serbia Email addr ess : bozaj@ mi.sanu.ac. rs T. ˇ S, S.V: F acul ty of Mathema tics, University of Belgrade, Studentski trg 1 6, 1 1000 Bel- grade, Serbia Email addr ess : vsrdja n@matf.bg.a c.rs, tijana@matf .bg.ac.rs