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.

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

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 THE BURNSIDE RING-VALUED MORSE FORMULA FOR VECTOR FIELDS ON MANIFOLDS WITH BOUNDARY

2009 ◽  
Vol 01 (01) ◽  
pp. 13-27 ◽  
Author(s):  
GABRIEL KATZ
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Compact Lie Group ◽  
Manifolds With Boundary ◽  
Burnside Ring ◽  
Polynomial Inequality ◽  
Invariant Vector ◽  
Generic Polynomial ◽  
Invariant Vector Field ◽  
Algebraic Fields

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula [Formula: see text] which takes its values in A(G). Here Ind G(v) denotes the equivariant index of the field v, [Formula: see text] the v-induced Morse stratification (see [10]) of the boundary ∂X, and [Formula: see text] the class of the (n - k)-manifold [Formula: see text] in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝn defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas ([3, 4]).

On the geometrical properties of hypercomplex four-dimensional Lie groups

2020 ◽  
Vol 27 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Mehri Nasehi ◽  
Mansour Aghasi
Keyword(s):  
Lie Groups ◽  
Vector Fields ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Invariant Vector ◽  
Geometrical Properties ◽  
Invariant Vector Field ◽  
Left Invariant Vector ◽  
Totally Geodesic Hypersurfaces ◽  
Yamabe Solitons

AbstractIn this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.

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 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.

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