Search

We describe the Z2-graded central polynomials for the matrix algebra of order two, M2(K), and for the algebras M1,1(E) and E⊗E over an infinite field K, charK≠2. Here E is the infinite-dimensional Grassmann algebra, and M1,1(E) stands for the algebra of the 2×2 matrices whose entries on the diagonal belong to E0, the centre of E, and the off-diagonal entries lie in E1, the anticommutative part of E. It turns out that in characteristic 0 the graded central polynomials for M1,1(E) and E⊗E are the same (it is well known that these two algebras satisfy the same polynomial identities when charK=0). On the contrary, this is not the case in characteristic p>2. We describe systems of generators for the Z2-graded central polynomials for all these algebras.Finally we give a generating set of the central polynomials with involution for M2(K). We consider the transpose and the symplectic involutions.

In this paper we consider the algebra M1,1(E) endowed with the involution ∗ induced by the transposition superinvolution of the superalgebra M1,1(F) of 2×2-matrices over the field F. We study the ∗-polynomial identities for this algebra in the case of characteristic zero. We describe a finite set generating the ideal of its ∗-identities. We also consider Mn(E), the algebra of n×n matrices over the Grassmann algebra E. We prove that for a large class of involutions defined on it any ∗-polynomial identity is indeed a polynomial identity. A similar result holds for the verbally prime algebra Mk,l(E).

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-10T17:30:02Z No. of bitstreams: 1 DAVID LEVI DA SILVA MACÊDO - DISSERTAÇÃO PPGMAT 2015..pdf: 982236 bytes, checksum: eeb47d97976467c33db1c843ee7e5f90 (MD5) Made available in DSpace on 2018-08-10T17:30:02Z (GMT). No. of bitstreams: 1 DAVID LEVI DA SILVA MACÊDO - DISSERTAÇÃO PPGMAT 2015..pdf: 982236 bytes, checksum: eeb47d97976467c33db1c843ee7e5f90 (MD5) Previous issue date: 2015-08 Capes Para ler o resumo deste trabalho recomendamos o download do arquivo, uma vez que o mesmo possui fórmulas e caracteres matemáticos que não foram possíveis trascreve-los aqui. To read the summary of this work we recommend downloading the file, since it has formulas and mathematical characters that were not possible to transcribe them here.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-05T13:16:57Z No. of bitstreams: 1 SIRLENE TRAJANO ALVES - DISSERTAÇÃO PPGMAT 2012..pdf: 543242 bytes, checksum: 8ace2f30dc5a59df9bafcf55b8e7147b (MD5) Made available in DSpace on 2018-08-05T13:16:57Z (GMT). No. of bitstreams: 1 SIRLENE TRAJANO ALVES - DISSERTAÇÃO PPGMAT 2012..pdf: 543242 bytes, checksum: 8ace2f30dc5a59df9bafcf55b8e7147b (MD5) Previous issue date: 2012-03 O tema central desta dissertação é a descrição das identidades polinomiais graduadas da álgebra Mn(K). Métodos diferentes são empregados conforme a característica do corpo: se Char K = 0, à descrição das identidades graduadas se reduz a descrição das identidades multilineares, o que foi feito no Capítulo 2, onde são descritas as identidade de Mn(K) com uma classe ampla de graduações elementares; se Char K =p>0 e K é in nito, a descrição das identidades graduadas é reduzida à descrição das identidades multi-homogêneas, que torna o problema mais difícil, e técnicas como a construção de álgebras genéricas são necessárias. No Capítulo 3 são descritas as identidades Z e Zn-graduadas de Mn(K) para um corpo in nito K. The main theme of this dissertation is the description of the graded polynomial identities of the algebra Mn(K). Diferent methods are used depending on the characteristic of the field: if Char K = 0, the description of the graded identities is reduced to the description of the multilinear graded identities, what was done in Chapter 2, where the identities of Mn(K) are described for a wide class of elementary gradings; if Char K =p>0 and K is in nite, the description of the graded identities is reduced to the study of the multi-homogeneous identities, wich makes it harder, and techniques such as the construction of generic algebras are necessary. In Chapter 3 the Z and Zn-graded identities of Mn(K) are described for an infinite field K

