Structure theory for finite-dimensional Jordan algebras

1968 ◽  
pp. 189-212
Keyword(s):  
Jordan Algebras ◽  
Structure Theory ◽  
Finite Dimensional

scholarly journals Holomorphic Cohomology of Complex Analytic Linear Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 531-542 ◽  
Author(s):  
G. Hochschild ◽  
G. D. Mostow
Keyword(s):  
Linear Group ◽  
Structure Theory ◽  
Linear Groups ◽  
Dimensional Representation ◽  
Lie Algebra Cohomology ◽  
Analytic Group ◽  
Finite Dimensional Representation ◽  
Semidirect Products ◽  
Finite Dimensional ◽  
Rational Cohomology

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

scholarly journals Automorphism Groups of Jordan Algebras

1968 ◽  
Vol 32 ◽  
pp. 227-235
Author(s):  
E. Ray Bobo
Keyword(s):  
Vector Space ◽  
Automorphism Groups ◽  
Infinite Family ◽  
Jordan Algebras ◽  
Structure Theory ◽  
Important Class ◽  
Natural Basis ◽  
Generators And Relations ◽  
Characteristic Two ◽  
Central Simple

In his development of a structure theory for Jordan algebras of characteristic two, E.C. Paige [1] introduces an important class of central simple Jordan algebras S[2n], It is the purpose of this paper to completely determine the automorphism groups of the algebras S[2n]. The automorphisms will be represented as matrices operating on a natural basis for the underlying vector space of the algebra. Using this characterization, generators and relations will be obtained for each of the automorphism groups. In this way, we will produce an infinite family of finite 2-groups.

A Structure Theory for Jordan Algebras

1947 ◽  
Vol 48 (3) ◽  
pp. 546 ◽  
Author(s):  
A. A. Albert
Keyword(s):  
Jordan Algebras ◽  
Structure Theory

A Generalization of Degree Two Simple Finite-Dimensional Noncommutative Jordan Algebras

1980 ◽  
Vol 32 (2) ◽  
pp. 480-493
Author(s):  
Mary Ellen Conlon
Keyword(s):  
Vector Space ◽  
Simple Algebra ◽  
Jordan Algebras ◽  
Prime Characteristic ◽  
Nilpotent Elements ◽  
Finite Dimensional ◽  
Image Position ◽  
Algebra Over A Field

Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose1Then is noncommutative Jordan.The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.

Hilbert space methods in the theory of Jordan algebras. II

1976 ◽  
Vol 79 (2) ◽  
pp. 307-319 ◽  
Author(s):  
C. Viola Devapakkiam ◽  
P. S. Rema
Keyword(s):  
Hilbert Space ◽  
Jordan Algebra ◽  
Canonical Basis ◽  
Classification Problem ◽  
Jordan Algebras ◽  
Type I ◽  
Type Ii ◽  
Canonical Bases ◽  
Finite Dimensional ◽  
Jordan Isomorphism

In this paper we consider the classification problem for separable special simple J*-algebras (cf. (8)). We show, using a result of Ancochea, that if is the (finite-dimensional) Jordan algebra of all complex n × n matrices and ø a Jordan isomorphism of onto a special J*-algebra J then An can be given the structure of an H*-algebra such that ø is a *-preserving isomorphism of the J*-algebra onto J. This result enables us to construct explicitly a canonical basis for a finite-dimensional simple special J*-algebra isomorphic to a Jordan algebra of type I from which we also obtain canonical bases for special simple finite-dimensional J*-algebras isomorphic to Jordan algebras of type II and III.

Sign in / Sign up

Export Citation Format

Share Document