ON ROTA-BAXTER NIJENHUIS TD ALGEBRA

Abstract

On Rota-Baxter Nijenhuis TD Algebra By Monica Aggarwal Dissertation Director: Professor Li Guo There was a long standing problem of G. C. Rota regarding the classification of all linear operators on associative algebras that satisfy algebraic identities. Initially, only very few of such operators were known, for example, the derivative operator, average operator, difference operator and Rota-Baxter operator. Recently, in a paper by L. Guo, W. Sit and R. Zhang, the authors revisited Rota’s problem 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 Rota-Baxter Nijenhuis TD (RBNTD) operator which puts together the terms of the well-known Rota-Baxter operator, Nijenhuis operator and Leroux’ TD operator. In this dissertation, we initiate a systematic study of the RBNTD operator, extending the previous works on the Rota-Baxter, Nijenhuis and TD operators. After giving basic properties and examples, we construct free commutative and then free (non-commutative) RBNTD algebras. We then use free RBNTD algebras to obtain an extension of the renowned dendriform algebra with five binary operations.