Lie Algebras and Differentiations in Rings of Power Series

1950 ◽  
Vol 72 (1) ◽  
pp. 58 ◽  
Author(s):  
G. Hochschild
Keyword(s):  
Power Series ◽  
Lie Algebras ◽  
Rings Of Power Series

scholarly journals On F-integrable actions of the restricted Lie algebra of a formal group F in characteristic p > 0

1989 ◽  
Vol 115 ◽  
pp. 125-137 ◽  
Author(s):  
Andrzej Tyc
Keyword(s):  
Power Series ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Formal Power Series ◽  
Integral Domain ◽  
Formal Group ◽  
Characteristic P ◽  
Restricted Lie Algebra ◽  
Formal Power

Let k be an integral domain, let F = (F1X, Y),…, Fn(X, Y)), X = (X1,…, Xn), Y = (Y1,…, Yn), be an n-dimensional formal group over k, and let L(F) be the Lie algebra of all F-invariant k-derivations of the ring of formal power series k[X] (cf. § 2). If A is a (commutative) k-algebra and Derk (A) denotes the Lie algebra of all k-derivations d: A → A, then by an action of L(F) on A we mean a morphism of Lie algebras φ: L(F) → Derk (A) such that φ(dp) = φ(d)p, provided char (k) = p > 0. An action of the formal group F on A is a morphism of k-algebras D: A-→A[X] such that D(a)≡a mod (X) for a ∊ A, and FA ∘ D = DY ∘ D, where FA: A[X] → A[X, Y], DF: A[X] → A[X, Y] are morphisms of k-algebras given by FA(g(X)) = g(F), DY(Σa aaXa) = Σa D(aa)Y; for a motivation of this notion, see [15]. Let D: A → A[X] be such an action.

Automorphisms of the completion of relatively free Lie algebras

2018 ◽  
Vol 28 (06) ◽  
pp. 1091-1100
Author(s):  
C. E. Kofinas
Keyword(s):  
Automorphism Group ◽  
Power Series ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Free Lie Algebra ◽  
Central Series ◽  
Dense Subgroup ◽  
Free Lie Algebras ◽  
Finite Set ◽  
Formal Power

Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.

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