The left greedy Lie algebra basis and star graphs

Walter, Ben; Shiri, Aminreza

doi:10.2140/involve.2016.9.783

Cited by 3 publications

(4 citation statements)

References 10 publications

(14 reference statements)

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“…It is well known that dim Lie(m−1) = (m − 2)!, with a basis of left-normed monomials in which the leftmost generator is the first generator [15,52]. In our setting, these are the monomials 3) The monomials in the well known Lyndon basis [12] or Hall bases are generally not left-normed, though there is relatively recent work on bases of left-normed [60] and right-normed [13] monomials for a free Lie algebra.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…We use the direct-sum decomposition (1.8), which also applies over Z to the non-torsion part of the group π * (Conf(m, R n )). Fix a basis of left-normed monomials b j;I for each free graded Lie algebra summand; such a basis exists by for example [60]. Let B be the union of these bases, with elements corresponding to multi-indices (j; I).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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Diagrams for primitive cycles in spaces of pure braids and string links

Komendarczyk,

Koytcheff,

Volić

2024

Annales De l'Institut Fourier

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The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of Brunnian spherical links and certain Chen integrals. Finally we show that graphing induces injections of a certain submodule of the homotopy of configuration spaces into the homotopy of many spaces of string links. We conjecture that graphing is injective on all rational homotopy classes.Résumé. -L'espace des lacets basés d'un espace de configurations des points dans un espace euclidien peut être vu comme un espace de tresses pures dans un espace euclidien d'une dimension superieure. Nous continuons notre travail sur ces espaces du point de vue de l'ADGC de Kontsevich des diagrammes et des intégrales de Chen. Nous construisons une connexion série puissance qui donne une isomorphisme d'algèbre de Hopf entre l'homologie de l'espace des tresses pures et la construction cobar sur les diagrammes. Il mappe les classes primitives aux arbres trivalents modulo la relation IHX. En conséquence, nous établissons une bijection entre les invariants de Milnor des entrelacs sphériques bruniens et certaines intégrales de Chen. Enfin nous montrons que le graphe induit des injections d'un certain sous-module de l'homotopie des espaces de configurations dans l'homotopie des espaces de longs entrelacs. Nous conjecturons qu'il induit des injections de toute l'homotopie rationelle des espaces de configurations.

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Section: Annales De L'institut Fouriermentioning

confidence: 99%

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Diagrams for primitive cycles in spaces of pure braids and string links

Komendarczyk,

Koytcheff,

Volić

2024

Annales De l'Institut Fourier

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“…Using this terminology, our previously defined distinct-vertex Eil graphs are those for whom which all edges are heterogeneous. In [SW11] it is noted that linear graphs span E n , and in [WS16] a new basis is constructed using "star graphs". A different spanning set is crucial for our work.…”

Section: In Our Example Considering Two Different Reductions Ofmentioning

confidence: 99%

“…Thus the Hall basis, which in this case is also the Lyndon basis, is not Kronecker in pairing with letter-linking invariants. It would be interesting to see such a dual basis in general, since it seems like it would have symmetry properties which bases which use orderings on the generating set, both classical bases as well as new one such as those in [WS16], do not have.…”

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confidence: 99%

Linking of letters and the lower central series of free groups

Monroe¹,

Sinha²

2020

Preprint

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We develop invariants of the lower central series of free groups through linking of letters, showing they span the rational linear dual of the lower central series subquotients. We build on an approach to Lie coalgebras through operads, setting the stage for generalization to the lower central series Lie algebra of any group. Our approach yields a new co-basis for free Lie algebras. We compare with the classical approach through Fox derivatives.

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Linking of letters and the lower central series of free groups

Monroe¹,

Sinha²

2022

Communications in Algebra

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The left greedy Lie algebra basis and star graphs

Cited by 3 publications

References 10 publications

Diagrams for primitive cycles in spaces of pure braids and string links

Diagrams for primitive cycles in spaces of pure braids and string links

Linking of letters and the lower central series of free groups

Linking of letters and the lower central series of free groups

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