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Identidades polinomiais e polinômios centrais para Álgebra de Grassmann. / Polynomial identities and central polynomials for Grassmann's Algebra.COSTA, Nancy Lima. 05 August 2018 (has links)
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Previous issue date: 2012-08 / Capes / Neste trabalho de dissertação estudamos as identidades polinomiais ordinárias
para a Álgebra de Grassmann com unidade, denotada por E, e sem unidade, denotada
por E 0, para corpos de característica diferente de 2. Além disso, também estudamos as
identidades Z2-graduadas da álgebra E no caso em que o corpo tem característica
positiva. Por fim, descrevemos o T-espaço dos polinômios centrais de E tanto
para corpos de característica zero, quanto para corpos de característica positiva
e descrevemos também os polinômios centrais de E 0 para corpos de característica
positiva. / In this dissertation we study the ordinary polynomial identities for the Grassmann
Algebra with unity, denoted by E, and without unity, denoted by E 0, for fields of characteristic di erent from 2. We also study the Z2-graded identities of the
algebra E over elds of positive characteristic. Finaly, we describe the T-space of the
central polynomials of E for fields of characteristic zero and also for fields of positive
characteristic, moreover we describe the T-space of the central polynomials of E
0 for fields of positive characteristic.
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Identidades polinomiais e polinômios centrais com involução. / Polynomial identities and involutional central polynomials.BEZERRA JÚNIOR, Claudemir Fidelis. 09 August 2018 (has links)
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Previous issue date: 2014-02 / Capes / Nesta dissertação são descritas bases para as identidades polinomiais e os polinômios
centrais com involução para a álgebra das matrizes 2 × 2 sobre um corpo in nito
K de característica p 6= 2, considerando-se a involução transposta, denotada por t, e
também a involução simplética, denotada por s. É conhecido que, como o corpo K é
in nito, se ∗ é uma involução em M2(K), então o ideal de identidades (M2(K), ∗) coincide
com (M2(K), t) ou com (M2(K), s). Consideramos também as álgebras Mn(E),
Mk,l(E) e M1,1(E) sobre corpos de característica 0. Para as álgebras Mn(E) e Mk,l(E),
provamos que para uma classe ampla de involuções as identidades polinomiais com
involução coincidem com as identidades ordinárias, e para a álgebra M1,1(E) com a involução ∗ induzida pela superinvolução transposta na superálgebra M1,1(K), exibimos
uma base nita para as ∗-identidades polinomiais. / In this dissertation we describe basis for the polynomial identities and central
polynomials with involution for the algebra of 2 × 2 matrices over an infinite field K
of characteristic p 6= 2 considering the transpose involution, denoted by t, and also
the symplectic involution, denoted by s. It is known that, since the field K is infinite,
if ∗ is an involution on M2(K), then the ideal of identities (M2(K), ∗) coincides with
(M2(K), t) or with (M2(K), s). We also consider the algebras Mn(E), Mk,l(E) and
M1,1(E) over fields of characteristic 0. For the algebras Mn(E) and Mk,l(E) we prove
that for a large class of involutions the polynomial identities with involution coincide
with the ordinary identities, and for the algebra M1,1(E) with the involution ∗ induced
by the transposition superinvolution of the superalgebra M1,1(K) we exhibit nite basis
for the ∗-polynomial identities.
