Pro-finite \(p\)-adic Lie algebras. (English) Zbl 1181.17009

McInnes, L.; Riley, D. M.

J. Algebra 319, No. 1, 205-234 (2008).

Let \(p\) be a prime number, \(\mathbb Z_p\) the ring of \(p\)-adic integers. A Lie ring \(L\) is called finite-\(p\) if it is finite nilpotent Lie \(\mathbb Z_p\)-algebra. A pro-\(p\) Lie ring \(L\) is the inverse limit of an inverse system of finite-\(p\) Lie rings. For example, as it is shown in the paper if \(L\) is a Lie \(\mathbb Z_p\)-algebra of finite rank then \(pL\) is a pro-\(p\) Lie ring. The goal of the paper declared by authors is to study pro-\(p\) Lie rings in the direction analogous to recent studies of pro-\(p\) groups and, thus, to develop understanding of groups as well as Lie rings.
The first part of the paper is devoted to the study of various properties of pro-\(p\) Lie rings concerning their topology, Prüfer rank, subring growth, \(p\)-adic module structure. According to results by Serre for finitely generated pro-\(p\) groups, Hartley, Segal for finitely generated pro-soluble groups, Segal and Nikolov for finitely generated pro-finite groups the topology of such groups is uniquely determined by their algebraic structure. In the second part of the paper it is proved that the topology of finitely generated pro-nilpotent Lie ring is uniquely determined by its algebraic structure. In the third part of the paper the Kurosh problem for pro-finite Lie rings is considered. It is proved that any Engelian pro-finite Lie ring is locally nilpotent. The proof is based on the techniques developed by Wilson and Zelmanov to prove that all periodic pro-finite groups are locally finite and that all Engelian pro-finite groups are locally nilpotent. The fundamental role in the proof plays Zelmanov’s theorem on a finitely generated modular Lie algebra with ad-nilpotent commutators in generators, satisfying non-trivial polynomial identity. At the end of the paper a series of open problems is formulated.

Reviewer: Michael Kuznetsov (Nizhnij Novgorod)

MSC:

17B60	Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B50	Modular Lie (super)algebras
20E18	Limits, profinite groups
20F40	Associated Lie structures for groups

Keywords:

Lie ring; \(p\)-adic Lie algebra; pro-\(p\) group; Kurosh problem

References:

[1]	Bahturin, J.A., Lectures on Lie algebras, Studien zur algebra und ihre anwendungen, vol. 4, (1978), Akademie-Verlag Berlin, lectures given at Humboldt University, Berlin and Lomonosov University, Moscow · Zbl 0396.17006
[2]	Bahturin, Y.A., Identical relations in Lie algebras, (1987), VNU Science Press b.v. Utrecht, translated from the Russian by Bakhturin
[3]	Dixon, J.D.; du Sautoy, M.P.F.; Mann, A.; Segal, D., Analytic pro-p groups, Cambridge stud. adv. math., vol. 61, (1999), Cambridge University Press Cambridge · Zbl 0934.20001
[4]	Golod, E.S., Some problems of Burnside type, (), 284-289 · Zbl 0206.32402
[5]	Hartley, B., Subgroups of finite index in profinite groups, Math. Z., 168, 1, 71-76, (1979) · Zbl 0394.20020
[6]	Hirano, Y., On extensions of rings with finite additive index, Math. J. okayama univ., 32, 93-95, (1990) · Zbl 0749.16020
[7]	Lazard, M., Groupes analytiques p-adiques, Inst. hautes études sci. publ. math., 26, 389-603, (1965) · Zbl 0139.02302
[8]	Lubotzky, A.; Mann, A., Powerful p-groups. I. finite groups, J. algebra, 105, 2, 484-505, (1987) · Zbl 0626.20010
[9]	Lubotzky, A.; Mann, A., Powerful p-groups. II. p-adic analytic groups, J. algebra, 105, 2, 506-515, (1987) · Zbl 0626.20022
[10]	Lubotzky, A.; Mann, A., On groups of polynomial subgroup growth, Invent. math., 104, 3, 521-533, (1991) · Zbl 0743.20040
[11]	Lubotzky, A.; Segal, D., Subgroup growth, Progr. math., vol. 212, (2003), Birkhäuser Verlag Basel · Zbl 1071.20033
[12]	Nikolov, N.; Segal, D., Finite index subgroups in profinite groups, C. R. math. acad. sci. Paris, 337, 5, 303-308, (2003) · Zbl 1033.20029
[13]	Ol’shanskii, A.Y., A simplification of Golod’s example, (), 263-265 · Zbl 0864.20024
[14]	Petrogradsky, V.M., Growth of subalgebras for restricted Lie algebras and transitive actions, Internat. J. algebra comput., 15, 5-6, 1151-1168, (2005) · Zbl 1150.17019
[15]	Ribes, L.; Zalesskii, P., Profinite groups, Ergeb. math. grenzgeb. (3), vol. 40, (2000), Springer-Verlag Berlin · Zbl 0949.20017
[16]	Riley, D.M.; Semple, J.F., Completion of restricted Lie algebras, Israel J. math., 86, 1-3, 277-299, (1994) · Zbl 0817.17020
[17]	Riley, D.M.; Tasić, V., On the growth of subalgebras in Lie p-algebras, J. algebra, 237, 1, 273-286, (2001) · Zbl 1028.17012
[18]	Segal, D., Ideals of finite index in a polynomial ring, Q. J. math. Oxford ser. (2), 48, 189, 83-92, (1997) · Zbl 0877.11064
[19]	Segal, D., On the growth of ideals and submodules, J. London math. soc. (2), 56, 2, 245-263, (1997) · Zbl 0923.13013
[20]	Segal, D., Closed subgroups of profinite groups, Proc. London math. soc. (3), 81, 1, 29-54, (2000) · Zbl 1030.20017
[21]	Serre, J.-P., Galois cohomology, (1997), Springer-Verlag Berlin, translated from the French by Patrick Ion and revised by the author
[22]	Shalev, A., Characterization of p-adic analytic groups in terms of wreath products, J. algebra, 145, 1, 204-208, (1992) · Zbl 0745.20029
[23]	Strade, H.; Farnsteiner, R., Modular Lie algebras and their representations, Monogr. textbooks pure appl. math., vol. 116, (1988), Marcel Dekker New York · Zbl 0648.17003
[24]	Ursul, M., Topological rings satisfying compactness conditions, Math. appl., vol. 549, (2002), Kluwer Academic Publishers Dordrecht · Zbl 1041.16037
[25]	Weigel, T., A remark on the ado – iwasawa theorem, J. algebra, 212, 2, 613-625, (1999) · Zbl 1043.17501
[26]	Wilson, J.S., On the structure of compact torsion groups, Monatsh. math., 96, 1, 57-66, (1983) · Zbl 0511.20020
[27]	Wilson, J.S., Profinite groups, London math. soc. monogr. (N.S.), vol. 19, (1998), Clarendon Press/Oxford University Press New York · Zbl 0909.20001
[28]	Wilson, J.S.; Zelmanov, E.I., Identities for Lie algebras of pro-p groups, J. pure appl. algebra, 81, 1, 103-109, (1992) · Zbl 0851.17007
[29]	Zelmanov, E.I., On periodic compact groups, Israel J. math., 77, 1-2, 83-95, (1992) · Zbl 0786.22008
[30]	Zelmanov, E.I., Lie ring methods in the theory of nilpotent groups, (), 567-585 · Zbl 0860.20031
[31]	Zelmanov, E.I., Nil rings and periodic groups, KMS lecture notes in math., (1992), Korean Mathematical Society Seoul, with a preface by Jongsik Kim · Zbl 0786.22008