The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras.

*(English)*Zbl 1037.17011In 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.

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.

Reviewer: Joaquín Jódar Reyes (Jaen)

##### 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 |