CYCLIC HOMOLOGY OF  $L_{\infty}$-ALGEBRAS

$A_{\infty}$-algebras, which are homotopy-invariant versions of associative algebras, play a crucial role in exploring complex topological structures and their categorical aspects. They are central to the study of Morse homotopy, Floer homologies, and mirror symmetry, with the Fukaya category-closely related to $A_{\infty}$-structures-being essential in categorical mirror symmetry. This category establishes a link between the Fukaya category of a symplectic manifold and the derived category of coherent sheaves on a Calabi-Yau manifold. Meanwhile, $L_{\infty}$-algebras extend Lie algebras by incorporating higher homotopy structures, vital for analyzing differential graded algebras and their homological properties. These advancements have opened new avenues in deformation theory, gauge theory, and string field theory. This paper examines key aspects of homological theory within the framework of infinity differential graded algebras (DGAs), focusing on the definitions and properties of Lie algebras and the cyclic homology theory of $L_{\infty}$-algebras. We analyze and prove the exact long sequence in the cyclic homology of $L_{\infty}$-algebras, highlighting the interplay between various algebraic structures. Our study also explores the trace and inclusion maps' roles and their influence through Morita equivalence, offering deeper insights into the relationships among different algebraic structures and their homological properties.