scholarly journals The Number of Sides of a Parallelogram

1999 ◽  
Vol Vol. 3 no. 2 ◽  
Author(s):  
Elisha Falbel ◽  
Pierre-Vincent Koseleff
Keyword(s):  
Nilpotent Group ◽  
Lie Group ◽  
Vector Fields ◽  
Invariant Vector ◽  
Free Nilpotent Group ◽  
Rational Series ◽  
Left Invariant Vector ◽  
International Audience ◽  
Invariant Vector Fields

International audience We define parallelograms of base a and b in a group. They appear as minimal relators in a presentation of a subgroup with generators a and b. In a Lie group they are realized as closed polygonal lines, with sides being orbits of left-invariant vector fields. We estimate the number of sides of parallelograms in a free nilpotent group and point out a relation to the rank of rational series.

scholarly journals C-Flows on a Lie Group for Euler Equations

1970 ◽  
Vol 40 ◽  
pp. 67-84
Author(s):  
Yoshihei Hasegawa
Keyword(s):  
Euler Equations ◽  
Lie Group ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Left Invariant Riemannian Metric ◽  
Invariant Vector Fields

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

scholarly journals MALLIAVIN CALCULUS FOR LIE GROUP-VALUED WIENER FUNCTIONS

2009 ◽  
Vol 12 (01) ◽  
pp. 67-89
Author(s):  
TAI MELCHER
Keyword(s):  
Lie Group ◽  
Malliavin Calculus ◽  
Compact Support ◽  
Smooth Function ◽  
Vector Fields ◽  
Wiener Measure ◽  
Invariant Vector ◽  
Stochastic Version ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

Let G be a Lie group equipped with a set of left invariant vector fields. These vector fields generate a function ξ on Wiener space into G via the stochastic version of Cartan's rolling map. It is shown here that, for any smooth function f with compact support, f(ξ) is Malliavin differentiable to all orders and these derivatives belong to Lp(μ) for all p > 1, where μ is Wiener measure.

The Maurer–Cartan Form

2020 ◽  
pp. 121-126
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Tangent Space ◽  
Vector Fields ◽  
Vector Spaces ◽  
Invariant Vector ◽  
Cartan Form ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The chapter describes the Maurer-Cartan form and the equation it satisfies, the Maurer-Cartan equation. The Maurer-Cartan form allows one to define a connection on the product bundle M × G → M for any manifold M. The Lie algebra g of a Lie group G is defined to be the tangent space at the identity. One will often identify the two vector spaces and think of elements of g as left-invariant vector fields on G.

Left-Invariant Vector Fields

2013 ◽  
pp. 45-49 ◽  
Author(s):  
Daniel Bump
Keyword(s):  
Vector Fields ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

scholarly journals VECTOR FIELDS ON QUANTUM GROUPS

1996 ◽  
Vol 11 (06) ◽  
pp. 1077-1100 ◽  
Author(s):  
PAOLO ASCHIERI ◽  
PETER SCHUPP
Keyword(s):  
Vector Field ◽  
Hopf Algebra ◽  
Quantum Group ◽  
Vector Fields ◽  
Lie Derivative ◽  
Tensor Fields ◽  
Invariant Vector ◽  
Invariant Vector Field ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left-invariant vector fields. We study the duality between vector fields and one-forms and generalize the construction to tensor fields. A Lie derivative along any (also non-left-invariant) vector field is proposed and a puzzling ambiguity in its definition discussed. These results hold for a generic Hopf algebra.

Left-Invariant Vector Fields

2004 ◽  
pp. 41-45
Author(s):  
Daniel Bump
Keyword(s):  
Vector Fields ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

scholarly journals The algebra of K -invariant vector fields on a symmetric space G/K

2003 ◽  
Vol 51 (3) ◽  
pp. 607-630
Author(s):  
Ilka Agricola ◽  
Roe Goodman
Keyword(s):  
Symmetric Space ◽  
Vector Fields ◽  
Invariant Vector ◽  
Invariant Vector Fields

Semilinear PDEs in the Heisenberg group: The role of the right invariant vector fields

2010 ◽  
Vol 72 (2) ◽  
pp. 987-997 ◽  
Author(s):  
Isabeau Birindelli ◽  
Fausto Ferrari ◽  
Enrico Valdinoci
Keyword(s):  
Heisenberg Group ◽  
Vector Fields ◽  
Invariant Vector ◽  
Semilinear Pdes ◽  
The Right ◽  
Invariant Vector Fields
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