کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6414774 | 1630515 | 2014 | 30 صفحه PDF | دانلود رایگان |
Let Q be a simple algebraic group of type A or C over a field of good positive characteristic. Let xâq=Lie(Q) and consider the centraliser qx={yâq:[xy]=0}. We show that the invariant algebra S(qx)qx is generated by the pth power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial [17] and we show that it remains so after reduction. Using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra, where the p-centre plays the role of the pth power subalgebra. In Zassenhausʼ foundational work [30], the invariant theory and representation theory of modular Lie algebras were shown to be explicitly intertwined. We exploit his theory to give a precise upper bound for the dimensions of simple qx-modules. An application to the geometry of the Zassenhaus variety is given.When g is of type A and g=kâp is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent eâk the invariant algebra S(pe)ke is generated by the pth power subalgebra and S(pe)Ke which is also shown to be polynomial.
Journal: Journal of Algebra - Volume 399, 1 February 2014, Pages 1021-1050