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.

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).

On some representations of the real number field

2011 ◽  
Vol 50 (2) ◽  
pp. 189-190
Author(s):  
A. S. Morozov
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Number Field

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.

A geometrical model for the real number field

1961 ◽  
Vol 57 (4) ◽  
pp. 722-727
Author(s):  
W. Greve
Keyword(s):  
Real Number ◽  
Number Field ◽  
Near Field ◽  
Geometrical Model ◽  
Order Structure ◽  
Affine Transformations ◽  
The Real ◽  
Abstract Group ◽  
Underlying Space ◽  
Real Number Field

Recently Cunningham and Valentine gave in (3) an axiomatic description of the one-dimensional real affine space in terms of its order structure and the (abstract) group of affine transformations It is the purpose of the present note to show that the system of axioms in (3) (cf. (L. 1)–(L. 5) of this note) leads in a natural way to a model of the real number field. Our method is suggested by a result of Hall ((4), p. 382), namely, that an infinite doubly transitive Frobenius group is isomorphic to the group of affine transformations in a near-field, provided that there is at most one transformation displacing all points and taking a given point a into a given point b. The salient point of our investigation is the redundancy of the latter condition in the case where the underlying space is endowed with a certain linear order structure which is invariant under the transformations of the given group.

scholarly journals Typical ranks ofm×n× (m− 1)ntensors with 3 ≤m≤nover the real number field

2014 ◽  
Vol 63 (5) ◽  
pp. 940-955 ◽  
Author(s):  
Toshio Sumi ◽  
Mitsuhiro Miyazaki ◽  
Toshio Sakata
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Number Field

Solvability by Real Radicals and Fermat Primes

2004 ◽  
Vol 47 (2) ◽  
pp. 229-236
Author(s):  
C. U. Jensen
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Roots ◽  
Fermat Primes ◽  
Real Number Field

AbstractWe give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

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]).

Sign in / Sign up

Export Citation Format

Share Document