G its cotangent bundle with its natural lie group structure obtained by performing a left trivialization of tg and endowing the resulting trivial bundle with the semidirect product, using the coadjoint action of g on the. In earlier work 1, we studied an extension of the canonical symplectic structure in the cotangent bundle of an affine space q r n, by additional terms implying the poisson noncommutativity of both configuration and momentum variables. For a connected lie group g, we show that a complex structure on the total space tg of the tangent bundle of g that is left invariant. Kaehler structures on t g having as underlying symplectic form the. On a cotangent bundle tg of a lie group g one can describe the standard liouville form.
Pdf lie groups of automorphisms of cotangent bundles of lie groups are completely characterized and interesting results are obtained. We suppose that a lie group g acts on qfrom the left via a free and proper action. Poisson structures on the cotangent bundle of a lie group or a principle bundle and their reductions. Characterization, in differential geometric terms, of the groups which can be interpreted as semidirect products of a lie group g by the group of translations of the dual space of its lie algebra. On the bks pairing for k ahler quantizations of the cotangent bundle of a lie group carlos florentino y, pedro matiasz, jos e mourao and joao p. Symplectic structures on cotangent bundles of principal bundles.
The cotangent bundle, and covariant vector fields 93. It is then clear that models whose carrier space is a lie group g can be very helpful in better. In this work we obtain a construction of the hamiltonian tube for a cotangentlifted. We have studied biinvariant metrics on cotangent bundles of lie groups and their isometries. Pdf automorphisms of cotangent bundles of lie groups. In this short section, as a prelude to discussing the cotangent bundle for coadjoint orbits, we examine the cotangent bundle for g. A natural oneparameter family of kahler quantizations of the cotangent bundle t.
Characterization of lie groups on the cotangent bundle of. Im not sure if what im doing is correct, so i was hoping i could get some feedback. Pdf poisson structures on the cotangent bundle of a lie. G of a lie group g and we describe explicitly the standard symplectic form. For x x a riemann surface of genus g g, the degree of the canonical bundle is 2 g. Nunesy march 18, 2011 abstract a natural oneparameter family of k ahler quantizations of the cotangent bundle tk of a compact lie group k, taking into account the halfform correction, was studied. In this article, we claim that such an extension can be done consistently when q is a lie group g. Poisson structures on the cotangent bundle of a lie group. Tangent and cotangent bundles automorphism groups and. To synthesize the lie algebra reduction methods of arnold 1966 with the techniques of smale 1970 on the reduction of cotangent bundles by abelian groups. The class of hopf algebras obtained from this procedure sometimes go by the name of quantum groups. This is possible because the cotangent bundle t g has two distinguished trivialisations, the left and right trivialisations 7 implemented respectively by the bases of the left and right invariant differential forms.
A hyperkahler structure on the cotangent bundle of a complex. Im trying to work out some calculations on the cotangent bundle of a lie group and id like to use the fact that on lie groups there are global frames. The cotangent bundle reduction scenario involves a manifold qand, of course, its cotangent bundle. In earlier work 1, we studied an extension of the canonical symplectic structure in the cotangent bundle of an af. Group eld theories gft are quantum eld theories which aim at describing the fundamental quantum structures that constitute spacetime. A hyperkahler structure on the cotangent bundle of a. Lecture 1 lie groups and the maurercartan equation january 11, 20. We have shown that the lie group of isometries of any biinvariant metric on the cotangent bundle of any semi. On the bks pairing for k ahler quantizations of the. Poisson structures on the cotangent bundle of a lie group or. Example 2 coadjoint action of a lie group on its lie algebra. The geometric nature of the flaschka transformation.
It is not true however that all spaces with trivial tangent bundles are lie groups. Second, the professor went on to say that because of the poisson bracket, we see the phase space of a physical system as the cotangent bundle of a manifold. What properties should tangent vectors and tangent spaces have. For example, it is the statement of the famous hairy ball theorem that every even dimensional sphere is not parallelizable and thus cant for any attempted definition of an operation. Intuitively this is the object we get by gluing at each point p.
This means it is divisible by 2 and hence there are theta characteristic square roots. Associated with this geometric reduction, we also develop the reduction of dynamics, by reducing a standard implicit lagrangian system as. In the present paper, it is shown that the associated blattnerkostantsternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work, from the point. Seearnal, currey, and dali2009,pedersen1988,pukanszky1990. Some relaxing of the assumptions on the action are possible, but this threatens to get into singular reduction. This will lead to the cotangent bundle and higher order bundles. We analyze the quantization commutes with reduction problem first studied in physics by dirac, and known in the mathematical literature also as the guilleminsternberg conjecture for the conjugate action of a compact connected lie group g on its own cotangent bundle tg. We give prominence to the fact that the lie groups of automorphisms of cotangent bundles of lie groups are super symmetric lie groups. In the present pa per, it is shown that the associated blattnerkostantsternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work. Example 1 symplectic structure on the cotangent bundle.
