Some Studies On Involutive Hom-associative Algebras And Rota-Baxter Operators

This thesis mainly studies the free involutive Hom-associative algebra, Rota-Baxter operators and classification of Rota-Baxter type operators, and consists of six chapters.The first one introduces the background and its recent development, and gives some basic notions and related notations, and then analyzes the motivations of this thesis.Chapter two first introduces the definition of Hom-semigroup, and further some examples are given to demonstrate that the class of semigroups is a proper subclass of the class of Hom-semigroups. Then we construct the free involutive Hom-semigroup by bracketed words. As a consequence, we obtain the explicit construction of free involutive Hom-associative algebras generated by a set.Chapter three studies the relative locations of subwords in free operated semi-groups. Firstly, we consider the relative locations of two subwords of a fixed word in a semigroup. Secondly, we establish a correspondence between relative locations of bracketed words and those of words by applying the Motzkin words. Finally, we obtain the classification of the relative of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested and intersecting cases.Chapter four applies the methods of rewriting systems and Grobner-Shirshov bases to give a unified approach to a class of linear operators on associative alge-bras. Since these operators resemble the classic Rota-Baxter operator, they are called Rota-Baxter type operators. Firstly, we introduce the term-rewriting systems on free modules and Rota-Baxter term-rewriting systems, and obtain some meaningful re-sults. Then we characterize a Rota-Baxter type operator by the convergency of a Rota-Baxter term-rewriting system. Secondly, by the theory of Grobner-Shirshov bases for free operated algebras, we obtain a canonical basis for the free algebras in the category of associative algebras with Rota-Baxter type operators. Finally, we construct a monomial order on the free operated semigroup. This allows us to show that these operators are indeed Rota-Baxter type operators as claimed by the conjec-ture mentioned in this chapter, and make some key progress towards solving Rota’s problem of classification of linear operators in some sense.Chapter five studies Rota-Baxter operators on the polynomial algebra, integra-tion and averaging operators. A Rota-Baxter operator is an algebraic abstraction and a generalization of integration. Following this classical connection, we study the relation between Rota-Baxter operators and integrals in the case of the polynomial algebra k[x]. We mainly consider two classes of Rota-Baxter operators, monomial ones and injective ones. For the first class, we apply averaging operators to deter-mine monomial Rota-Baxter operators. For the second case, we make use of the double product on Rota-Baxter algebras to determine a part of injective Rota-Baxter operators.Chapter six determines all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three. We first formulate the general setup for determining Rota-Baxter operators on a semigroup algebra in matrix form. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations. We also produce a Mathematica procedure to predict and verify these solutions.