Filtered multiplicative bases of restricted enveloping algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p

Differentiating the Weyl generic dimension formula and support varieties for quantum groups

The authors compute the support varieties of all irreducible modules for the small quantum group $u_\zeta(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $\zeta$ is a primitive $\ell$-th root of unity with $\ell$ larger than the Coxeter number of $\mathfrak{g}$. The calculation employs the prior calculations and techniques of Ostrik and of Nakano--Parshall--Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel $G_1$ of a reductive algebraic group scheme $G$ defined over the prime field $\mathbb{F}_p$.Comment: 10 pages, various typos corrected, references update

Representations of Lie colour algebras

The main goal of this paper is to lay the foundation for studying the representations of (restricted) Lie colour algebras and relate this to its structure and properties of their (restricted) universal enveloping algebras. Since there are not too many results in the literature, the paper starts more or less from the beginning but does not always give all the details. It also concentrates on questions related to complete sets and tensor products and the development of the arguments quite often is motivated by the ungraded case. In the first section several results on the Jacobson colour ideal are provided which are well-known in the ungraded case. Moreover, a colour version of a result of Steinberg which shows the significance of complete sets of modules is established. Here it is not necessary that the grading is commutative. In general, it was tried to adapt the assumptions on the grading to the investigated topics. The second section provides some abstract results on relating tensor products to being invariant under the comultiplication. The third and the fifth section are devoted to some elementary results on the structure and representation theory of Lie colour algebras resp. restricted Lie colour algebras. Both sections are followed by a first of complete sets. In particular, a theorem of Burnside is generalized to restricted Lie colour algebras and some applications to the blocks of supersolvable restricted Lie colour algebras are given. Finally, in the last section some of the methods are applied to characterize p-reductive restricted Lie colour algebras in several ways as well as to characterize finite-dimensional restricted Lie colour algebras whose semisimple restricted modules are closed under tensor products. (orig.)SIGLEAvailable from TIB Hannover: RR 9398(79) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman