Constants of derivations on free associative algebras
by
Vitor de Oliveira Ferreira
University of São Paulo, São Paulo, Brazil

A derivation on a ring R is an additive map d:R --> R which satisfies d(xy)=d(x)y+xd(y) for x, y in R. The set ker(d)={x in R : d(x)=0} is a subring of R, called the subring of constants of d. Subring of constants of derivations on free associative algebras have been widely studied (cf. [Ka81]). But some questions remain unanswered. For instance, in [Ka81] it is asked whether the subalgebra of constants of a finite-dimensional restricted Lie algebra of derivations on a free associative algebra over a field of positive characteristic is free.

Here we present a special case of the general problem where a positive answer can be given. We show that given a field k of positive characteristic p and a derivation d on k<x₁, ... , x_m> which satisfies d^q=0 (q=pⁿ), d(x₁) in k[x₁], d(x_i) in k<x₁, ... , x_i-1>, for i >= 2, and 1 in d^r(k[x₁]) (r=p^n-1), then the subalgebra of constants of d is free of rank (m-1)pⁿ+1.

This provides a proof for the well-known fact that the constants of each partial derivative on a free associative algebra of rank m over a field of positive characteristic p form a free subalgebra of rank (m-1)p+1 (cf. [Ge98]). The result also implies the freeness of the subalgebra of constants of the derivation d on the free associative algebra generated by {x₁, ... , x_p-1} over a field of positive characteristic p given by d(x_i)=x_i-1 for i >= 2 and d(x₁)=1. This is incidentally the crucial step in the proof of the following previous result by the author (see [Fe, Theorem 3.2]): Let k be a field of positive characteristic p and F=k(a) an extension such that a^p in k, but a not in k. Let R=F_k<x> and denote by U its universal field of fractions. Then R\otimes_k F is not a matrix ring over a fir, but the subring S of U obtained from R by adjoining the inverse of d^p-1(x), where d is the inner derivation of R determined by a, is such that S\otimes_k F is isomorphic to the p×p matrix ring over the ring F<x₁, ... , x_p²-p> * _F F[w, w^-1], which is a fir.

REFERENCES

[Fe] V. O. Ferreira, Tensor rings under field extensions, J. Algebra, to appear.

[Ge98] L. Gerritzen, Taylor expansion of noncommutative polynomials, Arch. Math. 71 (1998), 279-290.

[Ka81] V. K. Karchenko, Constants of derivations of prime rings, Math. USSR Izvestija 18 (1982), no. 2, 381-401.

Vitor O. Ferreira's Home Page

Date received: May 25, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-03.