
Deciding Identities in Finite Dimensional Algebras ?
D.P. Jacobs
Department of Computer Science
Clemson University
Clemson, SC 296341906 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 polynomialtime 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 coNP; while the
problem of recognizing finite dimensional, strictly power associative algebras is in coR.
Key words: polynomialtime, 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
? his research as artia su orte rant .