Nesta tese descrevemos as sub algebras coideais a direita que contêm todos elementos group-like dos grupos quânticos multiparâmetro U+ q (g) e Uq(g), onde g e uma algebra de Lie simples de tipo G2, no caso em que o parâmetro principal de quantização q não e raiz da unidade. Se q possui ordem multiplicativa nita t, t > 4, t (diferente) 6, a mesma classicação vale para as sub algebras coideais a direita homogêneas da versão multiparâmetro do grupo quântico de Lusztig uq(g). / In this thesis we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum groups U+ q (g) and Uq(g), where g is a simple Lie algebra of type G2, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is nite, t > 4, t (diferent) 6, then the same classi cation remains valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum group uq(g).

Nesta tese descrevemos as sub algebras coideais a direita que contêm todos elementos group-like dos grupos quânticos multiparâmetro U+ q (g) e Uq(g), onde g e uma algebra de Lie simples de tipo G2, no caso em que o parâmetro principal de quantização q não e raiz da unidade. Se q possui ordem multiplicativa nita t, t > 4, t (diferente) 6, a mesma classicação vale para as sub algebras coideais a direita homogêneas da versão multiparâmetro do grupo quântico de Lusztig uq(g). / In this thesis we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum groups U+ q (g) and Uq(g), where g is a simple Lie algebra of type G2, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is nite, t > 4, t (diferent) 6, then the same classi cation remains valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum group uq(g).

Nesta tese descrevemos as sub algebras coideais a direita que contêm todos elementos group-like dos grupos quânticos multiparâmetro U+ q (g) e Uq(g), onde g e uma algebra de Lie simples de tipo G2, no caso em que o parâmetro principal de quantização q não e raiz da unidade. Se q possui ordem multiplicativa nita t, t > 4, t (diferente) 6, a mesma classicação vale para as sub algebras coideais a direita homogêneas da versão multiparâmetro do grupo quântico de Lusztig uq(g). / In this thesis we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum groups U+ q (g) and Uq(g), where g is a simple Lie algebra of type G2, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is nite, t > 4, t (diferent) 6, then the same classi cation remains valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum group uq(g).

Let g be a Lie algebra. The subalgebra classification problem is to create a list of all subalgebras of g up to equivalence. The purpose of this thesis is to provide a software toolkit within the Differential Geometry package of Maple for classifying subalgebras of In particular the thesis will focus on classifying those subalgebras which are isomorphic to the Lie algebra sl(2) and those subalgebras of which have a basis aligned with the root space decomposition (regular subalgebras).

Motivated by A. Z. Petrov's classification of four-dimensional Lorentzian metrics, we provide an algebraic classification of the isometry-isotropy pairs of four-dimensional pseudo-Riemannian metrics admitting local slices with five-dimensional isometries contained in the Lorentz algebra. A purely Lie algebraic approach is applied with emphasis on the use of Lie theoretic invariants to distinguish invariant algebra-subalgebra pairs. This method yields an algorithm for identifying isometry-isotropy pairs subject to the aforementioned constraints.

No description available.

This dissertation is divided into two parts. The first part addresses the following problem: Suppose 𝑣 is a finitely additive probability measure defined on the power set 𝒜 of the integer Z so that each singleton set gets measure zero. Let X be a product space Π/β∈B * Zᵦ where each Zₐ is a copy of the integers. Let 𝒜ᴮ be the algebra of subsets of X generated by the subproducts Π/β∈B * Cᵦ where for all but finitely many β, Cᵦ = Zᵦ. Let 𝑣_B denote the product measure on 𝒜ᴮ which has each factor measure a copy of 𝑣. A subset E of X is said to be 𝑣_B -measurable iff [sic] there is only one finitely additive probability on the algebra generated by 𝒜ᴮ ∪ [E] which extends 𝑣_B. The set E ⊆ X is said to be universally product measurable (u.p.m.) iff [sic] for each finitely additive probability measure μ on 𝒜 which gives each singleton measure zero,E is μ_B -measurable. Two theorems are proved along with generalizations. The second part of this dissertation gives a proof of the following theorem and some generalizations: There are 2ᶜ nonisomorphic subalgebras of the power set algebra of the integers (where c = power of the continuum).

The differential geometry software package in Maple has the necessary tools and commands to automate the classification process for complex simple Lie algebras. The purpose of this thesis is to write the programs to complete the classification for real simple Lie algebras. This classification is difficult because the Cartan subalgebras are not all conjugate as they are in the complex case. For the process of the real classification, one must first identify a maximally noncompact Cartan subalgebra. The process of the Cayley transform is used to find this specific Cartan subalgebra. This Cartan subalgebra is used to find the simple roots for the given real simple Lie algebra. With this information, we can then create a Satake diagram. Then we match our given algebra's Satake diagram to a Satake diagram of a known algebra. The programs explained in this thesis complete this process of classification.

Nesse trabalho introduzimos as subálgebras de Mishchenko-Fomenko. Apresentamos o problema de Vinberg e a solução de Feigin, Frenkel e Toledano-Laredo em Feigin, Frenkel e Toledano-Laredo (2010) Também é mostrada a solução para as álgebras de Lie de tipo A apresentada em Futorny e Molev (2015). É estudado também o artigo Molev (2013) onde são apresentados geradores do centro de Feigin-Frenkel para as álgebras de Lie de tipo B, C e D. Também são introduzidas as subálgebras de Gelfand-Tsetlin, subálgebras das álgebras envolventes universais das álgebras de Lie de tipo A. Apresentamos uma definição de súbálgebra de Gelfand-Tsetlin para as álgebras de Lie de tipo C, introduzida em Molev e Yakimova (2017). São exibidas as variedades de Gelfand-Tsetlin de $\\mathfrak_$ e $\\mathfrak_$, sendo provado que a variedade de Gelfand-Tsetlin de $\\mathfrak_$ é equidimensional de dimensão 4. Também é demonstrado um novo resultado sobre a equidimensionalidade de $\\mathfrak_$. / In this dissertation, we introduce the Mishchenko-Fomenko subalgebras. We show Vinberg\'s problem and the solution given by Feigin, Frenkel and Toledano-Laredo in Feigin, Frenkel and Toledano-Laredo (2010). We also show a solution for Lie algebras of type A found in Futorny and Molev (2015). We study the article Molev (2013) where generators for the Feigin-Frenkel center are shown for Lie algebras of type B, C and D. We introduce the Gelfand-Tsetlin subalgebras, which are subalgebras of the universal enveloping algebras of Lie algebras of type A. We show a definition of Gelfand-Tsetlin for Lie algebras of type C, introduced in Molev and Yakimova (2017). We exhibit the Gelfand-Tsetlin varieties related to $\\mathfrak_$ and $\\mathfrak_$. We prove that the Gelfand-Tsetlin variety for $\\mathfrak_$ is equidimensional of dimension 4 and we prove a new result about the equidimensionality of $\\mathfrak_$.

<p>Definisane su neke klase polugrupa koje su uop&scaron;tenja tzv. antiinverznih polugrupa. Za neke op&scaron;te klase polugrupa nađene su baze u smislu Ljapina. Dati su identiteti algebri koje se potapaju u polumreže.</p> / <p>Some new classes of semigroups are defined which are generalizations of so called antiinverse semigroups. The Lyapin bases of some well known classes of semigroups are determined. The identities of the variety of subalgebras of semilattices are determined.</p>