On the multiplication ideal of a nonassociative algebra

Scott M. Farrand; David R. Finston

doi:10.1007/bf01195102

On the multiplication ideal of a nonassociative algebra

Archiv der Mathematik ◽

10.1007/bf01195102 ◽

1987 ◽

Vol 48 (4) ◽

pp. 298-302 ◽

Cited By ~ 3

Author(s):

Scott M. Farrand ◽

David R. Finston

Keyword(s):

Nonassociative Algebra ◽

Multiplication Ideal

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A nonassociative algebra having no finite basis for its laws

Siberian Mathematical Journal ◽

10.1007/bf00969242 ◽

1978 ◽

Vol 18 (6) ◽

pp. 1007-1008 ◽

Cited By ~ 3

Author(s):

Yu. N. Mal'tsev ◽

V. A. Parfenov

Keyword(s):

Finite Basis ◽

Nonassociative Algebra

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Norton Algebras of the Hamming Graphs via Linear Characters

The Electronic Journal of Combinatorics ◽

10.37236/10251 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Jia Huang

Keyword(s):

Group Theory ◽

Automorphism Group ◽

Nonassociative Algebra ◽

Bilinear Forms ◽

Distance Regular Graph ◽

Finite Abelian Groups ◽

Hamming Graph ◽

Large Subgroup ◽

Special Cases ◽

Hamming Graphs

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

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Real Flexible Division Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-039-x ◽

1982 ◽

Vol 34 (3) ◽

pp. 550-588 ◽

Cited By ~ 19

Author(s):

Georgia M. Benkart ◽

Daniel J. Britten ◽

J. Marshall Osborn

Keyword(s):

Jordan Algebra ◽

Nonassociative Algebra ◽

Division Algebras ◽

Hermitian Matrices ◽

Real Numbers ◽

The Real ◽

Precise Statement ◽

Finite Dimensional ◽

Dimension One

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

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