scholarly journals A Brief Study of Relation between Lie Derivative, Exterior Derivative and Contraction

2018 ◽  
Vol 6 (4) ◽  
pp. 805-806
Author(s):  
Navneet Garg
Keyword(s):  
Lie Derivative ◽  
Exterior Derivative

scholarly journals CARTAN CALCULUS VIA PAULI MATRICES

2003 ◽  
Vol 18 (28) ◽  
pp. 5231-5259
Author(s):  
D. MAURO
Keyword(s):  
Vector Field ◽  
Path Integral ◽  
Classical Mechanics ◽  
Tensor Products ◽  
Lie Derivative ◽  
Exterior Derivative ◽  
Integral Formalism ◽  
Jacobi Fields ◽  
Finite Dimensional ◽  
Pauli Matrices

In this paper we will provide a new operatorial counterpart of the path-integral formalism of classical mechanics developed in recent years. We call it new because the Jacobi fields and forms will be realized via finite dimensional matrices. As a byproduct of this we will prove that all the operations of the Cartan calculus, such as the exterior derivative, the interior contraction with a vector field, the Lie derivative and so on, can be realized by means of suitable tensor products of Pauli and identity matrices.

scholarly journals A PROPOSAL FOR A DIFFERENTIAL CALCULUS IN QUANTUM MECHANICS

1994 ◽  
Vol 09 (13) ◽  
pp. 2191-2227 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Hamiltonian System ◽  
Differential Calculus ◽  
Cotangent Bundle ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Derivative ◽  
Tensor Calculus ◽  
Exterior Calculus

In this paper, using, the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a quantum-deformed exterior calculus on the phase space of an arbitrary Hamiltonian system. Introducing additional bosonic and fermionic coordinates, we construct a supermanifold which is closely related to the tangent and cotangent bundle over phase space. Scalar functions on the supermanifold become equivalent to differential forms on the standard phase space. The algebra of these functions is equipped with a Moyal superstar product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.

scholarly journals Currents, charges and algebras in exceptional generalised geometry

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
David Osten
Keyword(s):  
Current Algebra ◽  
Hamiltonian Formulation ◽  
Building Blocks ◽  
Lie Derivative ◽  
Consistency Check ◽  
World Volume ◽  
Invariant Currents ◽  
Invariant Current ◽  
M Theory ◽  
A Current

Abstract A classical Ed(d)-invariant Hamiltonian formulation of world-volume theories of half-BPS p-branes in type IIb and eleven-dimensional supergravity is proposed, extending known results to d ≤ 6. It consists of a Hamiltonian, characterised by a generalised metric, and a current algebra constructed s.t. it reproduces the Ed(d) generalised Lie derivative. Ed(d)-covariance necessitates the introduction of so-called charges, specifying the type of p-brane and the choice of section. For p > 2, currents of p-branes are generically non- geometric due to the imposition of U-duality, e.g. the M5-currents contain coordinates associated to the M2-momentum.A derivation of the Ed(d)-invariant current algebra from a canonical Poisson structure is in general not possible. At most, one can derive a current algebra associated to para-Hermitian exceptional geometry.The membrane in the SL(5)-theory is studied in detail. It is shown that in a generalised frame the current algebra is twisted by the generalised fluxes. As a consistency check, the double dimensional reduction from membranes in M-theory to strings in type IIa string theory is performed. Many features generalise to p-branes in SL(p + 3) generalised geometries that form building blocks for the Ed(d)-invariant currents.

scholarly journals Gauged double field theory as an L∞ algebra

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Eric Lescano ◽  
Martín Mayo
Keyword(s):  
Field Theory ◽  
Algebraic Structure ◽  
Classical Field ◽  
Field Theories ◽  
Lie Derivative ◽  
Linear Perturbation ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Symmetry Transformations ◽  
Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

Some Properties of Symplectic Manifolds S on the Projective Tangent Bundle PTM

2011 ◽  
Vol 187 ◽  
pp. 483-486
Author(s):  
Yong He ◽  
Xiao Ying Lu ◽  
Wei Na Lu
Keyword(s):  
Vector Field ◽  
Symplectic Manifold ◽  
Tangent Bundle ◽  
Fiber Bundle ◽  
Symplectic Manifolds ◽  
Lie Derivative ◽  
The Relationship

In this paper, we show the relationship between 2-form of the two projective tangent bundle and the relationship between 2-form on projective tangent bundle and 1-form on by using the theory of fiber bundle and the properties of symplectic manifold of the projective tangent bundle . Moreover, we derived a simpler formula of Lie derivative of a special vector field, which is on the projective tangent bundle.

scholarly journals $$H^1$$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes

CALCOLO ◽  
2021 ◽  
Vol 58 (2) ◽  
Author(s):  
Francesca Bonizzoni ◽  
Guido Kanschat
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Arbitrary Order ◽  
Inner Product ◽  
Cochain Complex ◽  
Exterior Derivative ◽  
Product Construction ◽  
Cartesian Meshes ◽  
Conforming Finite Element ◽  
Interpolation Operators

AbstractA finite element cochain complex on Cartesian meshes of any dimension based on the $$H^1$$ H 1 -inner product is introduced. It yields $$H^1$$ H 1 -conforming finite element spaces with exterior derivatives in $$H^1$$ H 1 . We use a tensor product construction to obtain $$L^2$$ L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.

The exterior derivative as a Killing vector field

1996 ◽  
Vol 93 (1) ◽  
pp. 157-170 ◽  
Author(s):  
J. Monterde ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Exterior Derivative
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