Erwin Schrödinger International Institute of Mathematical Physics, Wien, Austria. Sept. 01, 1999

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1 Modern Physics Letters A 14, ) CONSTRUCTION OF COMPLETELY INTEGRABLE SYSTEMS BY POISSON MAPPINGS J. Grabowsi, G. Marmo, P. W. Michor Erwin Schrödinger International Institute of Mathematical Physics, Wien, Austria Sept. 01, 1999 Abstract. Pulling bac sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail. 1. Introduction The standard notion of complete integrability is the so called Liouville-Arnold integrability: a Hamiltonian system on a n-dimensional symplectic manifold M is said to be completely integrable if it has n first integrals in involution which are functionally independent on some open and dense subset of M. It is natural to extend the notion of complete integrability to sytems defined on Poisson manifolds N, Λ) by requiring that on each symplectic leaf such system defines a completely integrable system in the usual sense. This generalization implies that an integrable system is associated to a maximal abelian Poisson subalgebra of C N), {, } Λ ). The dynamical system Γ associated to a 1-form α on N via Γ = i α Λ will define a Hamiltonian system on a symplectic leaf S with embedding ε S : S N if we have ε sα = dh S. If this is the case for any symplectic leaf we may write α = f dg where the f are Casimir functions for Λ. When all the g s belong to a sufficiently large set of functions in involution which are functionally independent on each leaf, the dynamical system is completely integrable. For some cases, even if ε Sdα 0, we get an integrable system in the generalized sense of [1]. Of course, integrable systems are not easy to find. Recently, in the paper [3] we came accross a beautiful idea to construct completely integrable systems by using coproducts in Poisson-Hopf algebras. In this paper we put this construction into a geometric perspective in order to understand better which are the essential ideas 1991 Mathematics Subject Classification. 58F07. Key words and phrases. Completely integrable systems, Poisson mappings. J. Grabowsi was supported by KBN, grant Nr P03A Typeset by AMS-TEX

2 GRABOWSKI, MARMO, MICHOR that mae the construction possible. In addition to this, we construct a full family of Poisson-Hopf algebras associated with a parametrized family of Poisson-Lie structures on the group SB, C). The standard Lie-Poisson structures on SB, C) with SU) and SL, R) as dual groups are included in this scheme. This Poisson- Hopf algebras can be viewed as geometrical version of the corresponding quantum groups deformations of the universal enveloping algebra of the associated Lie algebras. We also present symplectic realizations of the corresponding commutation rules in the deformed algebras.. Constructing integrable systems by Poisson maps.1. Complete integrability on Poisson manifolds. If M, Λ) is a Poisson manifold, a Hamiltonian system H C M) is called completely integrable if it admits a complete set of first integrals in involution: There are f 1,..., f C M) with {f i, f } Λ = 0 and {f i, H} Λ = 0 such that on each symplectic leaf or an open dense set of symplectic leaves) H together with a suitable subset of f 1,..., f is Liouville-Arnold integrable... Constructing families of functions in involution by Poisson maps. Let Φ i : M i+1, Λ i+1 ) M i, Λ i ) be Poisson maps between manifolds, so that {f, g} i Φ i = {f Φ i, g Φ i } i+1. If we have a family of functions F 1 C M 1 ) in involution on M 1, Λ 1 ), we may consider the family F = F 1 Φ 1 ) C C M ) where C is a complete set of Casimir functions on M, Λ ), and so on: M 1, Λ 1 ) F 1, in involution Φ 1 M, Λ ) F = F 1 Φ 1 ) C Φ M 3, Λ 3 ) F 3 = F Φ ) C 3 Φ Poisson actions and multiplications. We shall apply the procedure of. mainly in the following situation: Consider M 1 M, Λ 1 Λ ). Then for the algebras of smooth functions we have C M 1 M ) = C M 1 ) C M ) for some suitable completed tensor product, where f 1 f )x, y) = f 1 x)f y). Then {f 1 f, g 1 g } Λ1 Λ = {f 1, g 1 } Λ1 g 1 g + f 1 f {g 1, g } Λ1. So if c 1 is a Casimir function of M 1, Λ 1 ), then c 1 1 is a Casimir function of M 1 M, Λ 1 Λ ). In this sense The Casimir functions of M 1, Λ 1 ) and those of M, Λ ) extend both to Casimir functions on M 1 M, Λ 1 Λ ). If Φ : M M, Λ Λ) M, Λ) is a Poisson map for example the multiplication of a Lie Poisson group) we may use it for the procedure of.. If Φ is associative then Φ : f f Φ is coassociative. But this is not essential for applying the

3 CONSTRUCTION OF COMPLETELY INTEGRABLE SYSTEMS 3 procedure in which M n = n M and Φ n is a Cartesian product of Φ with identities. For example, M Φ M M Φ Id M M M M Id M Φ Id M M M M M... We start with a set of functions F 1 C M) in involution and with a basis C of all Casimirs. Then F n C n M) is given recursively by F n+1 = F n Φ n ) {C 1 1, 1 C 1 1,... } and furnishes a family of functions in involution on n+1 M. If Φ is associative so Φ is coassociative) then the result does not depend on the path chosen to define the Φ n s. Another possibility is to consider a Poisson mapping Φ : M N, Λ M Λ N ) N, Λ N ) for example a Lie-Poisson action on N of a Lie Poisson group M) and to apply the procedure as follows: N Φ M N Id M Φ M M N Id M M Φ M M M N We may extend the procedure described in.3 as follows. We assume that we have furthermore Poisson manifolds e.g. symplectic ones) N 1,... N n and Poisson mappings ϕ i : N i M. The product map ϕ = ϕ 1... ϕ n : N 1... N n n M is a Poisson map. Let F n be the set of functions in involution on n M constructed in.3. Then F n ϕ is a set of functions in involution on i N i. Standard examples of Poisson maps ϕ i : N i M are the canonical embeddings of symplectic leaves N i of the Poisson manifold M. In this case, the Casimir functions 1 1 c 1 are constants on N 1... N n, but the coproducts Φ c, Φ Id M ) Φ c, etc., are usually no longer Casimirs and hence sometimes give rise to completely integrable systems on N 1... N n. See example Examples 3.1. Example. Let M = su) be the dual space of the Lie algebra su). It carries a Kostant-Kirillov-Souriau Poisson structure which is given in linear coordinates by Λ = z x y + x y z + y z x. Since Λ is linear, we have the obvious Poisson map Φ : M M M, Φx 1, y 1, z 1, x, y, z ) = x 1 + x, y 1 + y, z 1 + z ). A Casimir function for Λ is c = x + y + z. According to our procedure in.3 the functions c Φ = x 1 + x ) + y 1 + y ) + z 1 + z ), c 1 = x 1 + y 1 + z 1, 1 c = x + y + z, f Φ = fx 1 + x, y 1 + y, z 1 + z ),

4 4 GRABOWSKI, MARMO, MICHOR are functions in involution on M M, where f is an arbitrary function on M. If we tae for N the symplectic leaf N = c 1 1) which is a -dimensional sphere S, the Casimir functions c 1 and 1 c pull bac to constants on N N = S S. However, c Φ and f Φ are in involution and hence H = 1 c Φ) 1 = x 1x + y 1 y + z 1 z defines a completely integrable system on the symplectic manifold N N M M. The system defined by the Hamiltonian function H is, in fact, completely integrable on each symplectic leaf of M M, so that we get a completely integrable system on M M whose dynamics is given by the vector field Γ = z y 1 y z 1 ) x1 + z 1 y y 1 z ) x + x z 1 z x 1 ) y1 + x 1 z z 1 x ) y + y x 1 x y 1 ) z1 + y 1 x x 1 y ) z. This vector field is tangent to all products of spheres since c 1 = x 1 + y 1 + z 1 and 1 c = x + y + z are first integrals, and on N N it induces the motion which can be interpreted as associated with a spin-spin -interaction: J 1 = J 1 J, J = J J 1 ; The points on the spheres move in such a way that the velocity of each of them is the vector product of the two position vectors. Stationary solutions occupy the same or opposite points on the sphere. In sperical coordinates, the same system can be given a different interpretation: z i = sin β i, y i = cos β i sin α i, x i = cos β i cos α i. In canonical coordinates p i = sin β i, q i = α i we get the Hamiltonian function in the form H = p 1 p + 1 p 1 )1 p ) cosq 1 q ). Since H 1 = z ) c) is in involution with z ) we can consider the completely integrable system given by the Hamiltonian H 1 = p 1 + p 1 p 1 )1 p ) cosq 1 q ). Let us remar that our Hamiltonian H is a slight modification of the Hamiltonian H 0 = p 1 p p 1 p cosq 1 q ) obtained in [3]. We can inductively apply our procedure to get a completely integrable system on M with Hamiltonian H ) = x i x j + y i y j + z i z j ) i<j which reduces in canonical coordinates on N to H ) = i<j ) p i p j + 1 p i )1 p j ) cosq i q j ).

