The hypoelliptic Laplacian on X = G/K

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Bundle ◽  
Symmetric Space ◽  
Dirac Operator ◽  
Clifford Algebras ◽  
Second Order ◽  
Total Space ◽  
Hypoelliptic Operator ◽  
Heisenberg Algebras

This chapter constructs the hypoelliptic Laplacian ℒbX > 0 acting on the total space of a vector bundle TX ⊕ N ≃ g over the symmetric space X = G/K. The operator ℒbX is obtained using general constructions involving Clifford algebras and Heisenberg algebras, and also the Dirac operator of Kostant. The end result is the elliptic Laplacian 𝓛 X on X as well as the hypoelliptic Laplacian ℒbX, which is a second order hypoelliptic operator acting on X^. Among other things, this chapter gives a key formula relating 𝓛 XℒbX, as well as various formulas involving the operator ℒbX+La.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

scholarly journals Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras

2016 ◽  
Vol 766 ◽  
pp. 012001
Author(s):  
Sultan Catto ◽  
Yasemin Gürcan ◽  
Amish Khalfan ◽  
Levent Kurt ◽  
V. Kato La
Keyword(s):  
Clifford Algebras ◽  
Heisenberg Algebras

The concept of a filter

1967 ◽  
Vol 63 (1) ◽  
pp. 221-227 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Operator ◽  
Symmetric Space ◽  
Random Process ◽  
Response Function ◽  
Second Order ◽  
Closed Linear Operator ◽  
Direct Integral ◽  
Group Of Isometries ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractIt is proved that for a second-order, homogeneous, random process on a globally symmetric space a filter, that is a closed linear operator which is invariant under a group of isometries of the space, may be fully described through a response function, that is that it has a direct integral decomposition into components which are scalar multiples of the identity.

Vector bundles that fill n-space

1965 ◽  
Vol 61 (4) ◽  
pp. 869-875 ◽  
Author(s):  
S. A. Robertson ◽  
R. L. E. Schwarzenberger
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Total Space

The idea of exact filling bundle may be described roughly as follows. Suppose that ξk is a vector bundle with fibre Rk, total space E(ξk) and base X. We say that ξk is a real k-plane bundle on X. Let in be the trivial n-plane bundle on X so that E(in) = X × Rn. A bundle monomorphism j: ξk → in defines a map : E(ξk)→Rn obtained by composition of the embedding E(ξk)→E(in) and the product projection E(in) → Rn. The map represents each fibre of ξk as a k-plane in Rn.

scholarly journals First and second cohomology group of a bu ndle

2012 ◽  
Vol 20 (2) ◽  
pp. 71-78
Author(s):  
Adelina Manea
Keyword(s):  
Vector Bundle ◽  
Cohomology Group ◽  
Second Order ◽  
Base Space ◽  
Cohomology Groups ◽  
Trivial Bundle ◽  
Cech Cohomology

Abstract Let (E, π, M) be a vector bundle. We define two cohomology groups associated to π using the first and second order jet manifolds of this bundle. We prove that one of them is isomorphic with a Čech cohomology group of the base space. The particular case of trivial bundle is studied

scholarly journals Local index theory and the Riemann–Roch–Grothendieck theorem for complex flat vector bundles

2018 ◽  
Vol 12 (04) ◽  
pp. 941-987
Author(s):  
Man-Ho Ho
Keyword(s):  
Vector Bundle ◽  
Dirac Operator ◽  
Differential Form ◽  
Vector Bundles ◽  
Variational Formula ◽  
Index Theory ◽  
Index Theorem ◽  
Local Index ◽  
Chern Simons ◽  
Local Family

The purpose of this paper is to give a proof of the real part of the Riemann–Roch–Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut–Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger–Chern–Simons class of complex flat vector bundle.

scholarly journals Structures of spinors fiber bundles with special relativity of Dirac operator using the Clifford algebra

2021 ◽  
Vol 54 (1) ◽  
pp. 410-424
Author(s):  
Yousif Atyeib Ibrahim Hassan
Keyword(s):  
Dirac Operator ◽  
Clifford Algebra ◽  
Special Relativity ◽  
Riemannian Manifolds ◽  
Clifford Algebras ◽  
Cohomology Theory ◽  
Fiber Bundles

Abstract The purpose of this article is to demonstrate how to use the mathematics of spinor bundles and their category. We have used the methods of principle fiber bundles obey thorough solid harmonic treatment of pseudo-Riemannian manifolds and spinor structures with Clifford algebras, which couple with Dirac operator to study important applications in cohomology theory.

The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras

2011 ◽  
Vol 63 (6) ◽  
pp. 1364-1387 ◽  
Author(s):  
Eckhard Meinrenken
Keyword(s):  
Dirac Operator ◽  
Tensor Product ◽  
Lie Algebras ◽  
Clifford Algebras ◽  
Symmetric Bilinear Form ◽  
Affine Lie Algebras ◽  
Weil Algebra ◽  
Casimir Element ◽  
Infinite Dimensional ◽  
Graded Lie Algebra

AbstractLet be an infinite-dimensional graded Lie algebra, with dim , equipped with a non-degenerate symmetric bilinear form B of degree 0. The quantum Weil algebra is a completion of the tensor product of the enveloping and Clifford algebras of g. Provided that the Kac–Peterson class of g vanishes, one can construct a cubic Dirac operator D 2 , whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

scholarly journals Structure presque-transverse intégrable

1988 ◽  
Vol 37 (2) ◽  
pp. 173-177
Author(s):  
Tong Van Duc
Keyword(s):  
Vector Bundle ◽  
Total Space ◽  
Transversal Structure

We find conditions for which a manifold endowed with an integrable almost transversal structure is the total space of a vector bundle isomorphic to the transversal bundle of a foliation.

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