HNN-extensions of lie algebras

Authors:
A. I. Lichtman and M. Shirvani

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3501-3508

MSC (1991):
Primary 17B05; Secondary 16S10, 17B01

DOI:
https://doi.org/10.1090/S0002-9939-97-04124-5

MathSciNet review:
1423316

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Abstract | References | Similar Articles | Additional Information

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.

**1.**Yu. A. Bakhturin,*Identical Relations in Lie Algebras*, VNU Scientific Press, Utrecht, 1987. MR**88f:17032****2.**G.M. Bergman,*Modules over coproducts of rings*, Trans Amer. Math.Soc.**200**(1974), 1-32. MR**50:9970****3.**L.A. Bokut and G.P. Kukin,*Algorithmic and Combinatorial Algebra*, Kluwer, 1994. MR**95i:17002****4.**P.M. Cohn,*Skew fields: theory of general division rings*, Cambridge University Press, New York, 1995. MR**97d:12003****5.**P.M. Cohn,*On the free product of associative rings*Math. Z.**71**(1959), 380-398. MR**21:5648****6.**P.M. Cohn,*On the free product of associative rings, II*, Math. Z.**73**(1960), 433-456. MR**22:4747****7.**P.M. Cohn,*On the free product of associative rings, III*, J. Algebra**8**(1968), 376-383. MR**36:5170****8.**W. Dicks,*The HNN construction for rings*, J. Algebra**81**(1983), 434-487. MR**85c:16005****9.**G. Higman, B.H. Neumann and H. Neumann,*Embedding theorems for groups*, J. London Math. Soc.**24**(1949), 247-254. MR**11:322d****10.**N. Jacobson,*Lie Algebras*, Dover, New York, 1979. MR**80k:17001****11.**G.P. Kukin,*Subalgebras of a free Lie sum with an amalgamated subalgebra*, Algebra i Logika**11**(1972), 59-86 (In Russian). MR**46:9133****12.**R.C. Lyndon and P.E. Schupp,*Combinatorial Group Theory*, Springer, Berlin, 1977. MR**58:28182****13.**A. 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), 211-236. MR**81h:03092****14.**A.I. Malcev,*On representations of infinite algebras*, Mat. Sb.**13**(1943), 263-285 (In Russian). MR**6:116c****15.**G.B. Seligman,*Modular Lie Algebras*, Springer, Berlin, 1967. MR**39:6933****16.**J.-P. Serre,*Trees*, Springer, Berlin, 1980. MR**82c:20083****17.**A.I. Shirshov,*On free Lie rings*, Mat. Sb.**45**(1958), 13-21.

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Additional Information

**A. I. Lichtman**

Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
lichtman@cs.uwp.edu

**M. Shirvani**

Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
mazi@schur.math.ualberta.ca

DOI:
https://doi.org/10.1090/S0002-9939-97-04124-5

Received by editor(s):
March 22, 1996

Received by editor(s) in revised form:
July 9, 1996

Additional Notes:
The first author was partially supported by the NSF Grant No. 144-F1181, and the second author by NSERC, Canada.

Communicated by:
Ken Goodearl

Article copyright:
© Copyright 1997
American Mathematical Society