Academic literature on the topic 'Dimension de Gelfand-Kirillov'

A dimensão de Gelfand-Kirillov mede a taxa de crescimento assintótico de álgebras. Como fornece informações importantes sobre a sua estrutura, este invariante se tornou uma das ferramentas padrão no estudo de álgebras de dimensão infinita. Neste trabalho apresentamos as propriedades básicas da dimensão de Gelfand-Kirillov de álgebras e de módulos, e também mostramos o cálculo da dimensão de Gelfand-Kirillov de algumas álgebras e módulos, sendo o exemplo mais importante o cálculo da dimensão de Gelfand-Kirillov da álgebra de Weyl An.<br>The Gelfand-Kirillov dimension measures the asymptotic rate of growth of algebras. Since it provides important structural information, this invariant has become one of the standard tools in the study of innite dimensional algebras. In this work we present the basic properties of the Gelfand-Kirillov dimension of algebras and modules, and we also show the calculation of the Gelfand-Kirillov dimension of some algebras and modules, being the most important example the calculation of the Gelfand-Kirillov dimension of the Weyl algebra An.

Orientador: Plamen Emilov Kochloukov<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica<br>Made available in DSpace on 2018-08-25T04:30:00Z (GMT). No. of bitstreams: 1 Machado_GustavoGrings_D.pdf: 808427 bytes, checksum: 4482c43f5d1998040317e1873220ce8c (MD5) Previous issue date: 2014<br>Resumo: Neste trabalho estudamos o invariante denominado dimensão de Gelfand-Kirillov para álgebras com identidades polinomiais, sobretudo para álgebras não-associativas, com o objetivo de melhor compreender a estrutura das identidades polinomiais. Ultimamente este invariante tem ganhado importância, uma vez que ele é relativamente fácil de calcular e, de certa forma, é capaz de diferenciar o crescimento de duas álgebras. Para álgebras associativas a GK-dimensão mostrou-se muito útil ao detectar que álgebras que por um lado são PI-equivalentes sobre corpos de característica zero pelo Teorema do Produto Tensorial de Kemer, por outro lado não são PI-equivalentes quando a característica do corpo infinito é positiva. Isto aponta para o surgimento de novos ????-ideais, conjuntos de identidades satisfeitas por uma álgebra, que são ???? -primos para corpos infinitos de característica positiva. Ainda é um problema em aberto a classificação e a compreensão destes ????-ideais em característica positiva, embora seja bem compreendida para PI-Álgebras associativas em característica zero, segundo a teoria de Kemer. Entretanto a situação é ainda menos clara para variedades de álgebras não-associativas como Álgebras de Jordan ou Álgebras de Lie. Sabe-se muito pouco sobre resultados que apontem para uma classificação de ????-ideais fora do caso associativo, até mesmo sobre corpos de característica zero. Inclusive se conhece pouco sobre o comportamento dos ????-ideais, mesmo de álgebras simples. Aqui damos um passo, calculando algumas GK-dimensões para álgebras relativamente livres de posto finito a partir da expressão da série de Hilbert. Destacamos em especial que calculamos a dimensão de Gelfand-Kirillov da álgebra relativamente livre de qualquer posto finito da álgebra de Lie das matrizes 2 × 2 de traço zero sobre um corpo infinito de característica diferente de 2. Acreditamos que estes resultados permitirão ajudar a compreender melhor o comportamento dos ????-ideais em álgebras não-associativas<br>Abstract: In this thesis we study the invariant called Gelfand-Kirillov Dimension for algebras with polynomial identities, mainly for non-associative algebras, aiming at better understanding the structure of the polynomial identities. This invariant has gained importance lately since in many cases it is relatively easy to calculate and, surprisingly, it is capable of distinguishing the growth of two algebras. For associative algebras GK-dimension was found to be very useful to detect that algebras which on one hand are PI-equivalent over fields of characteristic zero, according to Tensor Product Theorem of Kemer, on the other hand are not PI-equivalent when the characteristic of the infinite base field is positive. This points towards the rise of new ????-ideals, sets of identities satisfied by an algebra, which are ????