Primordials and primordial Lie algebras

Abstract

Primordials \({d \in \mathcal{P}}\) are generalizations of ordinals \({\sigma \in \mathcal{O}}\) . Primordials are governed by their succession and precession. Primordials with their succession and precession are of interest in their own right. Remarkably, they also lead directly to certain primordial Lie algebras of set theory. Among these is the large primordial Lie algebra of set theory, whose basis is a class and not a set. The large primordial Lie algebra of set theory generalizes naturally to the large primordial Lie algebras of characteristic p ≥ 2. The simple primordial Lie algebras are the natural primordial Lie algebra \({\mathcal{L}^\natural}\) , the free primordial Lie algebras \({\mathcal{L}^c}\) for r ≥ 1 and r-tuples C of denumerable sequences Cj (1 ≤ j ≤ r) of elements of k, and, for p > 2, the normal sub Lie algebras of the \({\mathcal{L}^\natural,\mathcal{L}^c}\) as well. The split simple primordial Lie algebras are the Lie algebras L of type W—those which may be built directly from the natural primordial Lie algebra \({\mathcal{L}^\natural}\)—except when p = 2 and L is not free. Consequently, they are, up to isomorphism, the purely inseparable forms of the finite and infinite dimensional Lie algebras of type W. This sheds new light on, and adds interest to, the structure of these purely inseparable forms.