On the bks pairing for k ahler quantizations of the cotangent. Recently, ss constructed hamiltonian tubes for free actions of a lie group gand showed that this construction can be made explicit for g so3. I understand that we associate a symplectic form to the cotangent bundle, and that we want to think of phase space with a symplectic structure, but my second question is. Equivalently, the structure of g can be based on the maurercartan structure equations d. A lie group, as a di erentiable manifold, has a cotangent bundle and other associated bundles. Quantization of the cotangent bundle of a compact connected lie group 10. Modified symplectic structures in cotangent bundles of lie. Kof a compact lie group k, taking into account the halfform correction, was studied in fmmn. Suppose gis a compact, simply connected lie group, and an invariant inner product metric on its lie algebra g. For example, when the given lie group is an exponential solvable group, such as the group of lower triangular matrices, its simply connected coadjoint orbits are symplectomorphic to the canonical cotangent bundle of rn. Study of the canonical cotangent group of g corresponding to the coadjoint representation. We analyze the quantization commutes with reduction problem first studied in physics by dirac, and known in the mathematical literature also as the guilleminsternberg conjecture for the conjugate action of a compact connected lie group g on its own cotangent bundle t. Lecture 1 lie groups and the maurercartan equation. A gvalued pform is simply a section of the bundle p g 17 where v p is an abbreviation for v p t m.
In this work, we show that such an extension is achieved when q g is a lie group. Here the group g is the abelian lie group g, and the resulting poisson structure is the. The tangent bundle of the unit circle is trivial because it is a lie group under multiplication and its natural differential structure. Modified symplectic structures in cotangent bundles of lie groups. This means that if we regard tm as a manifold in its own right, there is a canonical section of the vector bundle ttm over tm. The structure of the lie algebra of prederivations of lie algebras of cotangent bundles of lie groups is explored and we have shown that the lie algebra of prederivations of lie algebras of cotangent bundle of lie groups are reductive lie algebras. Introduction the cotangent bundle t g of a lie group g with lie algebra g has a canonical lie group structure induced by the coadjoint action of g on g and also a.
Cotangent bundles for matrix algebras converge to the. In this case, the marsdenweinstein quotient is the cotangent bundle of the ordinarymanifoldquotient. Here we only need to assume that g is a connected compact lie group, with lie algebra g. K of a compact lie group k, taking into account the halfform correction, was studied in fmmn. Cotangent bundles for matrix algebras converge to the sphere. Opaque this 6 cotangent bundles in many mechanics problems, the phase space is the cotangent bundle t. Pdf on the bks pairing for kahler quantizations of the. Lie group and that dually on the cotangent bundle of g the canonical symplectic form is parallel with respect to the canonical connection. On the bks pairing for kahler quantizations of the cotangent bundle of a lie group. The lie algebra of the lie group of isometries of a biinvariant metric on a lie group is composed with prederivations of the lie algebra which are skewsymmetric with respect to the induced orthogonal structure on the lie algebra. Pdf on the geometry of cotagent bundles of lie groups. The resulting technique will be called dirac cotangent bundle reduction. Let mbe a symplectic manifold with the action of a lie group gand momentmapg.
We give explicit formulas for a product on the cotangent bundle t g of a lie group g. G g of a lie group g and we describe explicitly the standard symplectic form. An explicit product on the cotangent bundle of a lie group. One expects therefore by performing geometric quantization. Recall that the phase space is given by the cotangent bundle of the configuration space, where points of the configuration space represent possible instantaneous configurations of some system relative to an inertial frame. On the bks pairing for kahler quantizations of the cotangent. What are the tangent spaces to the line r and the plane r2 two of the most familiar. Associated with this geometric reduction, we also develop the reduction of dynamics, by reducing a standard implicit lagrangian system as well as its associated. Keywords cotangent bundle automorphism derivation lie group lie algebra biinvariant metric biinvariant tensor supersymmetric lie group lie superalgebra lie supergroup citation diatta, a manga, b. Pdf left invariant geometry of lie groups researchgate. Here g is a formal poisson lie group with lie algebra g. The cotangent bundle of an y lie group with its natural lie group structure, as above and in general any elemen t of the larger and interesting family of the socalled.
The kodaira vanishing theorem for complete kahler manifolds. A cotangent bundle hamiltonian tube theorem and its. The cotangent bundle is the set of pairs q, p, where q is an element of the configuration space and p is a covector at q. Quantization of the cotangent bundle of a compact connected lie group 10 3. The cotangent bundle, and covariant vector fields 93 4. May 02, 2015 the lie algebra of the lie group of isometries of a biinvariant metric on a lie group is composed with prederivations of the lie algebra which are skewsymmetric with respect to the induced orthogonal structure on the lie algebra. Global frame for the cotangent bundle of a lie group. The kodaira vanishing theorem for complete k ahler manifolds 11 3. Poisson structures on the cotangent bundle of a lie group or a principal bundle and their reductions article pdf available in journal of mathematical physics 359 november 1993 with 46 reads.
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