Leonid Krop

On the representations of the full matrix semigroup on homogeneous polynomials

This paper is the successor of [ 11. In the closing theorem there we determined the composition factors of w” for a regular I [ 1, 6.11. Here we are going to describe the structure of WA in the singular case. The second theme of this paper is the criterion for distributivity of the M -modules WCd’ Since the necessary conditions for that established in [ 1: 8.51 limit d to’d <p( p + 1) we just have to sort out the latter case. This will be done in the third section of this paper. For the convenience of the reader we recall some notation, terms, and definitions used in [ 11.

REPRESENTATION THEORY OF LIFTINGS OF QUANTUM PLANES

We systematically determine the regular representations, quivers and representation type of all liftings of two-dimensional quantum linear spaces.

Injective comodules for 2 × 2 quantum matrices

We study the injective comodules for the coordinate bialgebra of quantum 2 × 2 matrices at a root of unity. The related GL 2 and SL 2 theories are also specified.

Quantized hyperalgebras of rank 1

We study the algebraUζ obtained via Lusztig’s ‘integral’ form [Lu 1, 2] of the generic quantum algebra for the Lie algebrag = sl2 modulo the two-sided ideal generated byKl − 1. We show thatUζ is a smash product of the quantum deformation of the restricted universal enveloping algebraU ofg and the ordinary universal enveloping algebraU ofg, and we compute the primitive (= prime) ideals ofUζ. Next we describe a decomposition ofuζ into the simpleU-submodules, which leads to an explicit formula for the center and the indecomposable direct summands ofUζ. We conclude with a description of the lattice of cofinite ideals ofUζ in terms of a unique set of lattice generators.

Trace-Like Invariant for Representations of Nilpotent Liftings of Quantum Planes

We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module.

Spectra of quantized hyperalgebras

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflos Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztigs tensor product theorem.

The Second Indicator and the Trace of the Antipode for Representations of a Class of Hopf Algebras of Andruskiewitch-Schneider Type

We give definition of the second indicator for any finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0 with the square of antipode an inner automorphism. We derive a closed formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent type liftings of quantum linear spaces and compute the value of their second indicators. Uniqueness of indicators is examined.

Two-Sided Ideals of Some Finite-Dimensional Algebras

We set up the framework for discussing general properties of the lattice and the semigroup of ideals of a finite-dimensional algebra. We work out the examples u(sl 2), uς(sl 2) and kSL 2 (F p ) explicitly. Algebras with distributive lattice and commutative semigroup of ideals are classified.