5 CONSTRUCTION OF COMPLETELY INTEGRABLE SYSTEMS Example. We consider the following symplectic realization [8] of the Lie algebra su) in T R : x = 1 q 1q + p 1 p ), y = 1 p 1q q 1 p ), z = 1 4 p 1 + q 1 p q ). This defines a Poisson morphism ψ : T R su), q 1, q, p 1, p ) x, y, z), which is the momentum map of the corresponding Hamiltonian action of the group SU). As before, we consider the Casimir function c = x + y + z on su). This time, however, F = c ψ = 1 16 p 1 + p + q1 + q) is not a Casimir function on the symplectic manifold T R. The functions in involution on su) su) from example 3.1 give rise to functions in involution on T R T R = T R 4 as in.4, where q 1 etc. denote the functions on the second copy of T R : where F 1 = F q 1, q, p 1, p ), F = F q 1, q, p 1, p ), c) ψ = F 1 + F + H, H = 1 4 ) q 1 q + p 1 p ) q 1 q + p 1 p ) + p 1 q q 1 p ) p 1 q q 1 p ) and f) ψ = fg 1, G, G 3 ), where G 1 = 1 q 1q + p 1 p + q 1 q + p 1 p ), G = 1 p 1q q 1 p + p 1 q q 1 p ), p 1 + q 1 p q ) p 1 + q 1 p q ) G 3 = 1 4 p 1 + q 1 p q + p 1 + q 1 p q ). Hence, we have 4 independent functions in involution on T R 4 which define completely integrable systems. As Hamiltonian functions we can tae the pure interaction term H. The trajectories of the corresponding dynamics Γ H lie on the intersections of the level sets of F 1 and F which are, topologically, products of 3-dimensional spheres) and the level sets of H, and, say, G 1 which are, generically, 4-dimensional tori). Note that G and G 3 are additional constants of the motion. The whole set {F 1, F, H, G 1, G, G 3 } is, however, not independent, since G 1 + G + G 3 = F 1 + F + H. The dynamics Γ H on T R 4 = R 8 is described by a rather complicated vector field whose coefficients are polynomials of degree 3. The functions G 1, G, G 3 define the diagonal action of SU) on T R T R which preserves Γ H. The dynamics on S S from example 3.1 can be obtained via symplectic reduction with respect to this action.

