arXiv subjects

Leonid Krop

Publications and source records attributed to Leonid Krop.

7 recordsLinked to original sources

arXiv2015

On classification of finite-dimensional semisimple Hopf algebras

We develop a mechanism for classication of isomorphism types of non-trivial semisimple Hopf algebras whose group of grouplikes $G(H)$ is abelian of prime index $p$ which is the smallest prime divisor of $|G(H)|$. We describe structure of the second cohomology group of extensions of $\k C_p$ by $\k^G$ where $C_p$ is a cyclic group of order $p$ and $G$ a finite abelian group. We carry out an explicit classification for Hopf algebras of this kind of dimension $p^4$ for any odd prime $p$. The ground field is algebraically closed of characteristic $0$.

math.RA↗

arXiv2014

Isomorphism Types of Hopf Algebras in a Class of Abelian Extensions.I

There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which are extensions of the group algebra $\k C_p$ by ${\k}^G$ where $C_p$ is a cyclic group of prime order $p$ and ${\k}^G$ is the Hopf algebra dual of $\k G$, $G$ a finite abelian $p$-group and $\k$ is an algebraically closed field of characteristic $0$. We apply the method to calculate the number of isoclasses of commutative extensions and certain extensions of this kind of dimension $\le p^4$.

math.QA↗

arXiv2009

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.

math.QA↗

arXiv2009

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.

math.QA↗

arXiv2008

Representations of pointed Hopf algebras and their Drinfel'd quantum doubles

We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.

math.QA↗

arXiv2005

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. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.

math.QA↗

arXiv2004

Quantized hyperalgebras of rank 1

We study the algebra $U_ζ$ obtained via Lusztig's `integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra $\frak {g=sl}_2$ modulo the two-sided ideal generated by $K^l-1$. We show that $U_ζ$ is a smash product of the quantum deformation of the restricted universal enveloping algebra $\bold u_ζ$ of $\frak g$ and the ordinary universal enveloping algebra $U$ of $\frak g$, and we compute the primitive (= prime) ideals of $\Uz$. Next we describe a decomposition of $\bold u_ζ$ into the simple $U$- submodules, which leads to an explicit formula for the center and the indecomposable direct summands of $\Uz$. We conclude with a description of the lattice of cofinite ideals of $\Uz$ in terms of a unique set of lattice generators.

math.QA↗