λ-TD Algebras And Left Counital Hopf Algebras

The theory of operated algebras(algebras equipped with a linear operator)has played a pivotal role in mathematics and physics.Two primary instances of operated algebras are the Rota-Baxter algebra and TD-algebra.In this thesis,we introduce a λ-TD algebra that includes both the Rota-B axter algebra and the TDalgebra.This thesis consists of five chapters.Chapter one introduces the background of the project,and then gives the motivations and outline of the paper.Chapter two focuses on λ-TD algebras.Firstly,we introduce the notion of ATD algebra and then explore general properties of λ-TD algebras.Moreover,the explicit construction of free commutative λ-TD algebra on a commutative algebra is obtained by a generalized shuffle product,called the λ-TD shuffle product.Chapter three introduces the concepts of left counital cocycle bialgebra.We then show that the free commutative λ-TD algebra possesses a left counital bialgera structure by means of a suitable 1-cocycle condition.Chapter four,the classical result that every connected filtered bialgebra is a Hopf algebra,is extended to the context of left counital bialgebras.Given this result,we finally prove that the left counital bialgebra on the free commutativeλ-TD algebra is connected and filtered,and thus is a left counital Hopf algebra.Chapter five explores the structure of free noncommutative λ-TD algebras...