6 6 GRABOWSKI, MARMO, MICHOR 3.3. Example. Let us now consider the Lie group M = SB, C) of all matrices of the form ) e z/ x + iy A = 0 e z/, where 0 is fixed and z, x, y R may be viewed as global coordinates on SB, C). The coproduct corresponding to the group multiplication Φ : SB, C) SB, C) SB, C) is given by z) = z z, x) = x e z/ + e z/ x, y) = y e z/ + e z/ y. Φ is a Poisson map for each of the following Poisson structures parameterized by α, β, γ R): *) {z, x} = βy, {y, z} = αx, {x, y} = γ sinz). For α = β = γ = 1 we get the classsical realization of the quantum SU) group, and for α = β = γ = 1 we get the classsical realization of the quantum SL, R) of Drinfeld and Jimbo, [6]. For 0 we get the Lie algebras {z, x} = βy, {y, z} = αx, {x, y} = γz, with the standard cobracets u) = u u, corresponding to the addition in g. A Casimir function for the Poisson bracet *) is c = αx +βy + 4γ sinh z/), which for 0 goes to c 0 = αx + βy + γz, see [7]. The generic symplectic leaves of the Poisson structure are -dimensional except for the trivial case α = β = γ = 0). As in example 3.1 the Hamiltonian H = 1 c) defines a completely integrable system on SB, C) SB, C). In the coordinates x, y, z the Hamiltonian H has the form H = 1 αx 1 + βy1 + 4γ sinh z1 )) e z + 1 αx + βy + 4γ sinh z )) e z1 + αx 1 x + βy 1 y + 4γ sinh z1 z ) sinh )) e z1 z)/. In the limit for 0 we get H 0 = c c 0 + αx 1 x + βy 1 y + γz 1 z ) and for α = β = γ = 1 we are in the situation of example 3.1. In all generality, however, it is difficult to express the dynamics explicitly since we deal simultaneously with a parametrized family of structures for which even the topology of the symplectic leaves changes.

7 CONSTRUCTION OF COMPLETELY INTEGRABLE SYSTEMS 7 Even in the case α = β = γ = 1, where it is nown [] that SB, C), Λ) is equivalent, as a Poisson manifold, to su) with the Kostant-Kirillov-Souriau structure Λ 0 described in example 3.1, the dynamics described by H in the deformed case may differ from that of example 3.1. The reason is that SB, C), Λ) is not equivalent to su), Λ 0 ) as a Lie Poisson group, since SU, C) is not commutative. In particular, the deformed coproduct is not cocommutative and the interaction we obtain is not symmetric. In order to wor in canonical coordinates let us introduce a symplectic realization of the commutation rules *) with α = δ > 0 and β = 1: X = a 4γ sinh p ) sinδq), Y = δ a 4γ sinh p ) cosδq), Z = p, where a 0 and a > 0 if γ = 0. In particular, if γ > 0 we get the deformed SU), and if γ < 0 we get the deformed SL, R). In this realization the Casimir function is c = a and the Hamiltonian reads H = e p p1)/ a 4γ sinh p1 ) ) a 4γ sinh p ) ) cosδq 1 q ))+ + a cosh p1+p ) + 4γ sinh p1 ) sinh p ) ). This Hamiltonian is quite complicated. But if we put a = 0, γ = 1, and δ = 1 we get H 1 = 4e p p1)/ 1 sinh p1 p ) sinh )cosq 1 q ) 1) which is the Hamiltonian obtained in [3] for the deformed SL, R) Example. A slight modification of the previous example which, at least formally, is not dealing with Lie-Poisson groups, is the the following. Let M be the space of all upper triangular matrices of the form A = a x + iy 0 b where a, b, x, y R. We use these as coordinates on M. We consider the Poisson structure Λ on M with Poisson bracet ), **) {x, a} = βya, {x, b} = βyb, {x, y} = γb a ), {y, a} = αxa, {y, b} = αxb, {a, b} = 0, where α, β, γ R paramerterize a family of Poisson bracets. Matrix multiplication on M leads to the coproduct: a) = a a, b) = b b, x) = a x + x b, y) = a y + y b. The matrix multiplication turns out to be a Poisson mapping with respect to all bracets **). But M is not a Lie-Poisson group since it contains elements which are not invertible. On the other hand, all Λ are tangent to SB, C) M and give

