1-Parameter Subgroups of the Circle-Exponent Function in A-Convex Algebras (Report‪)‬

1. INTRODUCTION In differential geometry one speaks of integrable curves (solutions) of differential equations defined by (differentiable) vector fields on smooth manifolds. Yet, under the action of a Lie group on the particular manifold at issue ("Klein geometry"), one considers invariant vector fields, whose the Lie algebra is the "tangent space" of the Lie (action) group at the neutral element. We solve in the sequel analogous differential equations, by considering A-convex (topological) algebras, in connection with the underlying locally convex spaces, and the action of the (additive) Lie group R on the group of invertible and/or or quasi-invertible elements of the algebra. Precisely, the corresponding action of R on the latter group(s) is achieved, via the exponent/c-exponent function. So we arive at the "Klein geometry" in A-convex algebras (A. Mallios [3, 4]), through the latter functions.