Year: 2025
Type: article
Abstract: We give an account, in terms of synthetic differential geometry, of some of Sophus Lie's geometric theory of first order differential equations.This theory is, in modern terms, formulated in terms of vector fields on manifolds. Some algebra of nilpotent elementsWe consider a commutative ring R. Recall that a ∈ R is nilpotent if a k = 0 for some natural number k = 1, 2, 3, . ... We are mainly concerned with the case k = 2, and defineThe letter 'd' will hence forward be reserved to elements of D.Note that commutativity of R implies that d ∈ D ⇒ r • d ∈ D, for any r ∈ R, in particular -d ∈ D. So the subset D is closed under multiplication.However, it is not closed under addition; indeed (The 2-dimensional analogue of D ⊆ R is the subset D(2) ⊆ D × D given by. For x and y in R, we write x ∼ y if (x -y) 2 = 0.If R has the property that x + x = 0 implies x = 0 (which we henceforth assume), we therefore haveI owe credit to Gonzalo Reyes for long collaboration on the particular subject in the present note; some of this is documented in our joint preprint [10], as well as in our article [11].
Author: Anders Kock
Institution: Aarhus University
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4412209406