Inverse Limits in Representations of a Restricted Lie Algebra

• Published in: Acta mathematica Sinica. English series Vol. 28; no. 12; pp. 2463 - 2474
• Main Authors: Yao, Yu Feng, Shu, Bin, Li, Yi Yang
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.12.2012; Springer Nature B.V
• Subjects: Algebra; Categories; Decomposition; Inverse; Lie groups; Mathematical analysis; Mathematics; Mathematics and Statistics; Modules; Reciprocity; Representations; Running; Studies; Verma模; 代数闭域; 包络代数; 李代数; 标准形式; 范畴等价; 逆极限; 限制李超代数; inverse limit; reductive Lie algebra; Restricted Lie algebra; projective module; 17B20; 17B10; standard Levi form; 17B50
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-012-0665-3

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.