Generalized Johnson homomorphisms for extended N-series

Habiro, Kazuo; Massuyeau, Gwénaël

doi:10.1016/j.jalgebra.2018.05.031

Cited by 9 publications

(32 citation statements)

References 63 publications

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“…The morphism τ is then A 0 (K, G * )-equivariant. In fact, τ can then be extended to a morphism of extended Lie algebras, in the sense of [HM17]. Ideed, their construction of an algebra of extended derivations is exactly a construction of universal actions in the category of extended Lie algebras, and their version of the Johnson morphism is exactly the one we find if we replace N -series and Lie algebras by their extended version in the constructions above.…”

Section: Johnson's Morphismsmentioning

confidence: 94%

“…These methods include a description of Andreadakis-like filtrations via a categorical framework, allowing us to state and study a p-restricted version of the problem. We answer the questions asked in [HM17] about this problem, and use our answers to study the stable p-restricted Andreadakis problem. Also, we solve the stable q-torsion Andreadakis problem for Z n , getting a complete calculation of the Lie ring of the congruence group GL n (p Z) for n 5.…”

Section: Introductionmentioning

confidence: 99%

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On the stable Andreadakis problem

Darné

2019

Journal of Pure and Applied Algebra

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Let F n be the free group on n generators. Consider the group IA n of automorphisms of F n acting trivially on its abelianization. There are two canonical filtrations on IA n : the first one is its lower central series Γ * ; the second one is the Andreadakis filtration A * , defined from the action on F n . In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ * ) and L(A * ) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included. arXiv:1711.05991v2 [math.AT]

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Section: Johnson's Morphismsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

On the stable Andreadakis problem

Darné

2019

Journal of Pure and Applied Algebra

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“…Actions of extended graded Lie algebras on graded vector spaces. We briefly review the definition of extended N-series and extended graded Lie algebras defined in [9]. We see that the functor A d gives an action of an extended N-series Aut * (F n ) op on the filtered vector space A d, * (n), that the action induces an action of an extended graded Lie algebra gr(Aut(F n ) op ) on the graded vector space B d (n) and that the induced action is given by the functor B d with morphisms β r d,k .…”

Section: Bracket Mapmentioning

confidence: 99%

Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

Katada¹

2021

Preprint

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We study a filtered vector space A d (n) over a field k of characteristic 0, which consists of Jacobi diagrams of degree d on n oriented arcs for each n, d ≥ 0. We consider an action of the automorphism group Aut(Fn) of the free group Fn of rank n on the space A d (n), which is induced by an action of handlebody groups on bottom tangles. The action of Aut(Fn) on A d (n) induces an action of the general linear group GL(n, k) on the associated graded vector space of A d (n), which is regarded as the vector space B d (n) consisting of open Jacobi diagrams. Moreover, the Aut(Fn)-action on A d (n) induces an action on B d (n) of the associated graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of Fn with respect to its lower central series. We use an irreducible decomposition of B d (n) and computation of the gr(IA(n))-action on B d (n) to study the Aut(Fn)-module structure of A d (n). In particular, we consider the case where d = 2 in detail and give an indecomposable decomposition of A 2 (n). We also consider a functor A d , which includes the Aut(Fn)-module structure of A d (n) for all n.Contents 30 References 31

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“…By considering a descending series (K m ) m≥1 of normal subgroups of π (different from the lower central series) K. Habiro and G. Massuyeau introduced in [16] a filtration of the Lagrangian mapping class group L:…”

Section: Introductionmentioning

confidence: 99%

“…The group of symplectic degree m derivations of Lie(H) can be canonically identified with the kernel D m (H) of the Lie bracket [ , ] : H ⊗ Lie m+1 (H) → Lie m+2 (H). This way, for m ≥ 1 we have homomorphisms For the alternative Johnson homomorphisms [16], consider the graded Lie algebra Lie(B; A) freely generated by B in degree 1 and A in degree 2. The m-th alternative Johnson homomorphism τ a m : J a m M → Der m (Lie(B; A)) is defined on J a m M and it takes values in the group Der m (Lie(B; A)) of degree m derivations of Lie(B; A).…”

Section: Introductionmentioning

confidence: 99%

Alternative versions of the Johnson homomorphisms and the LMO functor

Vera

2019

Preprint

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Let Σ be a compact connected oriented surface with one boundary component and let M denote the mapping class group of Σ. By considering the action of M on the fundamental group of Σ it is possible to define different filtrations of M together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of M introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by Σ. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of 3-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor. Contents 1. Introduction 2. Spaces of Jacobi diagrams and their operations 2.1. Generalities 2.2. Operations on Jacobi diagrams 3. The Kontsevich integral and the LMO functor 3.1. Kontsevich Integral 3.2. The LMO functor 4. Johnson-type filtrations 4.1. Preliminaries 4.2. Alternative Torelli group 4.3. Alternative Johnson filtration 5. Johnson-type homomorphisms 5.1. Preliminaries 5.2. Alternative Johnson homomorphisms 5.3. Alternative Johnson homomorphism on L 5.4. Diagrammatic versions of the Johnson-type homomorphisms 6. Alternative Johnson homomorphisms and the LMO functor 6.1. The filtration on Lagrangian cobordisms induced by the alternative degree 6.2. First alternative Johnson homomorphism and the LMO functor 6.3. Higher alternative Johnson homomorphisms and the LMO functor References

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Generalized Johnson homomorphisms for extended N-series

Cited by 9 publications

References 63 publications

On the stable Andreadakis problem

On the stable Andreadakis problem

Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

Alternative versions of the Johnson homomorphisms and the LMO functor

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