Differential Groups and Formal Lie Theory for An Infinite Number of Parameters

the n series 2i will be called a differential group2 of order n.3 In Part II of the present paper, we prove a theorem on structure elements of d. g. which parallels Lie's third theorem. It is shown that n bilinear expressions in the uzj and vij can be used for the terms of the second degree in a d. g. if it is possible to add terms of the third degree in such a way as to get associativity through third degree terms. The d. g. problem belongs to the Lie theory4 of formal power series in two infinite sets of indeterminates. The operation of differentiation gives the problem its individuality. This is visible in the d. g. of order one; a paper which we are preparing on d. g. of order two, of which there are curious types, will make it plainer. For basic theory, we need an extension, to the formal case with an infinite number of parameters, of Lie's theorems on structure constants. This is the topic of Part I which, qualitatively, follows classical lines, supplying the considerations needed for the infinite case. Formal Lie theory for a finite number of parameters has been treated by Bochner.5