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Bivariate wavelet construction based on solutions of algebraic polynomial identitiesVan der Bijl, Rinske 03 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Multi-resolution analysis (MRA) has become a very popular eld of mathematical study
in the past two decades, being not only an area rich in applications but one that remains
lled with open problems. Building on the foundation of re nability of functions, MRA
seeks to lter through levels of ever-increasing detail components in data sets { a concept
enticing to an age where development of digital equipment (to name but one example)
needs to capture more and more information and then store this information in di erent
levels of detail. Except for designing digital objects such as animation movies, one of the
most recent popular research areas in which MRA is applied, is inpainting, where \lost"
data (in example, a photograph) is repaired by using boundary values of the data set
and \smudging" these values into the empty entries. Two main branches of application
in MRA are subdivision and wavelet analysis. The former uses re nable functions to
develop algorithms with which digital curves are created from a nite set of initial points
as input, the resulting curves (or drawings) of which possess certain levels of smoothness
(or, mathematically speaking, continuous derivatives). Wavelets on the other hand, yield
lters with which certain levels of detail components (or noise) can be edited out of a
data set. One of the greatest advantages when using wavelets, is that the detail data is
never lost, and the user can re-insert it to the original data set by merely applying the
wavelet algorithm in reverse. This opens up a wonderful application for wavelets, namely
that an existent data set can be edited by inserting detail components into it that were
never there, by also using such a wavelet algorithm. In the recent book by Chui and De Villiers (see [2]), algorithms for both subdivision and wavelet applications were developed
without using Fourier analysis as foundation, as have been done by researchers in earlier
years and which have left such algorithms unaccessible to end users such as computer
programmers. The fundamental result of Chapter 9 on wavelets of [2] was that feasibility
of wavelet decomposition is equivalent to the solvability of a certain set of identities
consisting of Laurent polynomials, referred to as Bezout identities, and it was shown how
such a system of identities can be solved in a systematic way. The work in [2] was done in
the univariate case only, and it will be the purpose of this thesis to develop similar results
in the bivariate case, where such a generalization is entirely non-trivial. After introducing
MRA in Chapter 1, as well as discussing the re nability of functions and introducing box
splines as prototype examples of functions that are re nable in the bivariate setting, our
fundamental result will also be that wavelet decomposition is equivalent to solving a set
of Bezout identities; this will be shown rigorously in Chapter 2. In Chapter 3, we give
a set of Laurent polynomials of shortest possible length satisfying the system of Bezout
identities in Chapter 2, for the particular case of the Courant hat function, which will
have been introduced as a linear box spline in Chapter 1. In Chapter 4, we investigate
an application of our result in Chapter 3 to bivariate interpolatory subdivision. With the
view to establish a general class of wavelets corresponding to the Courant hat function,
we proceed in the subsequent Chapters 5 { 8 to develop a general theory for solving the
Bezout identities of Chapter 2 separately, before suggesting strategies for reconciling these
solution classes in order to be a simultaneous solution of the system. / AFRIKAAANSE OPSOMMING: Multi-resolusie analise (MRA) het in die afgelope twee dekades toenemende gewildheid
geniet as 'n veld in wiskundige wetenskappe. Nie net is dit 'n area wat ryklik toepaslik
is nie, maar dit bevat ook steeds vele oop vraagstukke. MRA bou op die grondleggings
van verfynbare funksies en poog om deur vlakke van data-komponente te sorteer, of te
lter, 'n konsep wat aanloklik is in 'n era waar die ontwikkeling van digitale toestelle
(om maar 'n enkele voorbeeld te noem) sodanig moet wees dat meer en meer inligting
vasgel^e en gestoor moet word. Behalwe vir die ontwerp van digitale voorwerpe, soos
animasie- lms, word MRA ook toegepas in 'n mees vername navorsingsgebied genaamd
inverwing, waar \verlore" data (soos byvoorbeeld in 'n foto) herwin word deur data te
neem uit aangrensende gebiede en dit dan oor die le e data-dele te \smeer." Twee hooftakke
in toepassing van MRA is subdivisie en gol e-analise. Die eerste gebruik verfynbare
funksies om algoritmes te ontwikkel waarmee digitale krommes ontwerp kan word vanuit 'n
eindige aantal aanvanklike gegewe punte. Die verkrygde krommes (of sketse) kan voldoen
aan verlangde vlakke van gladheid (of verlangde grade van kontinue afgeleides, wiskundig
gesproke). Gol es word op hul beurt gebruik om lters te bou waarmee gewensde dataof
geraas-komponente verwyder kan word uit datastelle. Een van die grootste voordeel
van die gebruik van gol es bo ander soortgelyke instrumente om data lters mee te bou,
is dat die geraas-komponente wat uitgetrek word nooit verlore gaan nie, sodat die proses
omkeerbaar is deurdat die gebruiker die sodanige geraas-komponente in die groter datastel
kan terugbou deur die gol e-algoritme in trurat toe te pas. Hierdie eienskap van gol fies open 'n wonderlike toepassingsmoontlikheid daarvoor, naamlik dat 'n bestaande datastel
verander kan word deur data-komponente daartoe te voeg wat nooit daarin was nie,
deur so 'n gol e-algoritme te gebruik. In die onlangse boek deur Chui and De Villiers
(sien [2]) is algoritmes ontwikkel vir die toepassing van subdivisie sowel as gol es, sonder
om staat te maak op die grondlegging van Fourier-analise, soos wat die gebruik was in
vroe ere navorsing en waardeur algoritmes wat ontwikkel is minder e ektief was vir eindgebruikers.
