Simplicial homology of differential graded algebras of Lie algebras

In this paper, we study some important characteristics of the homological theory within infinity differential graded algebras (that denoted by DGA). Specifically, we explore definitions of Lie algebras and the simplicial homology theory of L_∞-algebras. Our primary objective is to improve the understanding of short exact sequence in simplicial homology of infinity algebras across specific classes of algebras. Additionally, we establish and prove the exact long sequence in simplicial homology of L_∞-algebras. Furthermore, we investigate the trace map and the inclusion map, clarifying their roles and connections within simplicial homology of L_∞-algebras. Notably, we show and apply the property of Morita equivalence in the context of simplicial homology of L_∞-algebras. Moreover, we establish that the trace map and the inclusion map are related inverse in simplicial homology of L_∞-algebras. Overall, this research improves our understanding of homological algebra within infinity differential graded algebras, focusing on their structures and practical applications.