-prime for infinite fields of positive characteristic. The classification and the understanding of such ????-ideals in positive characteristic are still open problems, although it is well understood for associative PI-Algebras in characteristic zero, using Kemer¿s theory. The situation is much less clear for varieties of non-associative algebras like Jordan Algebras or Lie Algebras. Very little is known about results towards a classification of ????-ideals outside the associative case, even over fields of characteristic zero. Accordingly little is known concerning the behavior of ????-ideals, even for simple algebras. Here we make a step towards this goal by computing some GK-dimensions of some relatively free algebras of finite rank by using the expression of the Hilbert series. In particular we compute the Gelfand-Kirillov dimension of the relatively free algebra of any finite rank generated by the Lie Algebra of the 2 × 2 traceless matrices over an infinite field of characteristic different from 2. We hope that results in this direction will contribute to a better understanding of the behavior of ????-ideals in non-associative algebras<br>Doutorado<br>Matematica<br>Doutor em Matemática

Let K be the field of complex numbers. In this thesis we construct new examples of noncommutative surfaces of GK-dimension 4 using the language of formal and infinitesimal deformations as introduced by Gerstenhaber. Our approach is to find families of deformations of a certain well known GK-dimension 4 birationally commutative surface defined by Zhang and Smith in unpublished work cited in [YZ06], which we call A. Let B* and K* be respectively the bar and Koszul complexes of a PBW algebra C = KhV / (R) . We construct a graph whose vertices are elements of the free algebra KhV i and edges are relations in R. We define a map m2 : B2 ! K2 that extends to a chain map m* : B* → K*. This map allows the Gerstenhaber bracket structure to be transferred from the bar complex to the Koszul complex. In particular, m2 provides a mechanism for algorithmically determining the set of infinitesimal deformations with vanishing primary obstruction. Using the computer algebra package 'Sage' [Dev15] and a Python package developed by the author [Cam], we calculate the degree 2 component of the second Hochschild cohomology of A. Furthermore, using the map m2 we describe the variety U ⊆ HH2/2 (A) of infinitesimal deformations with vanishing primary obstruction. We further show that U decomposes as a union of 3 irreducible subvarieties Vg, Vq and Vu. More generally, let C be a Koszul algebra with relations R, and let E be a localisation of C at some (left and right) Ore set. Since R is homogeneous in degree two, there is an embedding R ,↪ C⊗C and in the following we identify R with its (nonzero) image under this map. We construct an injective linear map ~⋀ : HH²(C) → HH²(E) and prove that if f ∈ HH²(E) satisfies f(R) ⊆ C then f ∈ Im(~⋀). In this way we describe a relationship between infinitesimal deformations of C with those of E. Rogalski and Sierra [RS12] have previously examined a family of deformations of A arising from automorphism of the surface P1 X P1. By applying our understanding of the map ~⋀ we show that these deformations correspond to the variety of infinitesimal deformations Vg. Furthermore, we show that deformations defined similarly by automorphisms of other minimal rational surfaces also correspond to infinitesimal deformations lying in Vg. We introduce a new family of deformations of A, which we call Aq. We show that elements of this family have families of deformations arising from certain quantum analogues of geometric automorphisms of minimal rational surfaces, as defined by Alev and Dumas. Furthermore, we show that after taking the semi-classical limit q → 1 we obtain a family of deformations of A whose infinitesimal deformation lies in Vq. Finally, we apply a heuristic search method in the space of Hochschild 2-cocycles of A. This search yields another new family of deformations of A. We show that elements of this family are non-noetherian PBW noncommutative surfaces with GK-dimension 4. We further show that elements of this family can have as function skew field the division ring of the quantum plane Kq(u; v), the division ring of the first Weyl algebra D1(K) or the commutative field K(u; v).

