Breadth and characteristic sequence of nilpotent Lie algebras

Abstract

Abstract. The notion of breadth of a nilpotent Lie algebra was introduced and used to approach problems of classification up to isomorphism in [6]. In the present paper, we study this invariant in terms of characteristic sequence, another invariant introduced by Goze and Ancochea in [2]. This permits to complete the determination of Lie algebras of breadth 2 studied in [6] and to begin the work for Lie algebras with breadth greater than 2.

Nilpotent Lie Algebras of breadth type $(0,3)$

“…In this section, we discuss the classification of breadth type (0, 2) nilpotent Lie algebras. This mostly follows from the known results about nilpotent Lie algebras of breadth 2 which have been classified in [KMS15] and [Rem17]. Note that a breadth 2 Lie algebra could be of type (0, 2) or (0, 1, 2).…”

Coadjoint Orbits of Lie Algebras and Cartan Class

“…This Lie algebra is nilpotent of dimension 3p + 2 and it has been introduced in [18] in the study of Pfaffian system of rank greater than 1 and of maximal class. To study the general case, we shall use the notion of characteristic sequence which is an invariant up to isomorphism of nilpotent Lie algebras (see for example [26] for a presentation of this notion). For any X ∈ g, let c(X) be the ordered sequence, for the lexicographic order, of the dimensions of the Jordan blocks of the nilpotent operator ad X.…”

“…This invariant was introduced in [2] in order to classify 7-dimensional nilpotent Lie algebras. A link between the notions of breath of nilpotent Lie algebra introduced in [22] and characteristic sequence is developed in [26]. If c(g) = (c 1 , c 2 , .…”

“…1. Among the Lie algebras g = t⊕I n−1 where I n−1 is an abelian ideal of dimension n−1 and t a 1-dimensional Lie subalgebra of g, there exist a family of nilpotent Lie algebras which are the "model" for a given characteristic sequence (see [20,26]). They are the nilpotent Lie algebras L n,c , c ∈ {(n − 1, 1), (n − 3, 2, 1), .…”

Rigid Lie algebras and algebraicity

“…It is also interesting to consider stable but not necessarily closed subsets W of V n or of a stable subvariety W of V n that is for every µ ∈ W then O(µ) ⊂ W. For example the subset N il n,k of N il n whose elements are k-step nilpotent, is stable for the action of GL(n, K). For this stable subset, there exists another invariant up an isomorphism which permits to describe it: the characteristic sequence of a nilpotent Lie algebras multiplication (see for example [11] for a detailled presentation of this notion). Let be µ ∈ N il n .…”