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-26T13:32:46Z No. of bitstreams: 1 JUSSIÊ UBALDO DA SILVA - DISSERTAÇÃO PPGMAT 2011..pdf: 609032 bytes, checksum: cb1a1234420f940ac2f6aa5c003e9d94 (MD5) Made available in DSpace on 2018-07-26T13:32:46Z (GMT). No. of bitstreams: 1 JUSSIÊ UBALDO DA SILVA - DISSERTAÇÃO PPGMAT 2011..pdf: 609032 bytes, checksum: cb1a1234420f940ac2f6aa5c003e9d94 (MD5) Previous issue date: 2011-07 Capes SendoG um grupo abeliano eR uma álgebraG-graduada, consideramos no produto tensorialR⊗E (sendoE a álgebra exterior de dimensão infinita) a (G×Z2)graduação natural, obtida a partir daG-graduação deR. Neste trabalho apresentamos resultados que relacionam as identidades graduadas e resultados que relacionam os polinômios centrais graduados das álgebrasR eR⊗E. Como aplicação obtemos a PI-equivalência entre as álgebrasM1,1(E)⊗E eM2(E), resultado que é parte do clássico Teorema do Produto Tensorial de Kemer. Também apresentamos descrições das identidades e dos polinômios centrais (Zn × Z2)-graduados da álgebra Mn(E), e das identidades e dos polinômios centrais Z2-graduados da álgebra E ⊗ E, considerando para esta última uma graduação diferente da usual. Para uma visualização mais confiáveis das formulas e sinais matemáticos deste resumo recomendamos o download do arquivo. LetG be an abelian group andR aG-graded algebra. We consider in the tensor product R ⊗ E, where E is the exterior algebra of infinite dimension, the natural (G×Z2)-grading, obtained fromG-grading ofR. In this work, we present results that relates the graded identities and also relates the graded central polynomials of the algebrasR andR⊗E. As an application we obtain the PI-equivalence between the algebras M1,1(E)⊗E and M2(E), which is a part of the Tensor Product Theorem of Kemer. We also present descriptions of the (Zn × Z2)-graded identities and central polynomials of the algebra Mn(E), as well as of theZ2-graded identities and central polynomials of the algebra E ⊗ E. In the last case, we consider a different grading from the usual one.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-23T14:58:20Z No. of bitstreams: 1 LEOMAQUES FRANCISCO SILVA BERNARDO - DISSERTAÇÃO PPGMAT 2009..pdf: 656966 bytes, checksum: 9ca0422e8cc572aa2c43d542260ef401 (MD5) Made available in DSpace on 2018-07-23T14:58:20Z (GMT). No. of bitstreams: 1 LEOMAQUES FRANCISCO SILVA BERNARDO - DISSERTAÇÃO PPGMAT 2009..pdf: 656966 bytes, checksum: 9ca0422e8cc572aa2c43d542260ef401 (MD5) Previous issue date: 2009-06 Capes Neste trabalho apresentamos um estudo sobre identidades e polinômios centrais para a álgebra das matrizes. Mais precisamente, apresentamos a descrição das identidades e polinômios centrais Zn-graduados e Z-graduados para a álgebra Mn(K) (matizes n x n sobre um corpo K), quando característica de K é zero. Depois, apresentamos a descrição dos polinômios centrais ordinários para a álgebra M2(K) (matrizes 2 x 2 sobre K), também para um corpo de característica zero. Finalmente, apresentamos duas construções clássicas de polinômios centrais para Mn(K), que surgiram como resposta a um problema sugerido por Kaplansky em 1956 sobre a existência de polinômios não triviais para esta álgebra. In this work we study polynomial identities and central polynomials for matrix algebras. More precisely, we present the description of the identities and Zn-graded and Z-graded central polynomials for the algebra Mn(K) (the n x n matrices over the field K) when the characteristic of K is zero. Afterwards we give the description or the ordinary (nongraded) central polynomials for the algebra m2(K), the 2 x 2 matrices over K, assuming the field of characteristic zero. Finally, we present two classical constructions of central polynomials for Mn(K). These appeared as an answer to a problem posed by Kaplansky in 1956 about the existence of nontrivial central polynomials for that algebra.