When computing the homology of Grassmannian manifolds, the first step is usually to look at the Schubert cell decomposition, and the chain complex associated with it. In the complex case and the real unoriented case with Z₂ coefficients the additive structure is obtained immediately (i.e., generated by the homology classes represented by the Schubert cells) because the boundary map is trivial. In the real unoriented case (with Z₂ coefficients) and the real oriented case, finding the additive structure is more complicated since the boundary map is nontrivial. In this paper, this boundary map is computed by cell orientation comparisons, using graph coordinates where the cells are linear, to simplify the comparisons. The integral homology groups for some low dimensional oriented and unoriented Grassmannians are determined directly from the chain complex (with the boundary map as computed). The integral cohomology ring structure for complex Grassmannians has been completely determined mainly using Schubert cell intersections (what is known as Schubert Calculus).. In this paper, a method using Schubert cell intersections to describe the Z₂ cohomology ring structure of the real Grassmannians is sketched. The results are identical to those for the complex Grassmannians (with coefficients), but the notation used for the cohomology generators is not the usual one. It indicates that the products are to a certain degree independent of the Grassmannian. / Science, Faculty of / Mathematics, Department of / Graduate

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In this thesis we study problems associated with the generalisation, to include Grassmann type variables, of the 'group theoretical' approach to quantisation of C.Isham [37]. Although a full generalisation of this quantisation scheme is not achieved, consideration of this problem leads us to make studies in four principle sectors: (A) Graded Poisson brackets and graded 'vector field like’ constructs. A graded version of the Hamiltonian vector field is defined and it is found that both left acting and right acting vector fields are necessary. Properties of these vector fields are investigated. (B) Local graded canonical transformations and graded function groups. Simple examples of these structures are studied. (C) The realisation of a general superalgebra by the use of graded 'functions' and the graded Poisson bracket. The graded generalisation of a standard classical result is presented. Also the - question of central, extensions to these algebras is studied and a partial generalisation of a classical result on this is given. (D) Investigations into a model of quantum mechanics on a2-sphere which incorporates fermions. This model is similarto that derived by Spiegelglas [56] and Barcelos-Neto et al.[6,7]from the 0(3) supersymmetrie sigma model first studied by Witten in [62,63], except that an additional primary constraint has been included. The graded Dirac brackets of this model are calculated.

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2016. / Submitted by Fernanda Percia França (fernandafranca@bce.unb.br) on 2016-08-09T17:27:27Z No. of bitstreams: 1 2016_BrunoTrindadeReis.pdf: 1080010 bytes, checksum: 32ebe9d37c7e36818b3f476daa917c2c (MD5) / Approved for entry into archive by Raquel Viana(raquelviana@bce.unb.br) on 2016-09-05T17:57:47Z (GMT) No. of bitstreams: 1 2016_BrunoTrindadeReis.pdf: 1080010 bytes, checksum: 32ebe9d37c7e36818b3f476daa917c2c (MD5) / Made available in DSpace on 2016-09-05T17:57:47Z (GMT). No. of bitstreams: 1 2016_BrunoTrindadeReis.pdf: 1080010 bytes, checksum: 32ebe9d37c7e36818b3f476daa917c2c (MD5) / Seja K um corpo de característica p >2. Sejam H a álgebra de Grassmannnão-unitária de dimensão infinita e Hna álgebra de Grassmann não unitária de um espaço vetorial de dimensão finita n, ambas sobre K. Seja A =K<Y,Z>/T2(H) a álgebra relativamente livre Z2-graduada da variedade de K-álgebras associativas Z2-graduadas não-unitárias determinada por H. Seja D = K<X>/T(H) a álgebra relativamente livre da variedade de álgebras associativas não-unitárias (sem graduação) determinada por H. Nesse trabalho construímos um mergulho de A em H, que determina um mergulho de D em H. Isso nos permite dar demonstrações simples e unificadas de resultados sobre identidades polinomiais e polinômios centrais de H e Hnobtidos anteriormente por vários autores. Os resultados obtidos também são válidos se K é um domínio de integridade de característica p >2. Estudamos também a álgebra de Grassmann unitária E de dimensão infinita sobre um corpo finito. Seja K um corpo finito e K1<X>a álgebra associativa livre unitária, livremente gerada por X. Damos uma representação de K1<X>/T (E) como produto tensorial da álgebra comutativa A = K[T]/I, onde I é o ideal de K[T] gerado por tq-t, e a álgebra B = K1<Y>/V , onde V é o T-ideal de K<Y>(ou seja, da álgebra associativa livre não-unitária) gerado por y1p e pelo comutador triplo [y1, y2;,y3]. Essa representação nos permite dar uma demonstração mais simples do resultado de Bekh-Ochire Rankin sobre uma base de identidades polinomiais de E sobre um corpo finito. ________________________________________________________________________________________________ ABSTRACT / Let K be a field of characteristic p >2. Let H be the infinite dimensional non-unitary Grassmann algebra and Hnthe non-unitary Grassmann algebra of a vector space of dimension n, both over K. Let A = K<Y,Z>/T2(H) be the Z2-graded relatively free algebra of the variety of Z2-graded non-unitary associative algebras determined by H. Let D = K<X>/T (H) be the relatively free algebra of the variety of non-unitary associative algebras (without grading) determined by H. In this work we construct an embedding of Ain H, determining an embedding of D in H. This allows us to give simple and unified proofs of results about polynomial identities and central polynomials of H e Hnobtained previously by several authors. The results obtained are also valid if K is an integral domain of characteristic p >2. We study also the infinite dimensional unitary Grassmann algebra E over a finite field. Let K be a finite field and K1<X>the unitary associative free algebra, freely generated by X. We give a representation of K1<X>/T (E) as a tensor product of the commutative algebra A = K[T]/I, where I is the ideal of K[T] generated by tq- t, and the algebra B = K1<Y>/V , where V is the T-ideal of K<Y>(that is, of the free associative non-unitary algebra) generated by y1pand [y1, y2,y3]. This representation allows us to give a simple proof of the result of Bekh-Ochir and Rankin on a basis of the polynomial identities of E over a finite field.

