On Free Differentials on Associative Algebras

Abstract

A differential d: R → _R M _R is called free if the differential of any element v has a unique presentation of the form dv = dx ⁱ · v _i, where x ^l,…, x ⁿ are generators of the unitary algebra R and dx ¹,…, dx ⁿ their differentials. Any free differential defines a commutation formula v dx ⁱ = dx ^k · A(v) ⁱ _k ,where A: v ↦ A(v) ⁱ _k is an algebra homomorphism A: R ↦ R _n×n It is easy to see that for any homomorphism R ↦ R _{n × n} there exists not more than one free differential. We are going to consider the existence problem of such a differential. We will show that for a given commutation rule vdx ⁱ = dx ^k · A(v) ⁱ _k a free algebra generated by the variables x ¹,…, x ⁿ has a related free differential. We will define an optimal algebra with respect to a fixed commutation rule. In the homogeneous case this algebra is characterized as the unique algebra which has no nonzero A-invariant subspaces with zero differentials. Finally, we will consider a number of examples of optimal algebras for different commutation rules. In particular, we will describe two variable commutation rules which define commutative optimal algebra.

Supported by the State Research Committee KBN

Supported by the Russian Fund of Fundamental Research No 93-011-16171