scholarly journals Analogues of Korn’s inequality on Heisenberg groups

2019 ◽  
Vol 485 (4) ◽  
pp. 405-409
Author(s):  
D. V. Isangulova
Keyword(s):  
Differential Operator ◽  
Lie Algebra ◽  
Isometry Group ◽  
Conformal Mappings ◽  
Symmetric Part ◽  
Heisenberg Groups ◽  
Korn’S Inequality ◽  
Coercive Estimate

Two analogues of Korn’s inequality on Heisenberg groups are constructed. First, the norm of the horizontal differential is estimated in terms of its symmetric part. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for this operator. Additionally, a coercive estimate is proved for a differential operator whose kernel coincides with the Lie algebra of the group of conformal mappings on Heisenberg groups.

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

WEYL TYPE ALGEBRAS FROM QUANTUM TORI

2006 ◽  
Vol 08 (02) ◽  
pp. 135-165 ◽  
Author(s):  
KAIMING ZHAO
Keyword(s):  
Differential Operator ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Operator Algebra ◽  
Laurent Polynomials ◽  
Associative Algebras ◽  
Defining Relations ◽  
Quantum Tori ◽  
Quantum Version ◽  
Algebra Over A Field

We introduce and study the quantum version of the differential operator algebra on Laurent polynomials and its associated Lie algebra over a field F of characteristic 0. The q-quantum torus Fq is the unital associative algebra over F generated by [Formula: see text] subject to the defining relations titj = qi,jtjti, where qi,i = 1, [Formula: see text]. Let D be a subspace of [Formula: see text] where ∂i is the derivation on Fq sending [Formula: see text] to [Formula: see text]. Then, the quantum differential operator algebra is the associative algebra Fq[D]. Assume that Fq[D] is simple as an associative algebra. We compute explicitly all 2-cocycles of Fq[D], viewed as a Lie algebra. More precisely, we show that the second cohomology group of Fq[D] has dimension n if D = 0, dimension 1 if dim D = 1, and dimension 0 if dim D > 1. We also determine all isomorphisms and anti-isomorphisms Fq[D] → Fq′[D′] of simple associative algebras, and all isomorphisms Fq[D]/F → Fq′[D′]/F of simple Lie algebras.

Parameter rigid actions of the Heisenberg groups

2007 ◽  
Vol 27 (6) ◽  
pp. 1719-1735 ◽  
Author(s):  
NATHAN M. DOS SANTOS
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Cohomology Group ◽  
Free Action ◽  
The Other ◽  
Heisenberg Groups

AbstractA locally free action of a Lie group is parameter rigid if for each other action with the same orbit foliation there exists a $C^\infty $ orbit-preserving diffeomorphism which conjugates the action to a reparametrization of the other by an automorphism of the Lie group. We show that for actions of the Heisenberg groups, if the first leafwise cohomology group of the orbit foliation is isomorphic to the first cohomology of the Lie algebra of the group, then the action is parameter rigid. Using this, we give examples of parameter rigid actions for all of the Heisenberg groups.

scholarly journals EXACTLY SOLVABLE DYNAMICAL SYSTEMS IN THE NEIGHBORHOOD OF THE CALOGERO MODEL

1999 ◽  
Vol 14 (03) ◽  
pp. 387-408 ◽  
Author(s):  
OLIVER HASCHKE ◽  
WERNER RÜHL
Keyword(s):  
Dynamical Systems ◽  
Differential Operator ◽  
Lie Algebra ◽  
Riemannian Metric ◽  
Second Order ◽  
Order Differential Operator ◽  
Calogero Model ◽  
Exactly Solvable ◽  
Quadratic Lie Algebra ◽  
Solvable Dynamical Systems

The Hamiltonian of the N-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For N = 3 and N = 4 such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all N.

Poincaré Lemma on Quaternion-like Heisenberg Groups

2018 ◽  
Vol 61 (3) ◽  
pp. 495-508
Author(s):  
Der-Chen Chang ◽  
Nanping Yang ◽  
Hsi-Chun Wu
Keyword(s):  
Differential Operator ◽  
Heisenberg Group ◽  
Existence Of Solutions ◽  
Partial Differential Operator ◽  
Heisenberg Groups ◽  
Smooth Functions ◽  
Operator System ◽  
Solution Function ◽  
Poincaré Lemma ◽  
Quaternion Heisenberg Group

AbstractFor smooth functions a1, a2, a3, a4 on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system X1f = a1, X2f = a2, X3f = a3, and X4f = a4. In addition, a formula for the solution function f is deduced, assuming solvability of the system.

Practical Correction of Three Fold Astigmatism in the Philips CM TEM

1996 ◽  
Vol 54 ◽  
pp. 418-419
Author(s):  
Arno J. Bleeker ◽  
Mark H.F. Overwijk ◽  
Max T. Otten
Keyword(s):  
Optical Properties ◽  
Phase Contrast ◽  
The Other ◽  
Objective Lens ◽  
Symmetric Part ◽  
A Posteriori ◽  
Contrast Transfer Function ◽  
High Brightness ◽  
Rotationally Symmetric ◽  
Brightness Field

With the improvement of the optical properties of the modern TEM objective lenses the point resolution is pushed beyond 0.2 nm. The objective lens of the CM300 UltraTwin combines a Cs of 0. 65 mm with a Cc of 1.4 mm. At 300 kV this results in a point resolution of 0.17 nm. Together with a high-brightness field-emission gun with an energy spread of 0.8 eV the information limit is pushed down to 0.1 nm. The rotationally symmetric part of the phase contrast transfer function (pctf), whose first zero at Scherzer focus determines the point resolution, is mainly determined by the Cs and defocus. Apart from the rotationally symmetric part there is also the non-rotationally symmetric part of the pctf. Here the main contributors are not only two-fold astigmatism and beam tilt but also three-fold astigmatism. The two-fold astigmatism together with the beam tilt can be corrected in a straight-forward way using the coma-free alignment and the objective stigmator. However, this only works well when the coefficient of three-fold astigmatism is negligible compared to the other aberration coefficients. Unfortunately this is not generally the case with the modern high-resolution objective lenses. Measurements done at a CM300 SuperTwin FEG showed a three fold-astigmatism of 1100 nm which is consistent with measurements done by others. A three-fold astigmatism of 1000 nm already sinificantly influences the image at a spatial frequency corresponding to 0.2 nm which is even above the point resolution of the objective lens. In principle it is possible to correct for the three-fold astigmatism a posteriori when through-focus series are taken or when off-axis holography is employed. This is, however not possible for single images. The only possibility is then to correct for the three-fold astigmatism in the microscope by the addition of a hexapole corrector near the objective lens.

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