887 research outputs found
Computing Nilpotent Quotients in Finitely Presented Lie Rings
A nilpotent quotient algorithm for finitely presented Lie rings over Z
(LieNQ) is described. The paper studies graded and non-graded cases separately.
The algorithm computes the so-called nilpotent presentation for a finitely
presented, nilpotent Lie ring. The nilpotent presentation consists of
generators for the abelian group and the products---expressed as linear
combinations---for pairs formed by generators. Using that presentation the word
problem is decidable in . Provided that the Lie ring is graded, it is
possible to determine the canonical presentation for a lower central factor of
. LieNQ's complexity is studied and it is shown that optimizing the
presentation is NP-hard. Computational details are provided with examples,
timing and some structure theorems obtained from computations. Implementation
in C and GAP 3.5 interface is available.Comment: DVI and Post-Script files onl
Chen Lie algebras
The Chen groups of a finitely-presented group G are the lower central series
quotients of its maximal metabelian quotient, G/G''. The direct sum of the Chen
groups is a graded Lie algebra, with bracket induced by the group commutator.
If G is the fundamental group of a formal space, we give an analog of a basic
result of D. Sullivan, by showing that the rational Chen Lie algebra of G is
isomorphic to the rational holonomy Lie algebra of G modulo the second derived
subalgebra. Following an idea of W.S. Massey, we point out a connection between
the Alexander invariant of a group G defined by commutator-relators, and its
integral holonomy Lie algebra.
As an application, we determine the Chen Lie algebras of several classes of
geometrically defined groups, including surface-like groups, fundamental groups
of certain classical link complements, and fundamental groups of complements of
complex hyperplane arrangements. For link groups, we sharpen Massey and
Traldi's solution of the Murasugi conjecture. For arrangement groups, we prove
that the rational Chen Lie algebra is combinatorially determined.Comment: 23 page
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