ALGEBRAIC STUDIES OF SYMMETRIC OPERATORS

Abstract

Algebraic Studies of Symmetric Operators By Zhiqin Shi Dissertation Director: Professor William Keigher There was an old problem of G. C. Rota regarding the classification of all linear operators on associative algebras that satisfy algebraic identities. We only know very few of such operators at the beginning, for example, the derivative operator, average operator, difference operator and Rota-Baxter operator. Recently L. Guo, W. Sit and R. Zhang revisited Rota’s problem in a paper by concentrating on two classes of operators: differential type operators and Rota-Baxter type operators. One of the Rota-Baxter type operators they found is the symmetric Rota-Baxter operator which symmetrizes the RotaBaxter operator. In this dissertation, we initiate a systematic study of the symmetric Rota-Baxter operator, extending the previous works on the original Rota-Baxter operator. After giving basic properties and examples, we construct free symmetric Rota-Baxter algebras on an algebra and on a set by bracketed words and rooted trees separately. We then use the free symmetric Rota-Baxter algebra to obtain an extension of the well known dendriform algebra and its free objects. Finally, we extend our study to differential algebras. We construct the free symmetric differential Rota-Baxter algebra based on the previous free symmetric Rota-Baxter algebra on a set and the free symmetric differential algebra.