scholarly journals Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group

2011 ◽  
Vol 52 (7) ◽  
pp. 073504 ◽  
Author(s):  
S. Capriotti ◽  
H. Montani
Keyword(s):  
Lie Group ◽  
Cotangent Bundle

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

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.

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

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 GAUGE-INVARIANT HAMILTONIAN FORMULATION OF LATTICE YANG-MILLS THEORY AND THE HEISENBERG DOUBLE

1995 ◽  
Vol 10 (34) ◽  
pp. 2619-2631 ◽  
Author(s):  
S.A. FROLOV
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Hamiltonian Formulation ◽  
Dimensional Torus ◽  
Gauge Invariant ◽  
Flat Connections ◽  
Yang Mills ◽  
Temporal Gauge ◽  
Heisenberg Double ◽  
Limiting Case

It is known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge A0=0 one should place on each link a cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so-called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double. The physical phase space of the (1+1)-dimensional γ-deformed Yang-Mills model is proved to be equivalent to the moduli space of flat connections on a two-dimensional torus.

scholarly journals Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

2021 ◽  
Author(s):  
L. Fehér
Keyword(s):  
Lie Group ◽  
Poisson Structure ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Poisson Reduction ◽  
Sutherland Model ◽  
Heisenberg Double ◽  
Real Forms

AbstractWe construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

scholarly journals Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 455
Author(s):  
Amitesh S. Jayaraman ◽  
Domenico Campolo ◽  
Gregory S. Chirikjian
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Diffusion Processes ◽  
Mathematical Finance ◽  
Alternative Interpretation ◽  
Affine Group ◽  
Black Scholes ◽  
Covariance Propagation ◽  
Alternative Means ◽  
Transformation Of Variables

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

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