Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras

Erik Darpö,Patrik Nystedt

doi:10.1007/s10468-020-09988-4

Erik Darpö, Patrik Nystedt

Open Access

https://doi.org/10.1007/s10468-020-09988-4

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Abstract

We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. Types of algebras that are studied include alternative, flexible, quadratic and involutive algebras, as well as algebras obtained by the Cayley–Dickson doubling process. Our results include precise criteria for von-Neumann finiteness and reversibility of involutive algebras in terms of isomorphism types of their 3-dimensional subalgebras.

Highlights

A unital ring A is called von-Neumann finite if every one-sided inverse in A is two-sided, in other words, if for all a, b ∈ A satisfying ab = 1, the relation ba = 1 holds
Observe that the classes of von-Neumann finite respectively reversible algebras are closed under taking subalgebras
(c) In view of (b), we need to show that a 3-dimensional quadratic algebra is vonNeumann finite and reversible if and only if it is commutative

Summary

Introduction

We consider von-Neumann finiteness and reversibility for some classes of unital rings which are not associative – apparently a new line of investigation. The notions of von-Neumann finiteness and reversibility are relevant by themselves in the structure theory of non-associative algebras. We will consider the problem of characterising von-Neumann finiteness and reversibility for the classes of nonassociative algebras defined below. A Hurwitz algebra A has zero-divisors if and only if its quadratic form n is isotropic If this is the case, A is said to be split. Given an involutive algebra B and a non-zero scalar μ, a new algebra CDμ(B) is constructed as follows: as a vector space, CDμ(B) = B × B, with multiplication defined by (a, b)(c, d) = (ac + μdb, da + bc) for all a, b, c, d ∈ B. A note on conventions: we use words as non-commutative and non-associative in the strict sense: a non-commutative algebra is one that does not satisfy the commutative law, etc

Summary of Our Results

Preliminaries

Proofs of Our Results

Examples