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تسجيل مستخدم جديدTwisting of affine algebraic groups, II
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تأليف
Shlomo Gelaki
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We use cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_Jmathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $mathbb{C}$. In particular, we show that ${}_Jmathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_Jmathcal{O}(G)_{J}cong U(g)$ as algebras, where $g={rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_Jmathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $Hsubset G$ of $J$. Finally, we work out several examples to illustrate our results.
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اقرأ أيضاً
We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first s
how that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2-cocycles for connected nilpotent G, hence of fiber functors Rep(G)to Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G)_J for arbitrary G.
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