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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A matrix representation for associative algebras. II


Author: Jacques Lewin
Journal: Trans. Amer. Math. Soc. 188 (1974), 309-317
MSC: Primary 16A64; Secondary 16A42
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-74-99943-7
PII: S 0002-9947(74)99943-7
Keywords: Free algebras, free subalgebras, PI algebras, free Lie algebras
Article copyright: © Copyright 1974 American Mathematical Society