Inverse Limits in Representations of a Restricted Lie Algebra  

作　　者：Yu Feng YAO Bin SHU Yi Yang LI 

机构地区：[1]Department of Mathematics,Shanghai Maritime University [2]Department of Mathematics,East China Normal University [3]School of Fundamental Studies,Shanghai University of Engineering Science

出　　处：《Acta Mathematica Sinica,English Series》2012年第12期2463-2474,共12页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China (Grant Nos. 11126062,11201293 and 11271130);the Innovation Program of Shanghai Municipal Education Commission (Grant Nos. 12ZZ038 and 13YZ077)

摘　　要：Let （g, [p]） be a restricted Lie algebra over an algebraically closed field of characteristic p 〉 O. Then the inverse limits of ＂higher＂ reduced enveloping algebras {uxs （g） I s ∈ N} with X running over g＊ make representations of g split into different ＂blocks＂. In this paper, we study such an infinite- dimensional algebra Ax （g） ：=lim Uxs （g） for a given X C g＊. A module category equivalence is built between subcategories of U（g）-rnod and Ax（g）-mod. In the case of reductive Lie algebras, （quasi） generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized x-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.Let （g, [p]） be a restricted Lie algebra over an algebraically closed field of characteristic p 〉 O. Then the inverse limits of ＂higher＂ reduced enveloping algebras {uxs （g） I s ∈ N} with X running over g＊ make representations of g split into different ＂blocks＂. In this paper, we study such an infinite- dimensional algebra Ax （g） ：=lim Uxs （g） for a given X C g＊. A module category equivalence is built between subcategories of U（g）-rnod and Ax（g）-mod. In the case of reductive Lie algebras, （quasi） generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized x-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.

关 键 词：Restricted Lie algebra reductive Lie algebra inverse limit projective module standard Levi form 

分 类 号：O152.5[理学—数学] O189.11[理学—基础数学]