Structure Of Cellular Algebras

This thesis first introduces the development background of cellular algebra,and some existing research results and research.And in the first chapter presents the basic theorem of algebra is introduced first,algebra homomorphism,algebra module,submodule,quotients,the concept of the ideal,and gives some examples to illustrate,and then introduces the concept of group of said representation of algebra,and irreducible representation and irreducible modules of some concepts,and gives the relevant examples to illustrate,finally introduces the short exact column and the concept of the semidirect product and the concept of convolution.Cellular algebra is a algebraic structure received considerable attention in recent years.There is index set(43)which shows complexity of the corresponding cellular algebra in some sense.We study in detail the so called one parameter cellular algebras whose index set(43)contain only one element.General cellular algebras can be viewed as been composed from one parameter cellular algebras.We observe a one parameter cellular algebra is the same thing with certain matrix algebra as a set,but equipped with a different multiplication rule determined by a symmetric matrix.We study in detail the one parameter cellular algebras corresponding to different symmetric matrices,especially their structure and representations.We also try to construct certain functions on those algebras which could be called determinant and trace.In addition,we discuss the structure of cellular algebra corresponding to symmetric matrices that are degenerate and nondegenerate,and find that when a non-degenerate matrix is isomorphic to matrix algebra,when a degenerate matrix is a degenerate matrix,we can construct a short positive concatenation containing kernel and image.