Indecomposable finite-dimensional representations of a class of Lie algebras and Lie superalgebras

In the article at hand, we sketch how, by utilizing nilpotency to its fullest extent (Engel, Super Engel) while using methods from the theory of universal enveloping algebras, a complete description of the indecomposable representations may be reached. In practice, the combinatorics is still formidable, though. It turns out that the method applies to both a class of ordinary Lie algebras and to a similar class of Lie superalgebras. Besides some examples, due to the level of complexity we will only describe a few precise results. One of these is a complete classification of which ideals can occur in the enveloping algebra of the translation subgroup of the Poincar\'e group. Equivalently, this determines all indecomposable representations with a single, 1-dimensional source. Another result is the construction of an infinite-dimensional family of inequivalent representations already in dimension 12. This is much lower than the 24-dimensional representations which were thought to be the lowest possible. The complexity increases considerably, though yet in a manageable fashion, in the supersymmetric setting. Besides a few examples, only a subclass of ideals of the enveloping algebra of the super Poincar\'e algebra will be determined in the present article.Comment: LaTeX 14 page

Logarithmic correction to BH entropy as Noether charge

We consider the role of the type-A trace anomaly in static black hole solutions to semiclassical Einstein equation in four dimensions. Via Wald's Noether charge formalism, we compute the contribution to the entropy coming from the anomaly induced effective action and unveil a logarithmic correction to the Bekenstein-Hawking area law. The corrected entropy is given by a seemingly universal formula involving the coefficient of the type-A trace anomaly, the Euler characteristic of the horizon and the value at the horizon of the solution to the uniformization problem for Q-curvature. Two instances are examined in detail: Schwarzschild and a four-dimensional massless topological black hole. We also find agreement with the logarithmic correction due to one-loop contribution of conformal fields in the Schwarzschild background.Comment: 14 pages, JHEP styl