Azumaya Algebras In Gr-categories

In this doctoral dissertation, we will research Azumaya algebra in Gr-categories. As a natural extension of associative algebras, algebras in categories have been studied in depth for many years. Many experts and scholars have been working in this field and many important and interesting result have been achieved. In this paper, we firstly study generalized Clifford algebras by viewing them as algebras in suitable symmetric linear Gr-categories; Secondly, we study the structure the-orem of Azumaya algebras in braided Gr-category, and prove that the Octonions form an Azumaya algebra in some suitable braided linear Gr-categories; thirdly, we give a classification of Azumaya algebras in a simple braided Gr-category.The dissertation consists of four chapters.In chapter 1, we firstly introduce some history and development about Azu-maya algebras and Gr-categories. Then, we present the main results and structure of this dissertation.In chapter 2, we introduce some definitions and basic facts about Azumaya algebras in monoidal categories, tensor categories and Gr-categories. we mainly focus on twisted group algebras and gauge transformations in Gr-categories which will be used in later chapters.In chapter 3, we study generalized Clifford algebras as algebras in suitable symmetric linear Gr-categories. By viewing Clifford algebras as twisted group algebra of Z2n, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. In this chap-ter, we study generalized Clifford algebras in a similar manner as twisted group algebras of Znm, get the periodicity of a class of generalized clifford algebras, in-troduce the generalized Clifford process, improve and generalizer the results of Albuquerque, Majid and Bulacu. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decom- position theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.In chapter 4, we study Azumaya algebras in general braided Gr-categories. Firstly, we prove that Azumaya algebrazs in braided linear Gr-category are center simple algebras in these categories, which is an extension of result about graded algebras. Secondly, by gauge transformation, we view the Octonions as an asso-ciative algebra in certain tensor categories, or more precisely as a twisted group algebra by a 2-cochain, we show that the Octonions form an Azumaya algebra in some suitable braided linear Gr-categories. Finally, we get concrete structure theorem and classification of Azumaya algebras in a simple braided linear Gr-category (Vecz2Φ,R).