scholarly journals The Representations of Lie Algebras of Prime Characteristic

1954 ◽  
Vol 2 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Quotient Ring ◽  
Irreducible Representations ◽  
Indecomposable Representation ◽  
Prime Characteristic ◽  
Representations Of Lie Algebras ◽  
Algebra Over A Field

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras

1971 ◽  
Vol 14 (1) ◽  
pp. 113-115 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
Space Decomposition ◽  
Finite Dimensional

Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals On Real Irreducible Representations of Lie Algebras

1959 ◽  
Vol 14 ◽  
pp. 59-83 ◽  
Author(s):  
Nagayoshi Iwahori
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Number Field ◽  
Irreducible Representations ◽  
The Real ◽  
Representations Of Lie Algebras ◽  
Real Matrices ◽  
Real Number Field

Let us consider the following two problems:Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.

Palindromes in the free metabelian Lie algebras

2019 ◽  
Vol 29 (05) ◽  
pp. 885-891
Author(s):  
Şehmus Fındık ◽  
Nazar Şahi̇n Öğüşlü
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Reverse Order ◽  
Generating Set ◽  
Linear Basis ◽  
Free Metabelian Lie Algebra ◽  
Definition Of ◽  
Algebra Over A Field

A palindrome, in general, is a word in a fixed alphabet which is preserved when taken in reverse order. Let [Formula: see text] be the free metabelian Lie algebra over a field of characteristic zero generated by [Formula: see text]. We propose the following definition of palindromes in the setting of Lie algebras: An element [Formula: see text] is called a palindrome if it is preserved under the change of generators; i.e. [Formula: see text]. We give a linear basis and an explicit infinite generating set for the Lie subalgebra of palindromes.

scholarly journals A remark on irreducible representations of Lie algebras

1979 ◽  
Vol 27 (3) ◽  
pp. 332-336 ◽  
Author(s):  
JU. A. Bahturin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Irreducible Representations ◽  
Bounded Degree ◽  
Precise Form ◽  
Representations Of Lie Algebras ◽  
Necessary And Sufficient

AbstractIn addition to the results of the paper (Bachturin (1974)) we give the precise form of the necessary and sufficient conditions ensuring that all irreducible representations of a Lie algebra were of finite bounded degree.

Uniqueness theorems for left-symmetric enveloping algebras

1996 ◽  
Vol 120 (2) ◽  
pp. 193-206
Author(s):  
J. R. Bolgar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Automorphism Groups ◽  
Uniqueness Theorems ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Uniqueness Results ◽  
Symmetric Algebras ◽  
Algebra Over A Field

AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

The Second Cohomology of Current Algebras of General Lie Algebras

2008 ◽  
Vol 60 (4) ◽  
pp. 892-922 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Friedrich Wagemann
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Cohomology Space ◽  
Trivial Module ◽  
Commutative Associative Algebra ◽  
Locally Convex ◽  
Algebra Over A Field

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

The Construction of Representations of Lie Algebras of Characteristic Zero

1962 ◽  
Vol 14 ◽  
pp. 304-312
Author(s):  
B. Noonan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Basis Element ◽  
A Value ◽  
Representations Of Lie Algebras ◽  
Explicit Representations ◽  
The Ideal

In this paper a procedure is given whereby, from a representation of an ideal contained in the radical, explicit representations of a Lie algebra by matrices can be constructed in an algebraically closed field of characteristic zero. The construction is sufficiently general to permit one arbitrary eigenvalue to be assigned to the representation of each basis element of the radical not in the ideal. The theorem of Ado is proved as an application of the construction. While Ado's theorem has several proofs (1; 3; 5; 6), the present one has a value in its explicitness and in the fact that the degree of the representation can be given.

scholarly journals Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

2014 ◽  
Vol 14 (02) ◽  
pp. 1550024
Author(s):  
Guillermo Ames ◽  
Leandro Cagliero ◽  
Mónica Cruz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Algebraic Version ◽  
Finite Dimensional ◽  
Rank Conjecture ◽  
Algebra Over A Field ◽  
Toral Rank

If 𝔫 is a [Formula: see text]-graded nilpotent finite-dimensional Lie algebra over a field of characteristic zero, a well-known result of Deninger and Singhof states that dim H*(𝔫) ≥ L(p) where p is the polynomial associated to the grading and L(p) is the sum of the absolute values of the coefficients of p. From this result they derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that dim H*(𝔫) ≥ 2 dim (ℨ) for any finite-dimensional nilpotent Lie algebra 𝔫 with center ℨ. The TRC is more than 25 years old and remains open even for [Formula: see text]-graded 3-step nilpotent Lie algebras. Investigating to what extent the bound given by Deninger and Singhof could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra 𝔫 with center ℨ: (A) If 𝔫 admits a [Formula: see text]-grading [Formula: see text], such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a ℤ+-grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? (B) If 𝔫 is r-step nilpotent admitting a grading 𝔫 = 𝔫1 ⊕ 𝔫2 ⊕ ⋯ ⊕ 𝔫k such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

scholarly journals SOME PROPERTIES ON THE TENSOR SQUARE OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250085 ◽  
Author(s):  
PEYMAN NIROOMAND
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Tensor Square

In the present paper we extend the results of [2, 4] for the tensor square of Lie algebras. More precisely, for any Lie algebra L with L/L2 of finite dimension, we prove L ⊗ L ≅ L □ L ⊕ L ∧ L and Z∧(L) ∩ L2 = Z⊗(L). Moreover, we show that L ∧ L is isomorphic to derived subalgebra of a cover of L, and finally we give a free presentation for it.

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