Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. (English) Zbl 0695.16006

Lusztig, George

J. Am. Math. Soc. 3, No. 1, 257-296 (1990).

Some finite-dimensional Hopf algebras over the field \({\mathcal F}_ p\) (p is prime) play an important role in the theory of modular representations. One can define a finite-dimensional Hopf algebra in terms of an indecomposable positive-definite symmetric Cartan matrix \((a_{ij})\) with \(1\leq i,j\leq n\). The present paper is aimed to link the following problems: (i) Finding the characters of the finite-dimensional simple modules of the algebraic group on \(\bar F_ p\) corresponding to the matrix \((a_{ij})\). (ii) Finding the characters of the finite- dimensional simple modules over the quantum group corresponding to \((a_{ij})\) at \({}^ p\sqrt{1}\).

Reviewer: E.Kryachko

MSC:

16W30	Hopf algebras (associative rings and algebras) (MSC2000)
17B35	Universal enveloping (super)algebras
20G05	Representation theory for linear algebraic groups
17B20	Simple, semisimple, reductive (super)algebras
22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties

Keywords:

finite-dimensional Hopf algebras; indecomposable positive-definite symmetric Cartan matrix; characters; finite-dimensional simple modules; algebraic group; quantum group