Chen Lie algebras

By Stefan Papadima, Alexander I. Suciu. 2003

70 citations recorded by the source corpus.

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 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 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 link complements in S 3 , and fundamental groups of complements of hyperplane arrangements in ℂ l . 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cd11301cf2b3cefed5c5ce89d46f517548a004ae

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Citations in the corpus

The spectral sequence of an equivariant chain complex and homology with local coefficients (2007, background): “ Sullivan in [43], for which much more can be said. This is the class of 1-formal groups, consisting of those ﬁnitely presented groups πfor which the Malcev completion is a quadratic Lie algebra (see [35] for a detailed treatment). Examples include ﬁnitely generated abelian groups and right-angled A”
Topology and geometry of cohomology jump loci (2009, unknown): “th the quotient ﬁltrations induced from E, and E′/E′′ carries the subspace ﬁltration induced from E/E′′. The adjoint action of X on E/E′′ induces a k[[X]]-module structure on E′/E′′. Theorem 3.5 from [65] gives a commutative diagram of groups, (5.5) 0//G ′/G ′′ j // κ0 G/G κ′′ p //G ab κ′ // 0 //”
Hyperplane arrangements and Milnor fibrations (2013, background): “Moreover, as shown in [51], the ranks of the Chen Lie algebra, θk = rank(grk(G/G )), are also combinatorially determined.”
The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points (2005, background): “Thus, Ln ⊗ Q is the rational holonomy Lie algebra of Mn in the sense of Chen, see [ PS ].”
Bieri–Neumann–Strebel–Renz invariants and homology jumping loci (2008, unknown): “Following Sullivan [53], the group G is said to be 1-formal if its Malcev completion (in the sense of Quillen [48]) is a quadratically presented, complete Lie algebra; we refer to [43] for more details.”
Algebraic invariants for right-angled Artin groups (2004, background): “G;R)). In the integral case, simply write H(G) = HZ(G), and note that H(G)⊗Q= HQ(G). Moreover, note that, in the case when H1(G) is torsion-free, the deﬁnition of H(G) coincides with the one given in [23]. 2.2. Formalityproperties. In Appendix A of [27], Quillen associates, in a functorial way, a Ma”
Groups and Lie algebras corresponding to the Yang–Baxter equations (2005, methodology): “ynomials with the following exponential generating functions: (3.1) 1 + X∞ n=1 Ptrn(t) un n! = ee tu−1 t , (3.2) 1+ X∞ n=1 Pqtr n (t) un n! = e u 1−tu. 3For basics about formality of groups, see e.g. [PS]. 5 (iii) The polynomials P (q)tr n (t) are given by the following explicit formulas: Ptrn(t) = ”