ACTION OF HOM SPACE AND TENSOR PRODUCT OF LIE ALGEBRAS

The primary goal of this study is to explore the Triple Action of Representation Lie Algebras (ATHS) with unique characteristics on Lie algebra using Schur's lemma , which states that the action of the tensor product of Lie algebras representation has interesting property. This involves examining the interaction of Lie algebra representations, focusing on specific actions within Lie algebras. A new approach to the tensor product and direct sum for Lie algebra representations has been introduced. Additionally, the study explores the relationship between the actions of Hom space and tensor product. The main goal is to achieve results on Lie algebra representations by utilizing a new structure comprising six representations and their corresponding actions.