##
**Locally nilpotent groups without torsion, complete over simple topological fields.
(Lokal nilpotente Gruppen ohne Torsion, die über einfachen topologischen Körpern vollständig sind.)**
*(Russian)*
Zbl 0068.25701

The author proposes the following object: A topological group \(G\) which is simultaneously a topological Lie algebra (possibly infinite-dimensional) over a topological field \(F\) of characteristic \(0\) in such a way that the Campbell-Hausdorff series converges to the (group) product. One example is provided by taking \(G\) to be a free Lie algebra over \(F\) with the obvious topology, defining the group operation via Campbell-Hausdorff. For a second example, take \(G\) to be a locally nilpotent Lie algebra; then the Campbell-Hausdorff series always expires in a finite number of terms and there are no convergence problems. In the first three sections these examples are studied and generalized, uniqueness theorems are proved, etc.

In §4 the author starts anew with a topological group \(G\) where it is assumed that there are sufficiently many continuous homomorphisms of the additive group of \(F\) so as to cover \(G\) (this is the completeness over \(F\) referred to in the title); it is further assumed that \(F\) is simple (has no proper closed subfields). Note that \(G\), as a consequence, is divisible. If one further assumes that \(G\) is locally nilpotent and torsion-free, it is shown that \(G\) is attached to a Lie algebra in the fashion described above. If \(F\) is the field of \(p\)-adic numbers, \(G\) is abelian and satisfies the second axiom of countability, then \(G\) is isomorphic to the direct sum of a finite number of copies of \(F\); in the proof the author makes use of the structure theorem of G. W. Mackey [Bull. Am. Math. Soc. 52, 940–944 (1946; Zbl 0063.03693)]. It is noted that one cannot drop the second axiom of countability.

In §4 the author starts anew with a topological group \(G\) where it is assumed that there are sufficiently many continuous homomorphisms of the additive group of \(F\) so as to cover \(G\) (this is the completeness over \(F\) referred to in the title); it is further assumed that \(F\) is simple (has no proper closed subfields). Note that \(G\), as a consequence, is divisible. If one further assumes that \(G\) is locally nilpotent and torsion-free, it is shown that \(G\) is attached to a Lie algebra in the fashion described above. If \(F\) is the field of \(p\)-adic numbers, \(G\) is abelian and satisfies the second axiom of countability, then \(G\) is isomorphic to the direct sum of a finite number of copies of \(F\); in the proof the author makes use of the structure theorem of G. W. Mackey [Bull. Am. Math. Soc. 52, 940–944 (1946; Zbl 0063.03693)]. It is noted that one cannot drop the second axiom of countability.

Reviewer: Irving Kaplansky (M. R. 17, 876)

### MSC:

22A05 | Structure of general topological groups |

22E05 | Local Lie groups |

22E20 | General properties and structure of other Lie groups |