An explicit product on the cotangent bundle of a lie group. In earlier work 1, we studied an extension of the canonical symplectic structure in the cotangent bundle of an af. A lie group, as a di erentiable manifold, has a cotangent bundle and other associated bundles. Group eld theories gft are quantum eld theories which aim at describing the fundamental quantum structures that constitute spacetime.
Here we only need to assume that g is a connected compact lie group, with lie algebra g. Seearnal, currey, and dali2009,pedersen1988,pukanszky1990. Cotangent bundles for matrix algebras converge to the sphere. 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. 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. Kaehler structures on t g having as underlying symplectic form the. On the bks pairing for kahler quantizations of the cotangent. Im not sure if what im doing is correct, so i was hoping i could get some feedback. The cotangent bundle reduction scenario involves a manifold qand, of course, its cotangent bundle.
Cotangent bundles for matrix algebras converge to the. 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. Here the group g is the abelian lie group g, and the resulting poisson structure is the. We suppose that a lie group g acts on qfrom the left via a free and proper action. We give prominence to the fact that the lie groups of automorphisms of cotangent bundles of lie groups are super symmetric lie groups. We have studied biinvariant metrics on cotangent bundles of lie groups and their isometries. Characterization of lie groups on the cotangent bundle of a. 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. What properties should tangent vectors and tangent spaces have. 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. 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. 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.
On the bks pairing for k ahler quantizations of the cotangent. G of a lie group g and we describe explicitly the standard symplectic form. G g of a lie group g and we describe explicitly the standard symplectic form. Intuitively this is the object we get by gluing at each point p. Kof a compact lie group k, taking into account the halfform correction, was studied in fmmn. Poisson structures on the cotangent bundle of a lie group. Quantization of the cotangent bundle of a compact connected lie group 10. Associated with this geometric reduction, we also develop the reduction of dynamics, by reducing a standard implicit lagrangian system as. 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. Lecture 1 lie groups and the maurercartan equation. The cotangent bundle of a lie group is the prototyp ical example of a symplectic groupoid.
This means it is divisible by 2 and hence there are theta characteristic square roots. The resulting technique will be called dirac cotangent bundle reduction. A natural oneparameter family of kahler quantizations of the cotangent bundle t. It is then clear that models whose carrier space is a lie group g can be very helpful in better. We have shown that the lie group of isometries of any biinvariant metric on the cotangent bundle of any semi. Equivalently, the structure of g can be based on the maurercartan structure equations d. Opaque this 6 cotangent bundles in many mechanics problems, the phase space is the cotangent bundle t. In this case, the marsdenweinstein quotient is the cotangent bundle of the ordinarymanifoldquotient. In this work, we show that such an extension is achieved when q g is a lie group.
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. 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 january 11, 20. 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. 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. Here g is a formal poisson lie group with lie algebra g.
It is not true however that all spaces with trivial tangent bundles are lie 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. 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. Recently, ss constructed hamiltonian tubes for free actions of a lie group gand showed that this construction can be made explicit for g so3. Modified symplectic structures in cotangent bundles of lie groups. The geometric nature of the flaschka transformation. This will lead to the cotangent bundle and higher order bundles. In this article, we claim that such an extension can be done consistently when q is a lie group g. The tangent bundle of the unit circle is trivial because it is a lie group under multiplication and its natural differential structure. Global frame for the cotangent bundle of a lie group. They are quantum eld theories of spacetime, rather than on spacetime. A gvalued pform is simply a section of the bundle p g 17 where v p is an abbreviation for v p t m. Poisson structures on the cotangent bundle of a lie group or. This precludes a lot of spaces from possessing a lie group structure at all.
Characterization of lie groups on the cotangent bundle of. Pdf on the geometry of cotagent bundles of lie groups. Tangent and cotangent bundles automorphism groups and. 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. K of a compact lie group k, taking into account the halfform correction, was studied in fmmn. 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 tg. The kodaira vanishing theorem for complete kahler manifolds. Example 1 symplectic structure on the cotangent bundle. A hyperkahler structure on the cotangent bundle of a. The cotangent bundle, and covariant vector fields 93 4. One expects therefore by performing geometric quantization. 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. Modified symplectic structures in cotangent bundles of lie. We give explicit formulas for a product on the cotangent bundle t g of a lie group g. On the geometry of cotangent bundles of lie groups core. Nunesy march 18, 2011 abstract a natural oneparameter family of k ahler quantizations of the. For x x a riemann surface of genus g g, the degree of the canonical bundle is 2 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.
A cotangent bundle hamiltonian tube theorem and its. In section 2 we apply these simple construction to the cotangent bundle. Let mbe a symplectic manifold with the action of a lie group gand momentmapg. 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 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. Associated with this geometric reduction, we also develop the reduction of dynamics, by reducing a standard implicit lagrangian system as well as its associated. In this work we obtain a construction of the hamiltonian tube for a cotangentlifted. What are the tangent spaces to the line r and the plane r2 two of the most familiar. Poisson structures on the cotangent bundle of a lie group or a principle bundle and their reductions. Suppose gis a compact, simply connected lie group, and an invariant inner product metric on its lie algebra g. 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. 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. 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 bks pairing for kahler quantizations of the.
Pdf automorphisms of cotangent bundles of lie groups. Pdf poisson structures on the cotangent bundle of a lie. 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. Symplectic structures on cotangent bundles of principal bundles. The class of hopf algebras obtained from this procedure sometimes go by the name of quantum groups. The kodaira vanishing theorem for complete k ahler manifolds 11 3.
Some relaxing of the assumptions on the action are possible, but this threatens to get into singular reduction. 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. 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. Pdf left invariant geometry of lie groups researchgate. On a cotangent bundle tg of a lie group g one can describe the standard liouville form.
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