In this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group conjecture. This conjecture predicts the existence of derived equivalences between categories of modules. Some attempts to prove Broue’s conjecture by lifting stable equivalences to derived equivalences highlight the importance of understanding the connection between transferring gradings via stable equivalences and transferring gradings via derived equivalences. The main idea that we use is the following. We start with an algebra which can be easily graded, and transfer this grading via derived or stable equivalence to another algebra which is not easily graded. We investigate the properties of the resulting grading. In the first chapter we list the background results that will be used in this thesis. In the second chapter we study gradings on Brauer tree algebras, a class of algebras that contains blocks of group algebras with cyclic defect groups. We show that there is a unique grading up to graded Morita equivalence and rescaling on an arbitrary basic Brauer tree algebra. The third chapter is devoted to the study of gradings on tame blocks of group algebras. We study extensively the class of blocks with dihedral defect groups. We investigate the existence, positivity and tightness of gradings, and we classify all gradings on these blocks up to graded Morita equivalence. The last chapter deals with the problem of transferring gradings via stable equivalences between blocks of group algebras. We demonstrate on three examples how such a transfer via stable equivalences is achieved between Brauer correspondents, where the group in question is a TI group.

Submitted by Luciana Sebin (lusebin@ufscar.br) on 2016-10-11T13:29:51Z No. of bitstreams: 1 DissFAN.pdf: 767462 bytes, checksum: 05054cc8952eed4e120838068aee80d8 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-17T18:50:57Z (GMT) No. of bitstreams: 1 DissFAN.pdf: 767462 bytes, checksum: 05054cc8952eed4e120838068aee80d8 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-17T18:51:05Z (GMT) No. of bitstreams: 1 DissFAN.pdf: 767462 bytes, checksum: 05054cc8952eed4e120838068aee80d8 (MD5) / Made available in DSpace on 2016-10-17T19:06:01Z (GMT). No. of bitstreams: 1 DissFAN.pdf: 767462 bytes, checksum: 05054cc8952eed4e120838068aee80d8 (MD5) Previous issue date: 2016-02-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / This work aims to give a description, under certain hypothesis, of the graded simple algebras and prove that they are determined by their graded identities. For this, we study the papers [3] and [19]. More precisely we will show the following: Let G be a group, F an algebraically closed eld, and R = L g2G Rg a finite dimensional G-graded F-algebra such that the order of each finite subgroup of G is invertible in F. Then R is a G-graded simple algebra if and only if R is isomorphic, as graded algebra, to the tensor product C = Mn(F) F [H], where H is a nite subgroup of G, is a 2-cocycle in H, Mn(F) has an elementary G-grading, F [H] has a canonical grading and C has an induced G-grading by the tensor product. Based on this result, admitting the same assumptions and adding that G is an abelian group, we prove that two graded simple algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras. / Este trabalho tem por objetivo dar uma descrição, sob certas hipóteses, das álgebras graduadas simples e demonstrar que elas são determinadas por suas identidades graduadas. Para isso, estudamos os artigos [3] e [19]. Precisamente mostraremos o seguinte: sejam G um grupo, F um corpo algebricamente fechado e R =Lg2GRg uma F-álgebra G-graduada de dimensão finita, tal que a ordem de todo subgrupo finito de G e invertível em F. Então R é uma álgebra G-graduada simples se, e somente se, R é isomorfa, como álgebra graduada, ao produto tensorial C = Mn(F) F[H], onde H e subgrupo finito de G, e um 2-cociclo em H, Mn(F) tem uma graduação elementar, F[H] tem uma graduação canônica e considera-se em C a G-graduação induzida pelo produto tensorial. Partindo deste resultado, admitindo as mesmas hipóteses e adicionando que G seja um grupo abeliano, provaremos que duas álgebras graduadas simples satisfazem as mesmas identidades graduadas se, e somente se, são isomorfas como álgebras graduadas.