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

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
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