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This dissertation uses techniques from the theory of categorical actions of Kac-Moody algebras to study the analog of the BGG category O for the queer Lie superalgebra. Chen recently reduced many questions about this category to its so-called types A, B, and C blocks. The type A blocks were completely described in joint work with Brundan in terms of the general linear Lie superalgebra. This dissertation proves that the type C blocks admit the structure of a tensor product categorification of the n-fold tensor power of the natural sp_\infty-module. Using this result, we relate the combinatorics for these blocks to Webster’s orthodox bases for the quantum group of type C_\infty, verifying the truth of a recent conjecture of Cheng-Kwon-Wang. This dissertation contains coauthored material.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23683号 / 理博第4773号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 玉川 安騎男, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Neste trabalho encontramos bases para as identidades T Z 32 e T Z 22 gradu- adas dos octônios. Utilizando a base obtida no T Z 22 , re-obtivemos uma base para as identidades Z 2 -graduadas das matrizes dois por dois. Também obti- vemos as identidades simultaneamente fracas e antissimétricas ou skew dos octônios na categorias de álgebras alternativas. Também obtivemos as identi- dades antissimétricas da álgebra de Malcev simples de dimensão sete, sl(O). Para ambos os casos estudados de identidades não graduadas dos octônios, mostramos positivamente a conjectura de Shestakov-Zhukavets: O T -ideal de identidades dos octônios coincide com o da álgebra alternativa quadrá- tica. / In this work we find bases for the T Z 32 and T Z 22 graded identities of the octonion algebra. Using the base obtained in the T Z 22 case, we re-obtain a basis for the Z 2 -graded identities of two by two matrices. We also obtained the simultaneously skew and weak identities of the octonions in the category of alternative algebras. In addition we find a basis of identities for the simple Malcev algebra of dimension seven, sl(O). For both skew cases of identities studied we positively show the Shestakov-Zhukavets conjecture: The T -ideal of identities of the octonions coincides with that of the quadratic alternative algebra.

The superline has one even and one odd coordinate. We consider the Lie superalgebra of contact vector fields on the superline. Its tensor density modules are a one-parameter family of deformations of the natural action on the ring of polynomials on the superline. They are parameterized by a complex number, and they are irreducible when this parameter is not zero. In this dissertation, we describe the annihilating ideals of these representations in the universal enveloping algebra of this Lie superalgebra by providing their generators. We also describe the intersection of all such ideals: the annihilator of the direct sum of the tensor density modules. The annihilating ideal of an irreducible non-zero left module is called a primitive ideal, and the space of all such ideals in the universal enveloping algebra is its primitive spectrum. The primitive spectrum is endowed with the Jacobson topology, which induces a topology on the annihilators of the tensor density modules. We conclude our discussion with a description of the annihilators as a topological space.

<p>This thesis, consisting of three papers and a summary, studies properties of stratified algebras and representations of Lie superalgebras.</p><p>In Paper I we give a characterization when the Ringel dual of an SSS-algebra is properly stratified.</p><p>We show that for an SSS-algebra, whose Ringel dual is properly stratified, there is a (generalized) tilting module which allows one to compute the finitistic dimension of the SSS-algebra, and moreover, it gives rise to a new covariant Ringel-type duality.</p><p>In Paper II we give a characterization of standardly stratified algebras in terms of certain filtrations of (left or right) projective modules, generalizing the corresponding theorem of V. Dlab. We extend the notion of Ringel duality to standardly stratified algebras and estimate their finitistic dimension in terms of endomorphism algebras of standard modules.</p><p>Paper III deals with the queer Lie superalgebra and the corresponding BGG-category O. We show that the typical blocks correspond to standardly stratified algebras, and we generalize Kostant's Theorem to the queer Lie superalgebra.</p>

