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**Sur les groupes de Lie formels à un paramètre.**
*(French)*
Zbl 0068.25703

The author develops, for one-parameter formal Lie groups, his “direct” method (as opposed to the “infinitesimal” method of the reviewer) which has already enabled him to prove that any such formal Lie group over a ring without nilpotent elements is abelian [C. R. Acad. Sci., Paris 239, 942–945 (1954; Zbl 0055.25602)] a result which has not yet been proved by “infinitesimal” methods. The author first defines the notion of \(q\)-bud: this is a formal power series \(f(x, y)\) with no constant term and coefficients in a given ring \(A\), such that:

a) \(f(x,y) - x - y\) begins with terms of total degree \(\ge 2\);

b) \(f(f(x,y), z) - f(x,f(y,z))\) begins with terms of total degree \(\ge q+1\); the \(q\)-bud is abelian if \(f(x,y) - f(y,x)\) begins with terms of total degree (\ge q+1\).

Then the problem of extending a \(q\)-bud into a \((q+1)\)-bud is studied in detail, and the author shows that such an extension is always possible if the \(q\)-bud is abelian (and can be done in such a way that the resulting \((q+1)\)-bud is also abelian); hence it follows that any abelian \(q\)-bud can be extended to a formal one-parameter Lie group.

In the course of his proof, the author also shows that there exists a well determined one-parameter formal Lie group law \(F(x, y)\) over the ring \(P\) of polynomials with integral coefficients in a denumerable set of indeterminates, such that any one-parameter group over a ring \(A\) can be obtained by mapping the coefficients of \(F(x, y)\) on elements of \(A\) by a homomorphism of \(P\) into \(A\).

He next takes up the classification of one-parameter groups over a ring \(K\) of characteristic \(p > 0\), and in the case in which \(K\) is an algebraically closed field, obtains a new proof of the reviewer’s classification theorem [Am. J. Math. 77, 218–244 (1955; Zbl 0064.25504)].

Finally, he discusses the automorphisms of these one-parameter groups, shows that they form a compact group, which in general is non-abelian; in the particular case in which the group is the multiplicative group, he obtains also the reviewer’s result that the group of automorphisms is in that case isomorphic to the multiplicative group of \(p\)-adic integers (loc. cit., p. 241). It remains to be seen if these interesting methods can be extended to all abelian formal Lie groups and yield new proofs of the recent results of the reviewer [ibid. 77, 429–452 (1955; Zbl 0064.25602)] and perhaps new insight in the more difficult theory of non-abelian formal Lie groups.

a) \(f(x,y) - x - y\) begins with terms of total degree \(\ge 2\);

b) \(f(f(x,y), z) - f(x,f(y,z))\) begins with terms of total degree \(\ge q+1\); the \(q\)-bud is abelian if \(f(x,y) - f(y,x)\) begins with terms of total degree (\ge q+1\).

Then the problem of extending a \(q\)-bud into a \((q+1)\)-bud is studied in detail, and the author shows that such an extension is always possible if the \(q\)-bud is abelian (and can be done in such a way that the resulting \((q+1)\)-bud is also abelian); hence it follows that any abelian \(q\)-bud can be extended to a formal one-parameter Lie group.

In the course of his proof, the author also shows that there exists a well determined one-parameter formal Lie group law \(F(x, y)\) over the ring \(P\) of polynomials with integral coefficients in a denumerable set of indeterminates, such that any one-parameter group over a ring \(A\) can be obtained by mapping the coefficients of \(F(x, y)\) on elements of \(A\) by a homomorphism of \(P\) into \(A\).

He next takes up the classification of one-parameter groups over a ring \(K\) of characteristic \(p > 0\), and in the case in which \(K\) is an algebraically closed field, obtains a new proof of the reviewer’s classification theorem [Am. J. Math. 77, 218–244 (1955; Zbl 0064.25504)].

Finally, he discusses the automorphisms of these one-parameter groups, shows that they form a compact group, which in general is non-abelian; in the particular case in which the group is the multiplicative group, he obtains also the reviewer’s result that the group of automorphisms is in that case isomorphic to the multiplicative group of \(p\)-adic integers (loc. cit., p. 241). It remains to be seen if these interesting methods can be extended to all abelian formal Lie groups and yield new proofs of the recent results of the reviewer [ibid. 77, 429–452 (1955; Zbl 0064.25602)] and perhaps new insight in the more difficult theory of non-abelian formal Lie groups.

Reviewer: Jean Dieudonné (M. R. 17, 508)

### MSC:

22E99 | Lie groups |