Abstract algebra, left and right translations, what are these good for?

[rayman123]

Homework Statement

Does anyone know what left and right translations are good for?
$$\begin{cases} R_{a}g=ga\\L_{a}g=ag\end{cases}$$ with $$a,g\in G$$ and G is a Lie group


How can we interpret these relations in the easiest way like we try to explain it to a student which if not familiar with abstract algebra? What do the left and the right translations do?




What applications do they have? where could I find some more infromartion about this applications?

Thanks

 

[lippyka]

in advance! Homework EquationsR_{a}g=ga\\L_{a}g=agThe Attempt at a SolutionLeft and right translations are important operations used in Lie groups. They are used to move elements of the group around within the group. In other words, they are used to transform elements of the group. A right translation is an operation that multiplies an element g of a Lie group G by another element a in the same group. Mathematically, this is expressed as R_{a}g=ga. This means that the element g is shifted to the right by the element a. A left translation is an operation that multiplies an element g of a Lie group G by another element a in the same group. Mathematically, this is expressed as L_{a}g=ag. This means that the element g is shifted to the left by the element a. The applications of left and right translations are found in many areas of mathematics, including geometry, topology, dynamical systems, and differential equations. For example, they can be used to generate symmetries of objects in Euclidean space or to study the stability of dynamical systems. For more information on the applications of left and right translations, you can read this article: https://math.stackexchange.com/ques...and-right-translations-in-lie-groups-good-for