On generators of $L/R^ 2$ Lie algebras
HTML articles powered by AMS MathViewer
- by Vladimir Shpilrain PDF
- Proc. Amer. Math. Soc. 119 (1993), 1039-1043 Request permission
Abstract:
Let $L$ be a free Lie algebra of finite rank $n$ and $R$ its arbitrary ideal. A necessary and sufficient condition for $n$ elements of the Lie algebra $L/{R^2}$ to be a generating set is given. In particular, we have a criterion for $n$ elements of a free Lie algebra of rank $n$ to be a generating set which is similar to the corresponding group-theoretic result due to Birman (An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41 (1973), 634-638).References
- Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
- Joan S. Birman, An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41 (1973), 634–638. MR 330295, DOI 10.1090/S0002-9939-1973-0330295-8
- Ralph H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560. MR 53938, DOI 10.2307/1969736
- A. F. Krasnikov, Generating elements of the group $F/[N,\,N]$, Mat. Zametki 24 (1978), no. 2, 167–173, 301 (Russian). MR 509900
- Christophe Reutenauer, Applications of a noncommutative Jacobian matrix, J. Pure Appl. Algebra 77 (1992), no. 2, 169–181. MR 1149019, DOI 10.1016/0022-4049(92)90083-R
- I. A. Yunus, The Fox problem for Lie algebras, Uspekhi Mat. Nauk 39 (1984), no. 3(237), 251–252 (Russian). MR 747808
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 119 (1993), 1039-1043
- MSC: Primary 17B01; Secondary 17B40
- DOI: https://doi.org/10.1090/S0002-9939-1993-1154249-X
- MathSciNet review: 1154249