8 8 GRABOWSKI, MARMO, MICHOR there the bracets *) with slightly modified coeficients α, β, γ, if we parameterize a = e z/, b = e z/. The Poisson tensor has independent Casimirs: c 1 = ab and c = αx + βy + γa + b ). The symplectic leaves are, generically, -dimensional we assume that α + β + γ 0), and as before, H = 1 c ) = 1 a 1αx + βy + γa + b ))+ 1 b αx 1 + βy 1 + γa 1 + b 1))+ a 1 b αx 1 x + βy 1 y γa 1 b ) describes a completely integrable system on M M which, on SB, C) = c 1 1 1), coincides with a system from example Example. This is of different type. Let D = G.G = G.G be a complete Drinfeld double group. The decompositions give us two Poisson projections π 1, π : D G onto the Lie-Poisson group G. Let F, F be two families of functions in involution on G. It is nown that pull bacs by π 1 and by π commute with respect to the symplectic structure on D. So we can tae F = F 1 π 1 ) F π ) as a set of functions in involution on D. For example, SL, C) = SU).SB, C) = SB, C).SU), where π 1 denotes the projection onto the left factor SB, C), and π onto the the right one. Let F consist of the Casimir C and some function f. Then the functions c π 1, f π 1, g π are in involution on D, where f and g are arbitrary in C SB, C)), and they are generically independent, so we have: H = f π 1 is a completely integrable system on the symplectic SL, C) for any f C SB, C)) Concluding remars. 1. We have shown that by using Poisson-compatible coproducts it is possible to generate interacting systems while preserving the complete integrability. We have given some examples. These can be extended to arbitrary Lie Poisson pairs.. The interaction we get is a -body interaction, one may wonder if it would not be possible to obtain non-factorizable n-body interactions by using n-ary bracets or n-ary operations [7], [8], [9]). 3. The composition procedure does not use the existence of an inverse for each element in the product, therefore the procedure may be extended to any algebra. Is it possible to obtain interacting systems with fermionic degrees of freedom by using graded algebras? 4. The procedure may clearly be extended to infinite dimensions. Can one use it to obtain interacting fields? We shall come bac to some of these questions in the near future.

9 CONSTRUCTION OF COMPLETELY INTEGRABLE SYSTEMS 9 References [1] Aleseevsy, D.; Grabowsi, J.; Marmo, G.; Michor, P.W., Completely integrable systems: a generalization, Modern Physics Letters A ), [] Aleseevsy, D.; Grabowsi, J.; Marmo, G.; Michor, P.W., Poisson structures on double Lie groups, J. Geom. Physics ), , math.dg/ [3] Ballesteros, A; Consetti, M.; Ragnisco, O., N-dimsional classical integrable systems from Hopf algebras, Czech. J. Phys ), [4] Ballesteros, A.; Ragnisco, O., A systematic construction of completely integrable Hamiltonians from coalgebras, J. Phys. A: Math. Gen ), [5] Ballesteros, A.; Ragnisco, O., N = Hamiltonians with sl)-coalgebra symmetry and their integrable deformations, Roma TRE -prepint, February [6] Drinfeld, V.G., Quantum groups, Proc. Intern. Congress Math., Bereley [7] J. Grabowsi and G. Marmo, Generalized n-poisson Bracets on a Symplectic Manifold, ESI preprint 669, math.dg/99019,, Modern Phys. Lett. A ), [8] J. Grabowsi and G. Marmo, On Filippov algebroids and multiplicative Nambu-Poisson structures, ESI preprint 668, math.dg/99017, to appear in Diff. Geom. Appl.. [9] J. Grabowsi and G. Marmo, Remars on Nambu-Poisson and Nambu-Jacobi bracets, ESI preprint 670, math.dg/99018, J. Phys. A: Math. Gen ), [10] Grabowsi, J.; Marmo, G.; Perelomov, A., Poisson structures: towards a classification, Mod. Phys. Lett. A ), [11] Man o, V.I; Marmo, G; Vitale, P; Zaccaria, F., A generalization of the Jordan-Schwinger map: the classical version and its q-deformations, Int. J. Mod. Phys. A ), J. Grabowsi: Institute of Mathematics, University of Warsaw, ul. Banacha, PL Warsaw, Poland; and Mathematical Institute, Polish Academy of Sciences, ul. Śniadecich 8, P.O. Box 137, PL Warsaw, Poland address: jagrab@mimuw.edu.pl G. Marmo: Dipart. di Scienze Fisiche - Università di Napoli, Mostra d Oltremare, Pad.19, I-8015 Napoli, Italy. address: gimarmo@na.infn.it P. W. Michor: Institut für Mathemati, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria; and Erwin Schrödinger International Institute of Mathematical Physics, Boltzmanngasse 9, A-1090 Wien, Austria address: peter.michor@univie.ac.at, peter.michor@esi.ac.at