Jordan matrix algebras defined by generators and relations

Matrix Algebras ◽

Number Of Generators ◽

<abstract><p>In the present paper we describe Jordan matrix algebras over a field by generators and relations. We prove that the minimun number of generators of some special Jordan matrix algebras over a field is $ 2 $.</p></abstract>

Reduced Jordan matrix algebras over complete local rings

Indagationes Mathematicae (Proceedings) ◽

10.1016/1385-7258(78)90027-6 ◽

1978 ◽

Vol 81 (1) ◽

pp. 97-109

Author(s):

Holger P. Petersson

Keyword(s):

Local Rings ◽

Matrix Algebras ◽

Generators and relations for matrix algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2006.01.043 ◽

2006 ◽

Vol 300 (1) ◽

pp. 134-159 ◽

Cited By ~ 1

Author(s):

Jon F. Carlson ◽

Graham Matthews

Keyword(s):

Matrix Algebras

A Comment on Finite Nilpotent Groups of Deficiency Zero

Canadian Mathematical Bulletin ◽

10.4153/cmb-1980-043-3 ◽

1980 ◽

Vol 23 (3) ◽

pp. 313-316 ◽

Cited By ~ 3

Author(s):

Edmund F. Robertson

Keyword(s):

Finite Group ◽

Finite Number ◽

Nilpotent Groups ◽

Metacyclic Groups ◽

Number Of Generators ◽

Nilpotent Class ◽

A Finite Group

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

An algorithm for analysis of the structure of finitely presented Lie algebras

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.243 ◽

1997 ◽

Vol Vol. 1 ◽

Author(s):

Vladimir P. Gerdt ◽

Vladimir V. Kornyak

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Nonlinear Partial Differential Equations ◽

Practical Importance ◽

Physical Models ◽

Defining Relations ◽

Number Of Generators ◽

Finite Set ◽

Finitely Presented

International audience We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

Peirce decompositions and Jordan matrix algebras

Structure and Representations of Jordan Algebras - Colloquium Publications ◽

10.1090/coll/039/03 ◽

1968 ◽

pp. 117-151

Keyword(s):

Matrix Algebras ◽

Some finite groups with zero deficiency

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700019121 ◽

1974 ◽

Vol 18 (1) ◽

pp. 73-75 ◽

Cited By ~ 3

Author(s):

J. W. Wamsley

Keyword(s):

Finite Groups ◽

Number Of Generators

AbstractWe introduce further finite groups which can be presented with an equal number of generators and relations.

Tensor Products of Jordan Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-009-4 ◽

1975 ◽

Vol 27 (1) ◽

pp. 60-74 ◽

Cited By ~ 3

Author(s):

Aubrey Wulfsohn

Keyword(s):

Tensor Product ◽

Jordan Algebra ◽

Kronecker Product ◽

Tensor Products ◽

Jordan Algebras ◽

Mapping Property ◽

Associative Algebras ◽

Matrix Algebras ◽

Special Jordan Algebra ◽

Let J1 and J2 be two Jordan algebras with unit elements. We define various tensor products of J1 and J2. The first, which we call the Kronecker product, is the most obvious and is based on the tensor product of the vector spaces. We find conditions sufficient for its existence and for its non-existence. Motivated by the universal mapping property for the tensor product of associative algebras we define, in Section 2, tensor products of J1 and J2 by means of a universal mapping property. The tensor products always exist for special Jordan algebras and need not coincide with the Kronecker product when the latter exists. In Section 3 we construct a more concrete tensor product for special Jordan algebras. Here the tensor product of a special Jordan algebra and an associative Jordan algebra coincides with the Kronecker product of these algebras. We show that this "special" tensor product is the natural tensor product for some Jordan matrix algebras.