A matrix representation for associative algebras. II
HTML articles powered by AMS MathViewer
- by Jacques Lewin PDF
- 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.References
- Ralph K. Amayo and Ian Stewart, Finitely generated Lie algebras, J. London Math. Soc. (2) 5 (1972), 697–703. MR 323850, DOI 10.1112/jlms/s2-5.4.697
- Gilbert Baumslag, Some subgroup theorems for free ${\mathfrak {v}}$-groups, Trans. Amer. Math. Soc. 108 (1963), 516–525. MR 154919, DOI 10.1090/S0002-9947-1963-0154919-1
- George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, DOI 10.1016/0001-8708(78)90010-5
- P. M. Cohn, On a generalization of the Euclidean algorithm, Proc. Cambridge Philos. Soc. 57 (1961), 18–30. MR 118743, DOI 10.1017/s0305004100034812
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- John P. Labute, Algèbres de Lie et pro-$p$-groupes définis par une seule relation, Invent. Math. 4 (1967), 142–158 (French). MR 218495, DOI 10.1007/BF01425247
- Jacques Lewin, A matrix representation for associative algebras. I, II, Trans. Amer. Math. Soc. 188 (1974), 293–308; ibid. 188 (1974), 309–317. MR 338081, DOI 10.1090/S0002-9947-1974-0338081-5
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 188 (1974), 309-317
- MSC: Primary 16A64; Secondary 16A42
- DOI: https://doi.org/10.1090/S0002-9947-74-99943-7