PRESENTATION BY BOREL SUBALGEBRAS AND CHEVALLEY GENERATORS FOR QUANTUM ENVELOPING ALGEBRAS

Proceedings of the Edinburgh Mathematical Society

Fabio Gavarini¹ 

关键词: Primary 17B37;    20G42;    Secondary 81R50;    quantum groups;    L-operators;    quantum root vectors;   

DOI  :  10.1017/S0013091504000689

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

We provide an alternative approach to the Faddeev–Reshetikhin–Takhtajan presentation of the quantum group $uqg$, with $L$-operators as generators and relations ruled by an $R$-matrix. We look at $uqg$ as being generated by the quantum Borel subalgebras $U_q(mathfrak{b}_+)$ and $U_q(mathfrak{b}_-)$, and use the standard presentation of the latter as quantum function algebras. When $mathfrak{g}=mathfrak{gl}_n$, these Borel quantum function algebras are generated by the entries of a triangular $q$-matrix. Thus, eventually, $U_q(mathfrak{gl}_n)$ is generated by the entries of an upper triangular and a lower triangular $q$-matrix, which share the same diagonal. The same elements generate over $Bbbk[q,q^{-1}]$ the unrestricted $Bbbk [q,q^{-1}]$-integral form of $U_q(mathfrak{gl}_n)$ of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for $mathfrak{g}=mathfrak{sl}_n$ too.

【 授权许可】

Unknown