14 research outputs found
Index and stable linear forms of Lie algebras
We characterize, in a purely algebraic manner, certain linear forms, called
stable, on a Lie algebra. As an application, we determine the index of a Borel
subalgebra of a semi-simple Lie algebra. Finally, we give an example of a
parabolic subalgebra of a semi-simple Lie algebra which does not admit any
stable linear form.Comment: This paper is written in Frenc
Automorphic properties of low energy string amplitudes in various dimensions
This paper explores the moduli-dependent coefficients of higher derivative
interactions that appear in the low-energy expansion of the four-graviton
amplitude of maximally supersymmetric string theory compactified on a d-torus.
These automorphic functions are determined for terms up to order D^6R^4 and
various values of d by imposing a variety of consistency conditions. They
satisfy Laplace eigenvalue equations with or without source terms, whose
solutions are given in terms of Eisenstein series, or more general automorphic
functions, for certain parabolic subgroups of the relevant U-duality groups.
The ultraviolet divergences of the corresponding supergravity field theory
limits are encoded in various logarithms, although the string theory
expressions are finite. This analysis includes intriguing representations of
SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and
two-loop string and supergravity amplitudes.Comment: 80 pages. 1 figure. v2:Typos corrected, footnotes amended and small
clarifications. v3: minor corrections. Version to appear in Phys Rev
Catenarity in quantum nilpotent algebras
In this paper, it is established that quantum nilpotent algebras (also known
as CGL extensions) are catenary, i.e., all saturated chains of inclusions of
prime ideals between any two given prime ideals have the same
length. This is achieved by proving that the prime spectra of these algebras
have normal separation, and then establishing the mild homological conditions
necessary to apply a result of Lenagan and the first author. The work also
recovers the Tauvel height formula for quantum nilpotent algebras, a result
that was first obtained by Lenagan and the authors through a different
approach.Comment: 11 page