Workshop on Quantum Field Theory and Representation Theory
University of São Paulo, August 21-24, 2007
|I. Dimitrov (Queen's, Canada)
Title: Cup product on homogeneous varieties and PRV.
|Abstract: Let G =GL(n,C) and let B ⊂ G
be a Borel
subgroup with corresponding homogeneous variety X =
G/B. If λ is a character of B, we denote by Lλ
corresponding line bundle on X. Given
λ2 we consider the
cup product map
(1) H q1 (X, L λ1) ⊗ H q2 (X, L λ2) → H q1+ q2 (X, L λ1 + λ2).
In this talk I will present necessary and sufficient conditions for (1) to be surjective. It turns out that whenever (1) is surjective, the G-module H q1+ q2 (X, L λ1 + λ2) is either trivial, or a generalized PRV component of the G-module H q1 (X, L λ1) ⊗ H q2 (X, L λ2). However not every generalized PRV component of the tensor product of two G-modules appears in this way. Finally, I will explain what combinatorial problems relate to the properities of (1), e.g., multiplicities of the generalized PRV components, etc. This talk is based on a joint work with Mike Roth.
| D.Gratcharov (San Jose State University, USA)
Title: Modules with bounded multiplicities.
Abstract: Let g be a finite dimensional simple Lie algebra. In this talk we will focus on the category B of all bounded weight g-modules, i.e. those that are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Benkart-Britten-Lemire implies that bounded weight g-modules exist only for g = sl(n) and g= sp(2n). In the second case the category B has enough projectives if and only if n>1 and is wild if and only if n>2. The case g= sl(n) is much more complicated because the subcategory of B consisted of all modules with equal weight multiplicities is never semisimple.
| R. Rocha (UFABC, Brazil)
Title: Clifford algebra-parametrized octonions and generalizations.
|Abstract: Introducing products between multivectors of Cl(0,7) and octonions,resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere, and the XY-product. We also present the formalism necessary to construct Clifford algebra-parametrized octonions. Finally we introduce a method to construct generalized octonionic algebras, where their octonionic units are parametrized by arbitrary Clifford multivectors. The products between Clifford multivectors and octonions, leading to an octonion, are shown to share graded-associative, supersymmetric properties. We also investigate the generalization of Moufang identities, for each one of the products introduced. The X-product equals twice the parallelizing torsion, given by the torsion tensor, and is used to investigate the S7 Kac-Moody algebra. The X-product has also been used to obtain triality maps and G2 actions, and it leads naturally to remarkable geometric and topological properties, for instance the Hopf fibrations and twistor formalism in ten dimensions. The paramount importance of octonions in the search for unification is based, for instance, in the fact that by extending the division-algebra-valued superalgebras to octonions, in D=11 an octonionic generalized Poincare superalgebra can be constructed, the so-called octonionic M-algebra that describes the octonionic M-theory where the octonionic super-2-brane and the octonionic super-5-brane sectors are shown to be equivalent. Also, there are other vast generalizations and applications of the octonionic formalism such as the classification of quaternionic and octonionic spinors and the pseudo-octonionic formalism.|
|D. Vassilevich (IF-USP, Brazil)
Title: Twisted symmetries in noncommutative field theories.
|Abstract: I explain how and why usual Lie algebra
being replaced on noncommutative spaces by Hopf algebra symmetries.