Deciding Identities in Finite Dimensional Algebras ?

D.P. Jacobs
Department of Computer Science
Clemson University
Clemson, SC 29634-1906 USA
[email protected]
April 20, 1994

Abstract
Several decision problems are examined involving the recognition of identities in finite dimensional, nonassociative algebras. We show, for example, that given any fixed nonassociative polynomial p, we can decide in polynomial-time if an algebra A over a finite field or Q satisfies p. We also show that over any finite field the problem of recognizing finite dimensional, power associative algebras is in co-NP; while the problem of recognizing finite dimensional, strictly power associative algebras is in co-R.

Key words: polynomial-time, random algorithm, power associative algebra.
AMS (MOS) subject classifications: 68Q15, 68Q25, 17A05

1 Introduction

A (nonassociative) algebra A over a field F is a vector space over F along with a multiplication operator for which
>=(ab) = (>=a)b = a(>=b) (1)

a(b + c) = ab + ac (2)

(a + b)c = ac + bc (3)

for all a; b; c 2 A and >= 2 F. Throughout this paper we assume A is an algebra of finite dimension n over F. e also assume A has been represented by a basis B = fb1; : : : ; bng, along with n3 structure constants ijk 2 F such that

bibj =
n
k=1
ijkbk
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