Gr(?)bner-Shirshov Basis Of The Degenerate And Non-Degenerate Affine Hecke Algebra Type A_n

Affine Hecke algebra is an active research field in modern algebra and there are various results about this algebra and its representation in the literature.The Gr(?)bner-Shirshov basis theory is a relatively new branch of algebra which was developed by Buchberger(for cormmutative algebra),Bergman(for associative algebra)and Shirshov(for Lie algebra)in order to solve the reduction problem in algebra.The core result in the Gr(?)bner-Shirshov 1?basis theory is the s?called Compositiorn-Diamond Lemma,which whenever we know the Composition-Diamond Lemma about an algebra,then by computing a Gr(?)bner-Shirshov basis of this algebra,we are able to give a linear basis of the algebra.So to give a Gr(?)bner-Shirshov basis of an algebra is very helpful to study the structure of the algebra.In this thesis,we first give a Gr(?)bner-Shirshov basis of the degenerate and non-degenerate affine Hecke algebra of type An by using the Shirshov algorithm and then,as an application,we give a linear basis for degenerate and non-degenerate affine Hecke algebra of type An by using the Composition-Diamond Lemma for associative algebra.