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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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