Nodal Non-Commutative Jordan Algebras

1960 ◽  
Vol 12 ◽  
pp. 488-492 ◽  
Author(s):  
Louis. A. Kokoris
Keyword(s):  
Associative Algebra ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Finite Dimensional

A finite dimensional power-associative algebra 𝒰 with a unity element 1 over a field J is called a nodal algebra by Schafer (7) if every element of 𝒰 has the form α1 + z where α is in J, z is nilpotent, and if 𝒰 does not have the form 𝒰 = ℐ1 + n with n a nil subalgebra of 𝒰. An algebra SI is called a non-commutative Jordan algebra if 𝒰 is flexible and 𝒰+ is a Jordan algebra. Some examples of nodal non-commutative Jordan algebras were given in (5) and it was proved in (6) that if 𝒰 is a simple nodal noncommutative Jordan algebra of characteristic not 2, then 𝒰+ is associative. In this paper we describe all simple nodal non-commutative Jordan algebras of characteristic not 2.

scholarly journals On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mohamed Benslimane ◽  
Abdelhadi Moutassim
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Alternative Algebra ◽  
Jordan Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
First Case ◽  
Hilbert Norm ◽  
Complex Powers

Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

Noetherian Banach Jordan pairs

2001 ◽  
Vol 130 (1) ◽  
pp. 25-36 ◽  
Author(s):  
N. BOUDI ◽  
H. MARHNINE ◽  
C. ZARHOUTI ◽  
A. FERNANDEZ LOPEZ ◽  
E. GARCIA RUS
Keyword(s):  
Recent Work ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jacobson Radical ◽  
Alternative Algebra ◽  
Chain Condition ◽  
Finite Capacity ◽  
Jordan Pair ◽  
Finite Dimensional ◽  
Ascending Chain Condition

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.Here we show the following results:(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

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.

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

2011 ◽  
Vol 10 (02) ◽  
pp. 319-333 ◽  
Author(s):  
J. BERNIK ◽  
R. DRNOVŠEK ◽  
D. KOKOL BUKOVŠEK ◽  
T. KOŠIR ◽  
M. OMLADIČ ◽  
...  
Keyword(s):  
Vector Space ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Linear Operators ◽  
Closed Field ◽  
Toeplitz Algebra ◽  
Nilpotent Operator ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Unital Associative Algebra

A set [Formula: see text] of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists [Formula: see text] such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

On Homomorphic Images of Special Jordan Algebras

1954 ◽  
Vol 6 ◽  
pp. 253-264 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Special Jordan Algebra ◽  
Image Position

A linear algebra is called a Jordan algebra if it satisfies the identities(1) ab = ba, (a2b) a = a2(ba).It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that1,where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

SYMMETRIES OF CERTAIN PHYSICAL THEORIES AND THE JORDAN ALGEBRAS

1992 ◽  
Vol 07 (15) ◽  
pp. 3623-3637 ◽  
Author(s):  
R. FOOT ◽  
G. C. JOSHI
Keyword(s):  
Jordan Algebra ◽  
Space Time ◽  
Jordan Algebras ◽  
Bosonic String ◽  
The Other ◽  
Structure Group ◽  
Division Algebras ◽  
Hermitian Matrices ◽  
Higher Dimensional ◽  
Algebraic Framework

It is shown that the sequence of Jordan algebras [Formula: see text], whose elements are the 3 × 3 Hermitian matrices over the division algebras ℝ, [Formula: see text], ℚ and [Formula: see text], can be associated with the bosonic string as well as the superstring. The construction reveals that the space–time symmetries of the first-quantized bosonic string and superstring actions can be related. The bosonic string and the superstring are associated with the exceptional Jordan algebra while the other Jordan algebras in the [Formula: see text] sequence can be related to parastring theories. We then proceed to further investigate a connection between the symmetries of supersymmetric Lagrangians and the transformations associated with the structure group of [Formula: see text]. The N = 1 on-shell supersymmetric Lagrangians in 3, 4 and 6-dimensions with a spin 0 field and a spin 1/2 field are incorporated within the Jordan-algebraic framework. We also make some remarks concerning a possible role for the division algebras in the construction of higher-dimensional extended objects.

scholarly journals Eisenstein Series Arising from Jordan Algebras

2019 ◽  
Vol 72 (1) ◽  
pp. 183-201 ◽  
Author(s):  
Marcela Hanzer ◽  
Gordan Savin
Keyword(s):  
Eisenstein Series ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Unipotent Radicals ◽  
Maximal Parabolic Subgroups

AbstractWe describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Equivalent norms on Banach Jordan algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 261-270 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Jordan Algebra ◽  
Sufficient Conditions ◽  
Equivalent Norm ◽  
Jordan Algebras ◽  
The Self ◽  
Equivalent Norms ◽  
Positive Elements ◽  
Definition Of ◽  
Necessary And Sufficient

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

Cayley Symmetries in Associative Algebras

1963 ◽  
Vol 15 ◽  
pp. 285-290 ◽  
Author(s):  
Earl J. Taft
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Wedderburn Decomposition ◽  
Finite Dimensional ◽  
Principal Theorem ◽  
Algebra Over A Field

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.

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