Bases, filtrations and module decompositions of free Lie algebras

Abstract

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.

Growth of structure constants of free Lie algebras relative to Hall bases

“…Staying within the scope of "monomial bases" (i.e. bases of the form e(A) for some A ⊂ Br(X), not necessarily a Hall set), one can use other constructions processes than the one yielding Hall sets, as illustrated by [12] or [41]. However, even if one uses an alternative construction to obtain a monomial basis of L(X), one could wonder if there exists a Hall set yielding, up to sign, the same basis.…”

The left greedy Lie algebra basis and star graphs

“…There are several methods to form bases for free Lie algebras using Lyndon-Shirshov words [7] [8] [4] [12]. For example, the standard bracketing [8] of an Lyndon-Shirshov word w is written [w], given by splitting the word w into two (nonempty) sub-words w = uv, such that the subword v is a maximally long Lyndon-Shirshov word, and then recursively defining Example 2.2.…”

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“…A linear basis for the free Lie algebra on an alphabet can be built on Lyndon-Shirshov words using standard bracketing [Reutenauer 1993]i (see [Melançon and Reutenauer 1989;Chibrikov 2006;Stöhr 2008] for other methods). Given a Lyndon-Shirshov word w, its standard bracketing [w] is recursively defined by…”