Die fundamentele resultaat oor gol es in Hoofstuk 9 in [2], verduidelik hoe
suksesvolle gol e-ontbinding ekwivalent is aan die oplosbaarheid van 'n sekere versameling
van identiteite bestaande uit Laurent-polinome, bekend as Bezout-identiteite, en dit is
bewys hoedat sodanige stelsels van identiteite opgelos kan word in 'n sistematiese proses.
Die werk in [2] is gedoen in die eenveranderlike geval, en dit is die doelwit van hierdie
tesis om soortgelyke resultate te ontwikkel in die tweeveranderlike geval, waar sodanige
veralgemening absoluut nie-triviaal is. Nadat 'n inleiding tot MRA in Hoofstuk 1 aangebied
word, terwyl die verfynbaarheid van funksies, met boks-latfunksies as prototipes van
verfynbare funksies in die tweeveranderlike geval, bespreek word, word ons fundamentele
resultaat gegee en bewys in Hoofstuk 2, naamlik dat gol e-ontbinding in die tweeveranderlike
geval ook ekwivalent is aan die oplos van 'n sekere stelsel van Bezout-identiteite. In
Hoofstuk 3 word 'n versameling van Laurent-polinome van korste moontlike lengte gegee
as illustrasie van 'n oplossing van 'n sodanige stelsel van Bezout-identiteite in Hoofstuk 2,
vir die besondere geval van die Courant hoedfunksie, wat in Hoofstuk 1 gede nieer word.
In Hoofstuk 4 ondersoek ons 'n toepassing van die resultaat in Hoofstuk 3 tot tweeveranderlike
interpolerende subdivisie. Met die oog op die ontwikkeling van 'n algemene klas
van gol es verwant aan die Courant hoedfunksie, brei ons vervolglik in Hoofstukke 5 {
8 'n algemene teorie uit om die oplossing van die stelsel van Bezout-identiteite te ondersoek,
elke identiteit apart, waarna ons moontlike strategie e voorstel vir die versoening van
hierdie klasse van gelyktydige oplossings van die Bezout stelsel.
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Identidades polinomiais para o produto tensorial de PI-álgebras. / Polynomial identities for the tensor product of PI-algebras.GALVÃO, Israel Burití. 05 August 2018 (has links)
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Previous issue date: 2012-03 / CNPq / Nesta dissertação foi feita uma abordagem sobre identidades polinomiais para o produto
tensorial de duas álgebras. Com base no crescimento da sequência de codimensões
de uma PI-álgebra, estudado inicialmente por Regev em 1972, apresentamos uma prova
de que o produto tensorial de duas PI-álgebras é ainda uma PI-álgebra. Depois, através
do produto de Kronecker de caracteres e do clássico Teorema do Gancho de Amitsur e
Regev, obtemos relações entre as codimensões e os cocaracteres de duas PI-álgebras e
as codimensões e cocaracteres do seu produto tensorial. Também através do estudo de
codimensões e cocaracteres, conseguimos exibir identidades polinomiais para o produto
tensorial. / In this dissertation we study polynomial identities for the tensor product of two algebras.
Based on the growth of the PI-algebra’s codimensions sequence, originally studied
by Regev in 1972, we present a proof that the tensor product of two PI-algebras is still
a PI-algebra. After this, using the Kronecker product of characters and the classic
Amitsur and Regev Hook Theorem, we obtained relations between the codimensions
and cocharacters of two PI-algebras and the codimensions and cocharacters of their
tensor product. With the study of codimensions and cocharacters, we also exhibit
polynomial identities for the tensor product.
