In this thesis, we firstly extend elements and periodicity properties of the theta function theory to functions that represent a wider domain of symmetries and properties, graded with different amounts of p ≥ 1, p ϵ N. Unlike theta functions, these generalised, "higher-level Appell functions" K(_p) satisfy open quasiperiodicity relations, with additive theta function terms emerging as violating terms of open quasiperiodic K(_p)’s. We evaluate the S and T modular transformations of these functions and show that the S-transform of K(_p) does not just give back K(_p), but also includes p additional 0-functions which are precisely those violating the quasiperiodicity of Appell functions. This sets a new pattern of modular group representations on functions that are not double quasiperiodic. While calculating the S-transform of K(_p), a newly arising function, namely ɸ(T, μ) will be also thoroughly analysed. As two interesting applications, we firstly study the modular group action on unitary and on an admissible class of non-unitary N = 2 characters which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. Secondly we continue this study for the admissible representation of the affine Lie superalgebra sl (2|l). We see in the final result for both cases that the functions A(T, V) are the "violating" terms of unitary calculations. We lastly confirm all our results by some sets of consistency checks including an essential residue calculation. We believe this new way of using Appell functions, could be used for any other algebraic structure whose characters can be rewritten in terms of higher-level Appell functions.

Zugl.: Darmstadt, Techn. University, Diss., 2008.

Hamburg, Univ., Diss., 2001. / Computerdatei im Fernzugriff.

Hamburg, Univ., Diss., 2001. / Computerdatei im Fernzugriff.

Hamburg, Universiẗat, Diss., 1999. / Dateiformat: zip, Dateien im PDF-Format.

Hamburg, Universiẗat, Diss., 2001.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-graduação em Matemática e Computação Científica / Made available in DSpace on 2012-10-24T17:26:52Z (GMT). No. of bitstreams: 1 264144.pdf: 489637 bytes, checksum: 8c0af27f0aa3c157bf281eec26ac4b59 (MD5) / Neste trabalho investigam-se os aspectos combinatoriais e algébricos da identidade de Sherman no caso genérico. Obtêm-se fórmulas para o cálculo do número de classes de equivalência de caminhos fechados não periódicos sobre o grafo onde a identidade está definida e, com base nelas, uma nova prova da identidade _e obtida. Ademais, as possíveis relações da identidade com as _álgebras de Lie são elucidadas. Neste contexto, prova-se que a identidade de Sherman é uma conseqüência da identidade de Witt generalizada de uma álgebra de Lie.

On considère la structure du Vect(R)-module sur les espaces des opérateurs différentiels bilinéaires agissant sur les espaces de densités. On calcule la première cohomologie différentielle des champs de vecteurs d’algèbre de Lie Vect(R) avec des coefficients dans l’espace des opérateurs différentiels bilinéaires agissant sur les densités pondérées.On considère l’action de Vect(S1) par la dérivée de Lie sur les espaces d’opérateurs pseudodifférentiels DO. On étudie les déformations h-triviales de l’intégration standard de l’algèbre de Lie Vect(S1) de champs de vecteurs lisses sur le cercle, dans l’algèbre de Lie de fonctions sur le fibré cotangent T*S1. On classe les déformations de cette action qui deviennent triviales une fois limitées à h où h = aff(1) ou sl(2). Les conditions nécessaires et suffisantes pour l’intégrabilité des déformations infinitésimales sont données. / We consider the Vect(R)-module structure on the spaces of bilinear differential operators acting on the spaces of weighted densities.We compute the first differential cohomology of the vector fields Lie algebra Vect(R) with coefficients in space of bilinear differential operators acting on weighted densities. we consider the action of Vect(S1) by Lie derivative on the spaces of pseudodifferential operators . We study the h-trivial deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T∗S1. We classify the deformations of this action that become trivial once restricted to h, where h = aff(1) or sl(2). Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.