Multisymplectic structures and invariant tensors for Lie systems

Abstract

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory. MSC 2010: 34A26 (primary); 34A05, 34C14, 53C15, 16T15 (secondary) the above-mentioned articles in an ad-hoc manner or by solving systems of PDEs. Then, our work simplifies their derivation.Remarkably, we show that the coalgebra method to derive superposition rules for Lie-Hamilton systems developed in [6,8] can be retrieved as a particular case of our techniques when it concerns Lie-Hamilton systems related to symplectic forms. Moreover, our methods give rise to tensor field invariants for multisymplectic Lie systems from invariants of tensor algebras, which are more general than the invariant structures appearing in the standard coalgebra method, e.g. Casimir elements and invariant functions [6].More specifically, a multisymplectic Lie system (N, Θ, X), where X is a Lie system on a manifold N with a compatible multisymplectic form Θ, is endowed with a finite-dimensional Lie algebra M of Hamiltonian forms for one of its Vessiot-Guldberg Lie algebras: a so-called Lie-Hamilton algebra of X. If g is an abstract Lie algebra isomorphic to M, then the adjoint representation of g can be extended to a Lie algebra representation on the tensor algebra, T (g), which makes the latter into a g-module [61]. Similarly, the symmetric and Grassmann algebras, S(g) and Λ(g), can be considered as g-submodules of T (g). Moreover, we endow T (g), S(g), Λ(g) with coalgebra structures (see [6,26] for details), which are extended to the tensor products TPrevious structures are then represented as covariant tensor fields on N and N m in such a way that the g-invariants in T (g) (or its g-submodules Λ(g) and S(g)) give rise to tensor invariants for X and its diagonal prolongations [21]; see diagrams (5.3) and (5.4) for details. This is employed to obtain constants of motion and superposition rules for X [22].Our approach shows that invariants and superposition rules for multisymplectic Lie systems can be obtained through Casimir elements of universal enveloping algebras, which can be understood as symmetric tensors in T (g), or co-cycles of the Chevalley-Eilenberg cohomology of g (see [61]), which are understood as antisymmetric tensors of T (g). Moreover, this method gives rise to obt...

Geometric Models for Lie–Hamilton Systems on ℝ2

“…Although Lie-Hamilton systems on R 2 related to VG Lie algebras isomorphic to sl 2 and so 3 were very briefly studied in [4], our analysis here is much more detailed and it additionally shows, as a bonus, the existence of additional features of such Lie-Hamilton systems, which retrieves in a more natural and general manner results given in [13,16]. In the given basis, system (4) takes the form…”

“…Our previous construction is similar to the tensor algebra structure given in [13]. In this paper, we will use such an approach to study Lie-Hamilton systems on the plane or higher-dimensional manifolds.…”

“…is tangent to F k . The coefficients of T so 3 in the local coordinates {e 1 , e 2 }, which are well defined on a point with k 2 > e 2 1 + e 2 2 of every leaf F k for k > 0, read det(T so 3 ) = det e 2 3 + e 2 2 −e 1 e 2 −e 2 e 1 e 2 3 + e 2 1 = (k 2 − e 2 1 − e 2 2 )k 2 > 0 (13) and T so 3 becomes a non-degenerate tensor field. Its inverse, g so 3 , becomes a Riemmanian metric since the determinant of its coefficients is positive.…”

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

“…Vice versa, every abstract finite-dimensional Lie algebra g can be thought of as the Lie algebra of left-invariant vector fields of a Lie group [39]. Meanwhile, V L G can be identified with the Grassmann algebra Λg, namely the algebra relative to the exterior product spanned by all the multivectors of the Lie algebra, g, of G [40]. Moreover, [•, •] can be restricted to V L G, leading to the algebraic Schouten bracket on Λg [21,31].…”

Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras

“…Meanwhile, V L G can be identified with the Grassmann algebra Λg, namely the algebra relative to the exterior product spanned by all the multivectors of the Lie algebra, g, of G [26]. Moreover, [•, •] can be restricted to V L G leading to the algebraic Schouten bracket on Λg [35,51].…”