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Identidades polinomiais para álgebras e matrizes triangulares superiores em blocos. / Polynomial identities for upper algebras and triangular arrays in blocks.ARAÚJO, Laise Dias Alves. 13 August 2018 (has links)
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Previous issue date: 2017-06 / Capes / Nesta dissertação estudamos as graduações elementares (ou boas graduações) e
as identidades polinomiais graduadas correspondentes em álgebras de matrizes triangulares superiores em blocos. Uma graduação elementar por um grupo G na álgebra
A = UT(α1, α2, ..., αr) de matrizes triangulares superiores em blocos é determinada por
uma n-upla em Gn, onde n = α1+· · ·+αr. Mostraremos que as graduações elementares
em A determinadas por duas n-uplas em Gnsão isomorfas se, e somente se, as n-uplas
estão na mesma órbita da bi-ação canônica em Gn com o grupo Sα1 × · · · × Sαr agindo
à esquerda e G à direita. Em seguida utilizamos estes resultados para mostrar que, sob
certas hipóteses (por exemplo, se o grupo G tem ordem prima), duas álgebras de matrizes
triangulares superiores em blocos, graduadas pelo grupo G, satisfazem as mesmas
identidades graduadas se, e somente se, são isomorfas (como álgebras graduadas). / In this dissertation we study elementary (or good) gradings in upper block triangular
matrix algebras and the corresponding graded polynomial identities. An elementary
grading by a group G on the algebra A = UT(α1, α2, ..., αr) of upper block triangular matrices is determined by an n-tuple in Gn, where n = α1 + · · · + αr. It will
be proved that the elementary gradings on A determined by two n-tuples in Gn are
isomorphic if and only if the n-tuples are in the same orbit in the canonical bi-action
on Gn with the group Sα1 × · · · × Sαr acting on the left and the group G acting on the
right. These results will be used to prove that under suitable hypothesis (for example
if the group G has prime order) two upper block triangular matrix algebras, graded by
the group G, satisfy the same graded identities if and only if they are isomorphic (as
graded algebras).
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O teorema do gancho e aplicações. / The hook theorem and applications.ROCHA, Josefa Itailma da. 02 August 2018 (has links)
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Previous issue date: 2011-12 / Capes / Neste trabalho usamos a Teoria de Young para representações dos grupos simétricos
no estudo de PI-álgebras. Amitai Regev (1972) introduziu os conceitos de codimensão
e cocaracter de uma PI-álgebra, os quais foram as principais ferramentas desse estudo.
Apresentamos inicialmente o Teorema do Gancho, que foi demonstrado por Amitsur
e Regev em 1982. Esse teorema refere-se ao comportamento da sequência de cocaracteres de uma PI-álgebra, dando condições para que um caracter irredutível do grupo Sn apare¸ca com multiplicidade n˜ao nula na decomposição do n-ésimo cocaracteres dessa PI-álgebra. Apresentamos também três aplicações desse teorema, entre elas o Teorema de Amitsur, que garante que toda PI-álgebra satisfaz uma potência de algum polinˆomio standard. Por fim, estudamos resultados de Amitsur e Regev de 1982 sobre um tipo de identidade que generaliza as identidades de Capelli. / In this work we use Young’s Theory for representations of the symmetric groups in
the study of PI-algebras. Amitai Regev (1972) introduced the concepts of codimension
and cocharacter of PI-algebras, which are the main tools in this study. We first present
the Hook Theorem, which was proved by Amitsur and Regev in 1982. This theorem
refers to the behavior of the sequence of cocharacters of a PI-algebra, giving conditions
for an irreducible character of the group Sn to appear with nonzero multiplicity in the
decomposition of the cocharacter of this PI-algebra. We also present three applications
of this theorem, including the Amitsur’s theorem, which ensures that all PI-algebra
satisfies a power of a standard polynomial. Finally, we study the results of Amitsur
and Regev (1982) about a type identity that generalizes the Capelli identities
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Grafos eulerianos e identidades polinomiais na álgebra Mn(K)Gonçalves, Fernanda Scabio 27 August 2013 (has links)
