Lusztig, George Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. (English) Zbl 0695.16006 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 Cited in 7 ReviewsCited in 177 Documents 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 PDF BibTeX XML Cite \textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 1, 257--296 (1990; Zbl 0695.16006) Full Text: DOI OpenURL