Abelian lie group

Abelian lie group

An Abelian Lie group is a type of Lie group where the group operation is commutative, meaning that the order in which the elements are multiplied does not affect the result. In this specific example, the group of translations in N dimensions is an Abelian Lie group with N generators, meaning that it is a commutative group that can be generated by N elements that correspond to translations in each dimension.From: Mathematical Methods for Physics and Engineering [2018]

Symmetries and Group Theory

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Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018

Mattias Blennow

The abstract group of translations in N dimensions is an Abelian Lie group with N generators. The general translation Tℓ→ by a vector ℓ→ transforms the position vector x→ according to () Tℓ→x→=x→+ℓ→

6-DOF consensus control of multi-agent rigid body systems using rotation matrices

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Published in International Journal of Control, 2022

Mohammad Maadani, Eric A. Butcher

The 6-dimensional configuration (accounting for translation and orientation) space of rigid body pose is the (non-Abelian) Lie group SE(3), a nonlinear manifold which can be expressed as the semi-direct product , where is the configuration space of the position of the centre of mass. The Lie algebra associated with , represented as , denotes the space of real skew-symmetric matrices which is isomorphic to a 3-dimensional vector space. Also, the Lie algebra of SE(3), denoted by , is a six-dimensional vector space that is tangent to SE(3) at the identity element.

A review on some classes of algebraic systems

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Published in International Journal of Control, 2020

Víctor Ayala, Heriberto Román-Flores

From the general theory of Lie groups, we know that any Abelian Lie group G has the form for some nonnegative integers d, n. Then, for an invariant system Σ on an Abelian Lie group, we cannot expect the uniform time property.

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