SU\G(𝔸)/K·U

Restricted representations of classical Lie algebras of types 𝐴₂ and 𝐵₂

Bart Braden

Type: Article

Publication Date: 1967-01-01

Citations: 30

DOI: https://doi.org/10.1090/s0002-9904-1967-11788-9

Abstract

The dimensions of the finite dimensional irreducible restricted modules for a Lie algebra of classical type have never been determined.C. W. Curtis ([5], [6]) has given sufficient, but not necessary, conditions that the dimension be given by Weyl's formula.In this paper, we give results which determine the dimensions of a certain class of finite dimensional irreducible restricted modules for a simple algebra of type A 2 or B 2 over a field of characteristic p>3.Counterexamples to a conjecture of N. Jacobson regarding the complete reducibility of certain modules for an algebra of classical type are given.Our method involves rather lengthy, though elementary, calculations (given in detail in [3]), and depends on a character formula for algebras of types A 2 or B 2 over the complex field, which is due to J. P. Antoine ([l], [2]).R. Steinberg has mentioned that the results for A 2 were obtained by Mark ([9]).

Locations

  • Bulletin of the American Mathematical Society - View - PDF

Similar Works

Action Title Year Authors
+ PDF Chat The restricted simple Lie algebras with a two-dimensional Cartan subalgebra 1979 Richard E. Block
Robert Lee Wilson
+ Representations of restricted Lie algebras and families of associative ℒ-algebras 1999 Alexander Premet
Serge Skryabin
+ PDF Chat A non-restricted counterexample to the first Kac–Weisfeiler conjecture 2016 Lewis Topley
+ Some Problems in the Representation Theory of Simple Modular Lie Algebras 2015 Georgia Benkart
Jörg Feldvoss
+ PDF Chat Some Problems in the Representation Theory of Simple Modular Lie Algebras 2015 Georgia Benkart
Jörg Feldvoss
+ A non-restricted counterexample to the first Kac-Weisfeiler conjecture 2016 Lewis Topley
+ A non-restricted counterexample to the first Kac-Weisfeiler conjecture 2016 Lewis Topley
+ PDF Chat Representations of restricted Lie algebras of characteristic 𝑝 1954 G. Hochschild
+ Dimensions of the Simple Restricted Modules for the Restricted Contact Lie Algebra 1994 Randall R. Holmes
+ The restricted simple Lie algebras are of classical or Cartan type 1984 Richard E. Block
Robert Lee Wilson
+ Restricted representations of classical lie algebras of prime characteristics 1973 Kwok Chi Wong
+ PDF Chat The radical of the restricted universal enveloping algebra of $A_1$ 1983 Kwok Chi Wong
+ PDF Chat Weyl groups for non-classical restricted Lie algebras and the Chevalley Restriction Theorem 2012 Jean-Marie Bois
Rolf Farnsteiner
Bin Shu
+ NILPOTENT VARIETIES OF SOME FINITE DIMENSIONAL RESTRICTED LIE ALGEBRAS 2019 Cong Chen
+ Nilpotent varieties of some finite dimensional restricted Lie algebras 2019 Cong Chen
+ Varieties of elementary subalgebras of submaximal rank in type A 2017 Yang Pan
+ PDF Chat Finite $W$-Superalgebras and Dimensional Lower Bounds for the Representations of Basic Lie Superalgebras 2017 Yang Zeng
Bin Shu
+ Classifying representations of finite classical groups of Lie type of dimension up to $\ell^4$ 2024 Luis Gutiérrez Frez
Adrián Zenteno
+ Simple Modules for the Restricted Cartan Type Lie Algebras 2006 Randall R. Holmes
Chaowen Zhang
+ PDF Chat On representations of restricted Lie superalgebras 2014 Yu-Feng Yao

Works That Cite This (28)

Action Title Year Authors
+ On the modular representations of the general linear and symmetric groups 1974 Roger W. Carter
G. Lusztig
+ On the third cohomology of algebraic groups of rank two in positive characteristic 2014 Askar Dzhumadil’daev
Sh. Sh. Ibraev
+ PDF Chat Introduction to a Quantum Theory over a Galois Field 2010 Felix M. Lev
+ On the block structure of regular unipotent elements from subsystem subgroups of type A 1 × A 2 in representations of the special linear group 2012 И. Д. Супруненко
+ Some Weyl modules and cohomology for algebraic groups 2020 Sh. Sh. Ibraev
+ PDF Chat Nonsplit extensions of modular Lie algebras of rank 2 2002 Askar Dzhumadil’daev
Sh. Sh. Ibraev
+ PDF Chat Modular representations of classical Lie algebras 1970 James E. Humphreys
+ Representations of 𝔰𝔩3 in characteristic 3 2016 Xin Wen
+ The Block Structure of the Images of Regular Unipotent Elements from Subsystem Symplectic Subgroups of Rank 2 in Irreducible Representations of Symplectic Groups. I 2020 Т. С. Бусел
И. Д. Супруненко
+ Some computations of cartan invariants for finite groups of lie type 1973 James E. Humphreys

Works Cited by This (10)

Action Title Year Authors
+ PDF Chat On the dimensions of the irreducible modules of Lie algebras of classical type 1960 Charles W. Curtis
+ PDF Chat Automorphisms of classical Lie algebras 1961 Robert Steinberg
+ A Note on Three Dimensional Simple Lie Algebras 1989 Nathan Jacobson
+ Characters of Irreducible Representations of the Simple Groups. II. Application to Classical Groups 1964 Jean-Pierre Antoine
David Speiser
+ PDF Chat On projective representations of certain finite groups 1960 Charles W. Curtis
+ Characters of Irreducible Representations of the Simple Groups. I. General Theory 1964 Jean-Pierre Antoine
David Speiser
+ PDF Chat Representations of Lie Algebras of Classical Type with Applications to Linear Groups 1960 Charles W. Curtis
+ PDF Chat A Note on Three Dimensional Simple Lie Algebras 1958 Nathan Jacobson
+ PDF Chat On the Dimensions of the Irreducible Modules of Lie Algebras of Classical Type 1960 Charles W. Curtis
+ PDF Chat On Projective Representations of Certain Finite Groups 1960 Charles W. Curtis