An Innovative Study on Lie Groups and Lie Algebras

Abstract

This research article mainly explores on matrix Lie groups admitting the Cayley Construction and presents innovative proofs of the following propositions.

• If a matrix group admits the Cayley construction and so is a matrix Lie group then the corresponding vector space coincides with the Cayley image of it.
• Every matrix Lie group possesses an in-image.

Furthermore three most important lemmas and one proposition in Lie Groups and Lie Algebras are presented with very simple and innovative proofs. One of three lemmas gives the necessary and sufficient condition for a topological group to be Hausdorff.As well the condition for a topological group to be connected is also derived.