scholarly journals Group gradings on the Lie algebra psln in positive characteristic

2009 ◽  
Vol 213 (9) ◽  
pp. 1739-1749 ◽  
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov
Keyword(s):  
Lie Algebra ◽  
Positive Characteristic ◽  
Group Gradings

scholarly journals The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1032
Author(s):  
Raúl Durán Díaz ◽  
Víctor Gayoso Martínez ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Linear Algebraic Group ◽  
Invariant Functions ◽  
Arbitrary Characteristic ◽  
Independent Invariant ◽  
Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Dual Space Derivations and H2(L, F) of Modular Lie Algebras

1987 ◽  
Vol 39 (5) ◽  
pp. 1078-1106 ◽  
Author(s):  
Rolf Farnsteiner
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Dual Space ◽  
Positive Characteristic ◽  
Base Field ◽  
Central Extensions ◽  
Simple Lie Algebras ◽  
Early Results ◽  
Modular Lie Algebras

It is well-known that the classical vanishing results of the cohomology theory of Lie algebras depend on the characteristic of the underlying base field. The theorems of Cartan and Zassenhaus, for instance, entail that non-modular simple Lie algebras do not admit non-trivial central extensions. In contrast, early results by Block [3] prove that this conclusion loses its validity if the underlying base field has positive characteristic.Central extensions of a given Lie algebra L, or equivalently its second cohomology group H(L, F), can be conveniently described by means of derivations φ:L → L*.

scholarly journals Wall-crossings and a categorification of K-theory stable bases of the Springer resolution

2021 ◽  
Vol 157 (11) ◽  
pp. 2341-2376
Author(s):  
Changjian Su ◽  
Gufang Zhao ◽  
Changlong Zhong
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Quantum Cohomology ◽  
Hecke Algebra ◽  
Verma Modules ◽  
Affine Hecke Algebra ◽  
Wall Crossing ◽  
K Theory ◽  
Monodromy Matrices

Abstract We compare the $K$ -theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$ -theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

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