HNN-extensions of Lie algebras

Lichtman, A.; Shirvani, M.

doi:10.1090/S0002-9939-97-04124-5

HNN-extensions of Lie algebras
HTML articles powered by AMS MathViewer

by A. I. Lichtman and M. Shirvani PDF

Proc. Amer. Math. Soc. 125 (1997), 3501-3508 Request permission

Abstract:

We define HNN-extensions of Lie algebras and study their properties. In particular, a sufficient condition for freeness of subalgebras is obtained. We also study differential HNN-extensions of associative rings. These constructions are used to give short proofs of Malcev’s and Shirshov’s theorems that an associative or Lie algebra of finite or countable dimension is embeddable into a two-generator algebra.

References

Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
George M. Bergman, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1–32. MR 357502, DOI 10.1090/S0002-9947-1974-0357502-5
L. A. Bokut′ and G. P. Kukin, Algorithmic and combinatorial algebra, Mathematics and its Applications, vol. 255, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1292459, DOI 10.1007/978-94-011-2002-9
P. M. Cohn, Skew fields, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995. Theory of general division rings. MR 1349108, DOI 10.1017/CBO9781139087193
P. M. Cohn, On the free product of associative rings, Math. Z. 71 (1959), 380–398. MR 106918, DOI 10.1007/BF01181410
Paul M. Cohn, On the free product of associative rings. II. The case of (skew) fields, Math. Z. 73 (1960), 433–456. MR 113916, DOI 10.1007/BF01215516
P. M. Cohn, On the free product of associative rings. III, J. Algebra 8 (1968), 376–383. MR 222118, DOI 10.1016/0021-8693(68)90066-5
Warren Dicks, The HNN construction for rings, J. Algebra 81 (1983), no. 2, 434–487. MR 700294, DOI 10.1016/0021-8693(83)90198-9
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
G. P. Kukin, Subalgebras of the free Lie sum of Lie algebras with a joint subalgebra, Algebra i Logika 11 (1972), 59–86, 121 (Russian). MR 0310030
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Angus Macintyre, Combinatorial problems for skew fields. I. Analogue of Britton’s lemma, and results of Adjan-Rabin type, Proc. London Math. Soc. (3) 39 (1979), no. 2, 211–236. MR 548978, DOI 10.1112/plms/s3-39.2.211
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR 0245627
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
A.I. Shirshov, On free Lie rings, Mat. Sb. 45 (1958), 13-21.