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The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras. (English) Zbl 1037.17011
In this work, the authors start by considering a general triple $R\subseteq Z\subseteq H$ of $$k$$-algebras ($$k$$ an algebraically closed field) where $$H$$ is a Hopf algebra with centre $$Z$$, $$Z$$ being an affine domain, and $$R$$ is a sub-Hopf algebra of $$H$$, contained in $$Z$$, over which $$H$$ (and $$Z$$) are finite modules. The (finite dimensional) representation theory of $$H$$ is studied by studying the finite dimensional $$k$$-algebras $$H/\mathfrak{m}H$$, as $$\mathfrak{m}$$ ranges across the maximal ideal spectrum of $$R$$. The strategy adopted is the following: given a maximal ideal $$\mathfrak{m}$$ of $$R$$, the ramification behaviour of the maximal ideals of $$Z$$ lying over $$\mathfrak{ m }$$ is analysed. It is then investigated how this behaviour interacts with the representation theory of $$H/\mathfrak{m}H$$ and how it varies as $$\mathfrak{m}$$ varies through $$\text{Maxspec}(R)$$.
Once these general results are studied in the above setting, they are applied to the more specific contexts of the enveloping algebras $$U(\mathfrak{g})$$ of semisimple Lie algebras $$\mathfrak{g}$$ in positive characteristic and the quantised enveloping algebras $$U_\varepsilon (\mathfrak{g})$$ of semisimple Lie algebras at a root of unity $$\varepsilon$$. These classes of algebras have the above mentioned structure, and the general results obtained are now interpreted in them. A characterization of the unramified maximal ideals of $$Z$$ lying over $$\mathfrak{m}$$ is given, and this is related to the representation theory of $$H/\mathfrak{m}H$$. The blocks of $$H/\mathfrak{m}H$$ are also described for these classes of algebras.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B35 Universal enveloping (super)algebras 16G99 Representation theory of associative rings and algebras
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