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-22T14:49:45Z No. of bitstreams: 1 CARLOS DAVID DE CARVALHO LOBÃO - DISSERTAÇÃO PPGMAT 2009..pdf: 418073 bytes, checksum: b2deb42599e396408cd91ddf1721d8eb (MD5)<br>Made available in DSpace on 2018-07-22T14:49:45Z (GMT). No. of bitstreams: 1 CARLOS DAVID DE CARVALHO LOBÃO - DISSERTAÇÃO PPGMAT 2009..pdf: 418073 bytes, checksum: b2deb42599e396408cd91ddf1721d8eb (MD5) Previous issue date: 2009-03<br>As álgebras verbalmente primas são bem conhecidas em característica 0. Já sobre corpos de característica p > 2 pouco sabemos sobre elas. Apresentamos modelos genéricos e calcularemos a dimensão de Gelfand-kirillov para as álgebras E⊗E, Aa,b, Ma,b(E)⊗E e Ma,b(E)⊗E. Como consequência, obteremos a prova de não PI-equivalência entre álgebras importantes para PI-Teoria em características positiva.<br>The verbally prime algebras are well understood in characteristic 0 while over a field of characteristic p > 2 little is known about them. In this work we discuss some sharp differents between these two generics cases for the characteristc. We exhibit constructions of generic models. By using these models we compute the Gelfand-Kirillov dimension of the relatively free algebras of rank m in the varieties generated by E⊗E, Aa,b, Ma,b(E)⊗E e Ma,b(E)⊗E. As consequence we obtain the PI non equivalence of important algebras for the PI theory in positive characteristic.

Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings. This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5). In Chapter 1, we introduce the background and the goals of the thesis. In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp. In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras. In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm. Finally, in Chapter 5, we summarize our results and propose possible future work.

Cette thèse est consacrée au programme de Langlands modulo p pour GL₂. Dans la première partie, j'étudie la dimension de Gelfand-Kirillov des représentations π provenant de la cohomologie modulo p des courbes de Shimura. Soit p un nombre premier et F un corps de nombres totalement réel non ramifié en des places divisant p. Soit r une représentation modulo p modulaire de dimension 2 du groupe de Galois de F qui satisfait l'hypothèse de Taylor-Wiles et quelques hypothèses techniques de généricité. Pour v une place fixée de F divisant p, on montre que de nombreuses représentations modulo p lisses admissibles de GL₂(Fᵥ) associées à r dans les espaces propres de Hecke correspondants de la cohomologie modulo p ont une dimension de Gelfand-Kirillov [Fᵥ:Qp]. Ceci s'appuie sur et étend les travaux de Breuil-Herzig-Hu-Morra-Schraen et de Hu-Wang, en donnant une preuve unifiée pour tous les cas (r semisimple ou non à v). Dans la deuxième partie, j'étudie les (phi,Gamma)-modules étales D_A(π) associés aux représentations π provenant de la cohomologie modulo p des courbes de Shimura. Soit K une extension finie non ramifiée de Qp. Pour π une représentation modulo p lisse admissible de GL₂(K) satisfaisant certaines propriétés de multiplicité un, je calcule le rang du (phi,Gamma)-module étale D_A(π) associé défini par Breuil-Herzig-Hu-Morra-Schraen, ce qui étend leurs résultats. Dans la troisième partie, j'étudie les propriétés de compatibilité local-global des (phi,Gamma)-modules étales D_A(π). Pour ⍴ une représentation modulo p réductible quelconque de dimension 2 du groupe de Galois de K, je calcule explicitement le (phi,Gamma)-module étale D_A(⍴) défini par Breuil-Herzig-Hu-Morra-Schraen. Ensuite, soit π une représentation modulo p lisse admissible de GL₂(K) apparaissant dans certains espaces propres de Hecke de la cohomologie modulo p et ⍴ sa représentation modulo p sous-jacente de dimension 2 du groupe de Galois de K. En supposant que ⍴ est maximalement non-scindée, je montre sous certaines hypothèses de généricité que le (phi,Gamma)-module étale D_A(π) défini par Breuil-Herzig-Hu-Morra-Schraen est isomorphe à D_A(⍴). Ceci étend leurs résultats, où ⍴ était supposé semisimple<br>This