A matrix representation for associative algebras. II

Lewin, Jacques

doi:10.1090/S0002-9947-74-99943-7

A matrix representation for associative algebras. II
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Trans. Amer. Math. Soc. 188 (1974), 309-317 Request permission

Abstract:

The results of part I of this paper are applied to show that if F is a free algebra over the field K and W is a subset of F which is algebraically independent modulo the commutator ideal [F, F], then W again generates a free algebra. On the way a similar theorem is proved for algebras that are free in the variety of K-algebras whose commutator ideal is nilpotent of class n. It is also shown that if L is a Lie algebra with universal enveloping algebra F, and U, V are ideals of L, then $FUF \cdot FVF \cap L = [U \cap V,U \cap V]$. This is used to extend the representation theorem of part I to free Lie algebras.

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