HOLOMORPHIC SYMPLECTIC AND POISSON STRUCTURES ON THE HOLOMORPHIC COTANGENT BUNDLE OF A COMPLEX LIE GROUP AND OF A HOLOMORPHIC PRINCIPAL BUNDLE

2013 ◽  
Vol 06 (03) ◽  
pp. 1350029 ◽  
Author(s):  
Cristian Ida
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Principal Bundle ◽  
Poisson Structures

In this paper we introduce holomorphic symplectic and Poisson structures on the holomorphic cotangent bundle of a complex Lie group and of a holomorphic principal bundle.

scholarly journals Poisson structures on the cotangent bundle of a Lie group or a principle bundle and their reductions

1994 ◽  
Vol 35 (9) ◽  
pp. 4909-4927 ◽  
Author(s):  
D. Alekseevsky ◽  
J. Grabowski ◽  
G. Marmo ◽  
P. W. Michor
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Poisson Structures

scholarly journals Poisson Principal Bundles

2021 ◽  
Author(s):  
Shahn Majid ◽  
◽  
Liam Williams ◽  
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Poisson Structure ◽  
Differential Calculus ◽  
Principal Bundle ◽  
Poisson Manifold ◽  
Spin Connection ◽  
Poisson Geometry ◽  
Hopf Fibration ◽  
Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

scholarly journals On the BKS pairing for Kähler quantizations of the cotangent bundle of a Lie group

2006 ◽  
Vol 234 (1) ◽  
pp. 180-198 ◽  
Author(s):  
Carlos Florentino ◽  
Pedro Matias ◽  
José Mourão ◽  
João P. Nunes
Keyword(s):  
Lie Group ◽  
Cotangent Bundle

Сurvature-torsion tensor for Cartan connection

2019 ◽  
pp. 155-168
Author(s):  
Yu. Shevchenko
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Principal Bundle ◽  
Torsion Tensor ◽  
Projective Connection ◽  
Cartan Connection ◽  
Bianchi Identities ◽  
Covariant Derivatives ◽  
Structure Equations ◽  
Derivatives Of

A Lie group containing a subgroup is considered. Such a group is a principal bundle, a typical fiber of this principal bundle is the subgroup and a base is a homogeneous space, which is obtained by factoring the group by the subgroup. Starting from this group, we constructed structure equations of a space with Cartan connection, which generalizes the Cartan point projective connection, Akivis’s linear projective connection, and a plane projective connection. Structure equations of this Cartan connection, containing the components of the curvature-torsion object, allowed: 1) to show that the curvature-torsion object forms a tensor containing a torsion tensor; 2) to find an analogue of the Bianchi identities such that the curvature-torsion tensor and its Pfaff derivatives satisfy this analogue; 3) to obtain the conditions for the transformation of Pfaffian derivatives of the curvature-torsion tensor into covariant derivatives with respect to the Cartan connection.

scholarly journals n-Tuple principal bundles

2018 ◽  
Vol 15 (12) ◽  
pp. 1850211
Author(s):  
Katarzyna Grabowska ◽  
Janusz Grabowski
Keyword(s):  
Lie Group ◽  
Compatibility Condition ◽  
Principal Bundle ◽  
Principal Bundles

We develop the concept of a double (more generally [Formula: see text]-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.

scholarly journals Hyperkähler metrics associated to compact Lie groups

1996 ◽  
Vol 120 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Andrew Dancer ◽  
Andrew Swann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Left And Right ◽  
Hyperkähler Metrics

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

Cohomology of Flat Principal Bundles

2018 ◽  
Vol 61 (3) ◽  
pp. 869-877
Author(s):  
Yanghyun Byun ◽  
Joohee Kim
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Real Coefficient ◽  
Similar Argument ◽  
Flat Connection ◽  
De Rham Cohomology ◽  
Trivial Bundle ◽  
De Rham ◽  
Adjoint Bundle ◽  
Connected Lie Group

AbstractWe invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H*dR(G) itself. Then, we show that when a flat connection A exists on a principal G-bundle P, we may construct a homomorphism EA: H*dR(G)→H*dR(P), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we show that that both the flat principal bundle and its adjoint bundle have the real coefficient cohomology isomorphic to that of the trivial bundle.

Sign in / Sign up

Export Citation Format

Share Document