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Previous issue date: 2013-08-27 / Financiadora de Estudos e Projetos / In this work we present some applications of graph theory in problems involving polynomial identities for the algebra Mn (K). A brief presentation of PI-theory and some concepts of graph theory, such as the definition of Eulerian graphs, which are the basic elements of this work, were presented to make the text self- contained. We show two different proofs of the Amitsur-Levitzki theorem, the proof of Razmyslov and other due to Swan's theorem - an important result on Eulerian graphs. Finally, a similar result of the Amitsur-Levitzki's theorem for skew-symmetric matrices is proved using elements of graph theory. We emphasize that the understanding of the technique makes it possible to simplify many results and has been an important tool in the study of PI-algebras. / Neste trabalho apresentamos algumas aplicações de Teoria de Grafos em problemas envolvendo identidades polinomiais para a álgebra das matrizes Mn (K). Uma breve apresentação de PI-teoria e de alguns on eitos de Teoria de Grafos, como a de_- nição de grafos eulerianos, que são os elementos básicos desta abordagem, foram apresentadas para tornar o texto auto contido. São explicitadas duas demonstrações distintas do Teorema de Amitsur-Levitzki, a de Razmyslov e uma de corrente do Teorema de Swan - um resultado importante a respeito de grafos eulerianos. Por _m, um resultado semelhante ao Teorema de Amitsur-Levitzki para matrizes antis- simétricas é demonstrado utilizando elementos de Teoria de Grafos. Ressaltamos que o entendimento da técnica utilizada torna possível a simplificação de diversos resultados e tem se mostrado uma importante ferramenta no estudo de PI-álgebras.
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Identidades e polinômios centrais para álgebras de matrizes. / Identities and central polynomials for matrix algebras.BERNARDO, Leomaques Francisco Silva. 23 July 2018 (has links)
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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.
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Codimensões e cocaracteres de PI-Álgebras. / Codimensions and cocaracteres of PI-Algebras.OLIVEIRA, Antonio Igor Silva de. 27 July 2018 (has links)
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Previous issue date: 2011-09 / Capes / As ideias de codimensões e cocaracteres de uma PI-álgebra são de grande importância
e são centrais nas aplicações das representações dos grupos simétricos à PIteoria
(teoria das identidades polinomiais). Os conceitos de codimensão e cocaracter
começaram a ser estudados em 1972 por Amitai Regev em seu importante trabalho sobre
identidades polinomiais do produto tensorial de PI-álgebras. Ao longo das últimas
décadas muitos resultados importantes surgiram com o uso das representações e dos
métodos assintóticos na PI-teoria. Neste trabalho apresentaremos inicialmente ideias
e resultados básicos da Teoria de Young sobre as representações dos grupos simétricos.
De posse desses resultados, estudaremos as sequências limitadas de codimensões e as
sequências de cocaracteres de álgebras que satisfazem alguma identidade de Capelli.
Apresentaremos também os cálculos das codimensões e dos cocaracteres da álgebra de
Grassmann. / The ideas of codimensions and cocharacters of a PI-algebra are of great and central
importance in the applications of representations of symmetric groups to PI-theory
(theory of the polynomial identities). The study of the concepts of codimensions and
cocharacters started in 1972 by Amitai Regev in his important work about polynomial
identities of the tensor product of PI-algebras. During the last decades many important
results arose with the use of representations and asymptotic methods in PI-theory. In
this work we will present firstly ideas and basic results in the Young’s theory about
the representations of symmetric groups. With these results we shall study the limited
sequences of codimensions and the cocharacter sequences of algebras that satisfy some
of the Capelli identity. It will also be presented the calculation of the codimensions
and cocharacters of the Grassmann Algebra.
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Identidades polinomiais graduadas para álgebras de matrizes. / Graded polynomial identities for matrix algebras.ALVES, Sirlene Trajano. 05 August 2018 (has links)
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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
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