Orientador: Lucio Centrone / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T01:00:17Z (GMT). No. of bitstreams: 1 Manuel_AlexSandroFaria_M.pdf: 1576341 bytes, checksum: 9b70b637adbe03cbed89830126b23521 (MD5) Previous issue date: 2014 / Resumo: Esta dissertação foi escrita com a intenção de conter os seus principais pré-requisitos. Assim, inicialmente, recordaremos algumas definições básicas e alguns resultados da álgebra clássica. Então, listaremos alguns resultados clássicos da teoria de PI-álgebras, bem como alguns resultados sobre codimensões e série de Hilbert. Este último nos dará ferramentas para descrever, pelo menos parcialmente, as identidades polinomiais da álgebra de Grassmann em característica positiva (principalmente a álgebra de Grassmann unitária). No entanto, muitos dos resultados podem funcionar em característica zero. Levaremos em consideração dois casos: no primeiro, o corpo base será considerado infinito (de acordo com um artigo escrito por Giambruno e Koshlukov) enquanto que, no segundo, consideraremos que o corpo base seja finito (de acordo com um artigo escrito por Regev) / Abstract: This dissertation was written with the intent of containing its main prerequisites. So, initially, we will recall some basic definitions and some results from classical algebra. Then we will list some classical results of the theory of PI-algebras as well as the ones about codimensions and Hilbert series. The latter will give us tools to describe, at least partially, the polynomial identities of the Grassmann algebra in positive characteristic (mainly the unitary Grassmann algebra). Nevertheless, many of the results may work in characteristic zero too. We will take in consideration two cases: in the first one the ground field will be considered infinite (according to a paper written by Giambruno and Koshlukov) while in the second one we will consider the ground field to be finite (according to a paper written by Regev) / Mestrado / Matematica / Mestre em Matemática

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Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2006. / Includes bibliographical references (leaves 57-59). Also available in electronic version.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2014. / Submitted by Larissa Stefane Vieira Rodrigues (larissarodrigues@bce.unb.br) on 2014-10-14T17:49:06Z No. of bitstreams: 0 / Rejected by Raquel Viana(raquelviana@bce.unb.br), reason: Não há dados nesse item. on 2014-10-14T20:09:22Z (GMT) / Submitted by Larissa Stefane Vieira Rodrigues (larissarodrigues@bce.unb.br) on 2014-10-20T17:32:28Z No. of bitstreams: 1 2014_GabrielSilvaCarvalho.pdf: 876012 bytes, checksum: 75afdd25af1405ef5a56ecd6a6edf9c4 (MD5) / Approved for entry into archive by Tania Milca Carvalho Malheiros(tania@bce.unb.br) on 2014-10-20T18:02:49Z (GMT) No. of bitstreams: 1 2014_GabrielSilvaCarvalho.pdf: 876012 bytes, checksum: 75afdd25af1405ef5a56ecd6a6edf9c4 (MD5) / Made available in DSpace on 2014-10-20T18:02:49Z (GMT). No. of bitstreams: 1 2014_GabrielSilvaCarvalho.pdf: 876012 bytes, checksum: 75afdd25af1405ef5a56ecd6a6edf9c4 (MD5) / Neste trabalho introduzimos as noçoes básicas do estudo de PI-álgebras. Descrevemos um sistema de geradores das identidades polinomiais graduadas das álgebras do tipo Ma (E) ® Mg (E), em que E e a ílgebra de Grassmann e a e 3 são funcoes que induzem uma Z2-graduaçao sobre E. Apresentamos uma forma alternativa para a prova de uma das PI-equivalôencias do Teorema de Kemer. Apresentamos resultados que relacionam as identidades graduadas das algebras A e A ® E. Como resultado mostramos a PI-equivalencia entre M2(E) e Mi,i(E) ® E, um caso particular do Teorema de Kemer. _______________________________________________________________________________________ ABSTRACT / In this work we introduce the basics of the studies of PI-algebras. We describe a system of generators of graded polynomial identities of algebras of type Ma(E) ® Mg (E), where E is the Grassmann algebra and a e 3 are maps that induce a Z2- gradings. We show an alternative proof of some of the PI-equivalences of kemer’s theorem. We present results that relate the graded identities of the algebras A and A ® E. As a result, we show the PI-equivalence of M2(E) and M11(E) ® E, a particular case of Kemer’s Theorems.

In this thesis we introduce and study the (i) Grassmannian, (ii) Lagrangian Grassmannian, and (iii) double Lagrangian Grassmannian of subspaces in ( A &otimes; B )n, where A and B are normed division algebras, i.e. R,C,H or O . / This gives a simple and uniform description of all symmetric spaces. This is analogous to Tits magic square description for simple Lie algebras. / We show that every irreducible compact Riemannian symmetric space X must be one of these Grassmannian spaces (up to a finite cover) or a compact simple Lie group. Furthermore, its noncompact dual symmetric space is the open sub-manifold of X consisting of spacelike linear subspaces, at least in the classical cases. / Huang, Yongdong. / "July 2007." / Adviser: Naichung Conan Leung. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0353. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 64-65). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

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