Two theorems of Lie on infinitesimal symmetries of differential equations

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. 1. Some algebra of nilpotent elements We consider a commutative ring R. Recall that a ∈ R is nilpotent if ak = 0 for some natural number k = 1, 2, 3, . . .. We are mainly concerned with the case k = 2, and define D := {d ∈ R | d2 = 0}. The 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 ( d1 + d2)2 = d2 1 + d2 2 + 2d1 · d2 = 0 + 0 + 2d1 · d2, which is not necessarily 0. The 2-dimensional analogue of D ⊆ R is the subset D(2) ⊆ D × D given by D(2) := {(d1, d2) ∈ R × R | d2 1 = 0, d 2 2 = 0, d 1 · d2 = 0}. 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 have 1.1. Proposition. For d1 and d2 in D, d1 + d2 ∈ D iff d1 · d2 = 0 iff d1 − d2 ∈ D iff (d1, d2) ∈ D(2) iff d1 ∼ d2. The n-dimensional analogue of D(2) is D(n) ⊆ Dn, described by D(n) := {(d1 . . . , dn) ∈ Rn | di · dj = 0 for all i, j = 1, . . . , n}. I 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]. Received by the editors 2024-03-29 and, in final form, 2024-06-18. Published on 2025-03-06 in the Lawvere Festschrift. 2020 Mathematics Subject Classification: 18F40, 53A17, 34A26, 18A40. Key words and phrases: infinitesimal transformation, vector field, synthetic differential geometry. © Anders Kock, 2025. Permission to copy for private use granted. 93 94 ANDERS KOCK By binomial expansion, one sees that ( d1, . . . , dn) ∈ D(n) implies P di ∈ D. We shall not in the present note have occasion to consider nilpotent elements of higher degree, such as D2 ⊆ R given by {x ∈ R | x3 = 0}. Note that we have D ⊆ D2. Also note that, for (d1, d2) ∈ D × D, we have d1 + d