This paper gives some su?cient conditions for the commutativity of quasi-toral restricted Lie algebras and characterizes some properties on semisimple quasi-toral restricted Lie algebras.
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＊ ma...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.展开更多
In [21], generalized restricted Lie algebras, defined over a field F of positive characteristic p, were introduced. In this note their cohomology, especially the so-called generalized restricted cohomology is studies....In [21], generalized restricted Lie algebras, defined over a field F of positive characteristic p, were introduced. In this note their cohomology, especially the so-called generalized restricted cohomology is studies. Some reduction properties are obtained. For graded Cartan type Lie algebras the author determines the first Lie-cohomology groups and the first generalized restricted cohomology groups with the coefficients in the highest weight modules from which all irreducible generalized restricted modules are derived.展开更多
In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-alg...In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A（m）-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.展开更多
We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3.We give a sufficient and necessary condition in terms of typical ...We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3.We give a sufficient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple.In the process,we also determine the socle for each restricted Kac module and the length for each simple restricted module.展开更多
基金Project supported by the Youth Science Foundation of Northeast Normal University (No.111494027)and the National Natural Science Foundation of China (No.10271076).
文摘This paper gives some su?cient conditions for the commutativity of quasi-toral restricted Lie algebras and characterizes some properties on semisimple quasi-toral restricted Lie algebras.
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)
文摘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.
文摘In [21], generalized restricted Lie algebras, defined over a field F of positive characteristic p, were introduced. In this note their cohomology, especially the so-called generalized restricted cohomology is studies. Some reduction properties are obtained. For graded Cartan type Lie algebras the author determines the first Lie-cohomology groups and the first generalized restricted cohomology groups with the coefficients in the highest weight modules from which all irreducible generalized restricted modules are derived.
基金Supported by ZJNSF(Grant Nos.LY17A010015 and LZ14A010001)NNSF(Grant No.11171296)
文摘In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A（m）-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.
基金supported in part by the National Natural Science Foundation of China(Grant No.11701158).
文摘We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3.We give a sufficient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple.In the process,we also determine the socle for each restricted Kac module and the length for each simple restricted module.
