foundations of the theory of graded bundles

This note is based on a minicourse given at the 6th workshop and seminar on topics oftheoretical physics' in tabriz, iran. we describe here the concept of a graded bundle which isa natural generalization of that of a vector bundle. a canonical example of such a structureis the higher tangent bundle tkm playing a fundamental role in higher order lagrangianformalism. graded bundles of degree k are particular graded manifolds of degree k in thesense that we can choose an atlas with local coordinates being homogeneous functions ofdegrees 0; 1; : : : ; k. note that graded bundles of degree 1 are just vector bundles. a littlemore specically, a vector bundle structure e ! m is encoded by assigning a weight of zero tothe base coordinates and one to the linear coordinates on the total space. the condition of theweight to be one for the bre coordinates is a restatement of linearity. thus philosophically,a graded bundle should be viewed as a on-linear or higher vector bundle'.in [5] grabowski and rotkiewicz established the one-to-one correspondence between manifoldsthat admit non-negatively graded local coordinates and manifolds equipped with anaction of the monoid of multiplicative reals, or homogeneity structures in the language developedin that paper. the main result states that each homogeneity structure admits an atlaswhose local coordinates are homogeneous.considering a natural compatibility condition of homogeneity structures we formulate,in turn, the concept of a double (r-fold, in general) graded bundle which gives a broad generalizationof the concept of a double vector bundle. double graded bundles are proven tobe locally trivial in the sense that we can nd local coordinates which are simultaneouslyhomogeneous with respect to both homogeneity structures.in this paper we report also on the key results of our collective works [3, 4, 6, 7]. the broadidea of these works can be stated as applying the notion of lie algebroid as a morphism ofdouble vector bundles to the setting of geometric mechanics in the spirit of w.m. tulczyjew.the challenge of describing mechanical systems congured on lie groupoids and theirreduction to lie algebroids was rst posted by weinstein [24] and libermann [12]. this challengewas taken up by many authors (e.g. martinez [15]) and various approaches developed, arather incomplete list is given in [2]. the notion of the tulczyjew triple for a lie algebroid, aswe shall understand it, was rst given in [6]. it was based on a framework for lagrangian andhamiltonian formalisms developed by tulczyjew [20, 21, 22] and a corresponding descriptionof lie algebroids [9, 10]. the motivation for extending the geometric tools of the lagrangianformalism on tangent bundles to lie algebroids comes from the fact that reductions usuallypush one out of the environment of tangent bundles and into the world of lie algebroids. ina similar way, reductions of higher order tangent bundles, which is where higher order mechanicallagrangians live', will push one into the environment of higher lie algebroids'.weighted lie algebroids turn out to give a clear geometric framework for reductions of higherorder tangent bundles [1].the structure of the note is as follows. in sections 2 and 3 we investigate the structure ofgraded spaces and graded bundles with particular examples of vector spaces and vector bundlesrespectively. we give also canonical examples of graded bundles such as higher tangentbundle and graded bundle associated to the wedge product of vector bundles. in section 4and 5 we pass to multiple graded structures concentrating mostly on double vector bundles,since they appear naturally in hamiltonian and lagrangian mechanics. in sections 6 and 7we describe the concept of lie and general algebroid. it is an interesting observation thatthe structure of lie algebroid can be given in many ways, e.g. as a bracket on sections of avector bundle, as a linear poisson structure on the dual vector bundle, as a dierential onsections of exterior products of the dual bundle and nally as a morphism of certain doublevector bundles. in this sense the lie bracket of vector elds on a manifold, the canonicalsymplectic structure of the cotangent bundle, de rham dierential and the tulczyjew mapm : ttm ! ttm all encode the same canonical lie algebroid structure on the tangentbundle tm. sections 8 and 9 are devoted to tulczyjew approach to lagrangian and hamiltonianmechanics known in the literature as the tulczyjew triple. in section 7 the classicalversion of the triple is introduced and illustrated by simple but instructive example of thedynamics of a free relativistic particle (see [19]). in section 8 the generalisation for the caseof kinematic configurations being an algebroid is described.