This thesis, consisting of three papers and a summary, studies properties of stratified algebras and representations of Lie superalgebras. In Paper I we give a characterization when the Ringel dual of an SSS-algebra is properly stratified. We show that for an SSS-algebra, whose Ringel dual is properly stratified, there is a (generalized) tilting module which allows one to compute the finitistic dimension of the SSS-algebra, and moreover, it gives rise to a new covariant Ringel-type duality. In Paper II we give a characterization of standardly stratified algebras in terms of certain filtrations of (left or right) projective modules, generalizing the corresponding theorem of V. Dlab. We extend the notion of Ringel duality to standardly stratified algebras and estimate their finitistic dimension in terms of endomorphism algebras of standard modules. Paper III deals with the queer Lie superalgebra and the corresponding BGG-category O. We show that the typical blocks correspond to standardly stratified algebras, and we generalize Kostant's Theorem to the queer Lie superalgebra.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-23T14:23:04Z No. of bitstreams: 1 RIVALDO DO NASCIMENTO JÚNIOR - DISSERTAÇÃO PPGMAT 2009..pdf: 371424 bytes, checksum: 6e808f19bfcee3712a8cc10f221c042b (MD5) / Made available in DSpace on 2018-07-23T14:23:04Z (GMT). No. of bitstreams: 1 RIVALDO DO NASCIMENTO JÚNIOR - DISSERTAÇÃO PPGMAT 2009..pdf: 371424 bytes, checksum: 6e808f19bfcee3712a8cc10f221c042b (MD5) Previous issue date: 2009-04 / Neste trabalho apresentamos um modelo para a superálgebra das matrizes triangulares superiores e mostraremos como obter o produto de dois T-ideais como núcleo de um homomorfismo de álgebras. em seguida, mostraremos como obter as identidades polinomiais para a álgebra das matrizes triangulares em blocos com Z2-graduação a partir das identidades ordinárias das álgebras de sua diagonal principal. / In this work we present a general model for the superalgebra of upper triangular matrices and show how to obtain the product of two T-ideals as the kernel of a homomorphism between two algebras. Next, we show how to obtain the polynomial identities for algebra of the block-triangular matrices with Z2-grading from the ordinary identities of the algebras of its main diagonal.

Les fonctions génératrices des coefficients de Clebsch Gordan pour la superalgèbre de Lie osp(1|2) sont dérivées en utilisant deux approches. Une première approche généralise une méthode proposée par Granovskii et Zhedanov pour l'appliquer dans le cas de osp(1|2), une algèbre dont le coproduit est torsadé. Une seconde approche repose sur la réalisation de osp(1|2) en tant qu'algèbre dynamique d'un oscillateur parabosonique et utilise une équivalence dans cette réalisation entre le changements de coordonnées polaires à cartésiennes et le problème de Clebsch-Gordan. Un chapitre moins formel précède ces dérivations et présente comment le problème de Clebsch-Gordan s'interprète en tant que réalisation d'une algèbre de fusion. La notion abstraite de fusion est introduite, soulignant son importance en physique, pour en venir au cas particulier du problème de Clebsch-Gordan. Un survol du cas de l'algèbre osp(1|2) et de ses utilisations en physique mathématique conclut ce chapitre. / The generating functions for the osp(1|2) Lie superalgebra Clebsch-Gordan coefficients are derived using two approaches. The first one consists of generalizing a method first proposed by Granovskii and Zhedanov to apply it to the case of osp(1|2), an algebra with a twisted coproduct. The second one is based on the realization of the osp(1|2) as the dynamical algebra for a parabosonic oscillator and used an equivalence in this realization between a change of basis from polar to cartesian coordinates and the Clebsch-Gordan problem. A less formal chapter precedes those derivations and present how the Clebsch-Gordan problem can be interpreted as a realization of a fusion algebra. The abstract notion of fusion is introduced, mentionning its importance in physics, and leads to the particular case of the Clebsch-Gordan problem. A brief review of the problem for the osp(1|2) algebra and its uses in mathematical physics concludes this chapter.