thesis is devoted to the mod p Langlands program for GL₂. In the first part, I study the Gelfand-Kirillov dimension of the representations π coming from the mod p cohomology of Shimura curves. Let p be a prime number and F a totally real number field unramified at places above p. Let r be a two-dimensional modular mod p representation of the Galois group of F that satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For v a fixed place of F above p, we prove that many of the admissible smooth mod p representations of GL₂(Fᵥ) associated to r in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand-Kirillov dimension [Fᵥ:Qp]. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen and Hu-Wang, giving a unified proof in all cases (r either semisimple or not at v). In the second part, I study the étale (phi,Gamma)-modules D_A(π) associated to the representations π coming from the mod p cohomology of Shimura curves and compute there ranks. Let K be a finite unramified extension of Qp. For π an admissible smooth mod p representation of GL₂(K) satisfying certain multiplicity-one properties, we compute the rank of the associatedétale étale (phi,Gamma)-modules D_A(π) defined by Breuil-Herzig-Hu-Morra-Schraen, extending their results. In the third part, I study the local-global compatibility properties of the étale (phi,Gamma)-modules D_A(π). For ⍴ any reducible two-dimensional mod p representation of the Galois group of K, we compute explicitly the associated étale (phi,Gamma)-module D_A(⍴) defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let π be an admissible smooth mod p representation of GL₂(K) occurring in some Hecke eigenspaces of the mod p cohomology and ⍴ be its underlying two-dimensional mod p representation of the Galois group of K. Assuming that ⍴ is maximally non-split, we prove under some genericity assumption that the associated étale (phi,Gamma)-module D_A(π) defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to D_A(⍴). This extends their results, where ⍴ was assumed to be semisimple

Soient g une k-algèbre de Lie, U(g) son algèbre enveloppante, K(g) le corps des fractions de U(g). L'objet de cette thèse est d'étudier des propriétés algébriques du corps gauche K(g) dans les deux cas suivants : d'une part si k est de caractéristique 0 et g est de dimension infinie ; d'autre part si k est de caractéristique positive et g est de dimension finie.<br /><br />On suppose k de caractéristique nulle. On définit d'abord la notion de "degré de transcendance de niveau q" pour les algèbres de Poisson. Cette notion est introduite à partir de la notion de dimension de niveau q définie par V. Pétrogradsky pour les algèbres associatives et les algèbres de Lie. On démontre, sous des hypothèses peu restrictives sur g, que le degré de transcendance de niveau q+1 de K(g) est égal à la dimension de niveau q de g.<br /><br />On s'attache ensuite à l'étude de la famille des algèbres de type Witt définies par R. Yu. On construit ainsi des familles infinies de corps gauches deux à deux non isomorphes mais de même degré de transcendance de niveau 3 donné. On étudie aussi la question des centralisateurs dans les corps enveloppants des parties positives des algèbres de type Witt. On établit en particulier le résultat suivant : il existe des algèbres de Lie non commutatives de dimension infinie g telles que le premier corps de Weyl ne se plonge pas dans K(g).<br /><br />Supposons maintenant k de caractéristique p>0. On étudie le cas particuliers des algèbres de Lie suivantes : les algèbres gl(n) ; les algèbres sl(n) lorsque p ne divise pas n ; l'algèbre de Witt modulaire W(1) et une sous-algèbre P de l'algèbre de Witt W(2) (s'identifiant à un produit tensoriel de l'algèbre de Lie W(1) avec une algèbre associative de polynômes tronqués). Dans tous les cas, on démontre que le corps enveloppant est isomorphe à un corps de Weyl. Pour les algèbres W(1) et P, on démontre en outre que le centre de l'algèbre enveloppante est un anneau factoriel, en accord avec une conjecture récente de A. Braun et C. Hajarnavis.

Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.<br>Esta tesis es un aporte a la clasificación de las álgebras de Hopf punteadas de dimensión de Gelfand-Kirillov finita sobre cuerpos algebraicamente cerrados y de característica cero. En una primera instancia nos concentramos en álgebras de Hopf punteadas de dimensión finita sobre grupos no abelianos y cuya trenza infinitesimal no es simple. En este contexto, estudiamos un espacio vectorial trenzado particular que puede realizarse como módulo de Yetter-Drinfeld sobre una familia de grupos no abelianos y que da lugar a un álgebra de Nichols de dimensión finita. Con el objetivo de clasificar las álgebras de Hopf punteadas que tiene esta trenza infinitesimal, seguimos los pasos propuestos por el método del levante. Encontramos una presentación minimal del álgebra de Nichols, crucial para demostrar la validez de la conjetura de generación en grado 1 en nuestro contexto. Introducimos un álgebra de pre-Nichols distinguida que tiene dimensión de Gelfand-Kirillov 2 y es una extensión del álgebra de Nichols por una subálgebra de Hopf trenzada normal. Finalmente describimos todas las álgebras de Hopf punteadas de dimensión finita cuya trenza infinitesimal es la trenza en cuestión; mas aún probamos que todas ellas son deformaciones por cociclo de la correspondiente bosonización del álgebra de Nichols. En la segunda parte de esta tesis consideramos dos familias de espacios vectoriales trenzados de tipo diagonal: los de tipo Cartan y los que tienen diagrama de Dynkin completamente disconexo. El objetivo es determinar, para cada una de estas trenzas, todas las álgebras de pre-Nichols de dimensión de Gelfand-Kirillov finita. Para ello introducimos la noción de álgebras de pre-Nichols eminentes. Mostramos que, salvo algunas excepciones, las álgebras de pre-Nichols distinguidas son eminentes. Este tratamiento se asienta en el conocimiento de las relaciones que definen, en cada caso, al álgebra de pre-Nichols distinguida, y las excepciones están relacionadas con fenómenos propios de los casos en los que intervienen raíces de la unidad de orden pequeño. Para dos de los casos excepcionales mencionados anteriormente, construimos álgebras de pre-Nichols eminentes que cubren propiamente a las correspondientes distinguidas.<br>In a first instance we focus on finite dimensional pointed Hopf algebras over non-abelian groups and with non-simple infinitesimal braiding.In this context we study a fixed braided vector space that can be realized as Yetter-Drinfeld module over a family of non-abelian groups, and it gives rise to a finite dimensional Nichols algebra. With the purpose of classifying finite dimensional pointed Hopf algebras with this fixed infinitesimal braiding, we follow the steps proposed by the Lifting method.We find a minimal presentation by generators and relations of the Nichols algebras, which will be crucial in our proof of the validity of the generation in degree one in this context. We introduce a distinguished pre-Nichols algebra, which has Gelfand-Kirillov dimension 2 and can be obtained as an extension of the Nichols algebra by a braided normal Hopf subalgebra. Finally, we describe all finite dimensional pointed Hopf algebras with this infinitesimal braiding, furthermore we show that all of them are cocycle deformations of the corresponding bosonization of the Nichols algebra. In the second part of this thesis we consider two families of braided vector spaces of diagonal type, namely: those of Cartan type and those with totally disconnected Dynkin diagram. The goal is to determine, for each of these braidings, all their pre-Nichols algebras with finite Gelfand-Kirillov dimension. With this purpose we introduce the notion of eminent pre-Nichols algebra. We show that, up to some exceptions, the distinguished pre-Nichols algebras are in fact eminent. This treatment is based on the knowledge of the defining relations of the distinguished pre-Nichols algebra, and the exceptions are related to particular phenomena that arise when roots of unity of small orders are involved. Eminent pre-Nichols algebras are constructed for two of the aforementioned exceptional cases.<br>Fil: Sanmarco, Guillermo Luis. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.<br>Fil: Sanmarco, Guillermo Luis. Consejo Nacional de Investigaciones Científicas y Técnicas - Universidad Nacional de Córdoba. Centro de Investigaciones y Estudios de Matemática; Argentina