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

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Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras \(U({\mathfrak g})\) of finite dimensional k-Lie algebras \({\mathfrak g}\) in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of \(U({\mathfrak g})\) for \({\mathfrak g}\) semisimple a conjecture of Humphreys [28] on the block structure of \(U({\mathfrak g})\) is confirmed. In the case of \(U_{\epsilon}({\mathfrak g})\) for \({\mathfrak g}\) semisimple and \(\epsilon\) an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of \(U_{\epsilon}({\mathfrak g})\) are determined, and a necessary condition (which may also be sufficient) for a baby Verma \(U_{\epsilon}({\mathfrak g})\)-module to be simple is obtained.

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Received: 24 June 1999; in final form: 30 March 2000 / Published online: 17 May 2001

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Brown, K., Gordon, I. The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras. Math Z 238, 733–779 (2001). https://doi.org/10.1007/s002090100274

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