Caracteres e cohomologia de módulos para álgebras de Kac-Moody afim e generalizações

Author(s)

Tiago Rodrigues Macedo

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2013-07-06

Examining board members

Adriano Adrega de Moura; Viktor Bekkert; Dessislava Hristova Kochloukova; Reimundo Heluani; Vyacheslav Futorny

• Advisor: Adriano Adrega de Moura; Daniel Ken Nakano

Abstract

In this thesis we consider two main problems. The first problem concerns extensions between simple modules for current algebras associated to complex, simple, finite-dimensional Lie algebras. To begin, we compute 1-extensions between finite-dimensional simple modules, partially recovering a result due to Kodera. Then we develop a technique aimed to compute higher extensions, and which we use to compute 2-extensions between certain simple modules. Finally we prove that cohomology groups of current algebras are isomorphic to the cohomology groups of its underlying simple Lie algebra, a result stated by Feigin. This part of the thesis arises from collaboration with B. Boe, C. Drupieski and D. Nakano. The second problem is concerned with the study of certain classes of modules for hyper algebras of current algebras. In the case that the underlying Lie algebra is simply laced, we show that local Weyl modules are isomorphic to certain Demazure modules, extending to positive characteristic a result due to Fourier-Littelmann. More generally, we extend a result of Naoi by proving that local Weyl modules admit a Demazure flag, i.e., a filtration with factors isomorphic to Demazure modules. Using this, we prove a conjecture of Jakelic-Moura stating that the character of local Weyl modules for hyper loop algebras are independent of the (algebraically closed) ground field (AU)

• FAPESP's process: 09/05887-8 - Characters and cohomology of modules for affine Kac-Moody algebras and generalizations
• Grantee: Tiago Rodrigues Macedo
• Support Opportunities: Scholarships in Brazil - Doctorate