Extensions of racks and quandles

Jackson, Nancy Ewald

doi:10.4310/hha.2005.v7.n1.a8

Cited by 16 publications

(18 citation statements)

References 11 publications

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“…The natural coefficients for Quillen cohomology in general are the Beck modules. In the case of racks and quandles, these Beck modules have already been discussed by Jackson [15]. In the prequel [34] to this paper, we showed that a canonical refinement of the Alexander module of a knot, the Alexander-Beck module, detects the unknot.…”

Section: Introductionsupporting

confidence: 54%

“…Let M be an X-module in the sense of Beck. (We refer again to [15] and [34] for the theory of Beck modules with an emphasis on racks and quandles.) In other words, that X-module M is an abelian group object in the slice category T X .…”

Section: Quillen Cohomology and Extmentioning

confidence: 99%

“…These invariants have found numerous applications, not only to knot theory. See [1,6,7] for instance, as well as [4,24,26,22,27,15] for various other approaches to rack and quandle (co)homology groups.…”

Section: Introductionmentioning

confidence: 99%

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Quandle cohomology is a Quillen cohomology

Szymik

2018

Trans. Amer. Math. Soc.

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Racks and quandles are fundamental algebraic structures related to the topology of knots, braids, and the Yang-Baxter equation. We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. This makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change).MSC: 18G50 (18C10, 20N02, 55U35, 57M27)

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Section: Introductionsupporting

confidence: 54%

Section: Quillen Cohomology and Extmentioning

confidence: 99%

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Quandle cohomology is a Quillen cohomology

Szymik

2018

Trans. Amer. Math. Soc.

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“…This complex is the same as the one defined in [Jac05], but in the left rack context. Adapting the proof given by N. Jackson in [Jac05], one easily sees that the second cohomology group HR 2 (X, A) is in bijection with the set of equivalence classes of abelian extensions of a pointed rack X by a X-module A. An abelian extension of a pointed rack X by a X-module A is a surjective pointed rack homomorphism E p X which satisfies the following axioms (E 0 ) for all x ∈ X, there is a simply transitively right action of A on p −1 (x).…”

Section: Pointed Rack Cohomologymentioning

confidence: 99%

The local integration of Leibniz algebras

Covez¹

2013

Ann. inst. Fourier

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This article gives a local answer to the coquecigrue problem. Hereby we mean the problem, formulated by J-L. Loday in [Lod93], is that of finding a generalization of the Lie's third theorem for Leibniz algebra. That is, we search a manifold provided with an algebraic structure which generalizes the structure of a (local) Lie group, and such that the tangent space at a distinguished point is a Leibniz algebra structure. Moreover, when the Leibniz algebra is a Lie algebra, we want that the integrating manifold is a Lie group. In his article [Kin07], M.K. Kinyon solves the particular case of split Leibniz algebras. He shows, in particular, that the tangent space at the neutral element of a Lie rack is provided with a Leibniz algebra structure. Hence it seemed reasonable to think that Lie racks give a solution to the coquecigrue problem, but M.K. Kinyon also showed that a Lie algebra can be integrated into a Lie rack which is not a Lie group. Therefore, we have to specify inside the category of Lie racks, which objects are the coquecigrues. In this article we give a local solution to this problem. We show that every Leibniz algebra becomes integrated into a local augmented Lie rack. The proof is inspired by E. Cartan's proof of Lie's third theorem, and, viewing a Leibniz algebra as a central extension by some center, proceeds by integrating explicitely the corresponding Leibniz 2-cocycle into a rack 2-cocycle. This proof gives us a way to construct local augmented Lie racks which integrate Leibniz algebras, and this article ends with an example of the integration of a non split Leibniz algebra in dimension 5.

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“…Many variations of this idea exist, including the quandle homology described in [3], the twisted quandle homology described in [4], the quandle homology with coefficients in a quandle module ( [1], [10]), the biquandle version known as YangBaxter homology ( [5]), and a unifying approach to the homology of quandles and Lie algebras ( [2]). …”

Section: Yang-baxter Cohomologymentioning

confidence: 99%

Virtual Yang-Baxter cocycle invariants

Ceniceros

Nelson

2009

Trans. Amer. Math. Soc.

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Abstract. We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

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Extensions of racks and quandles

Cited by 16 publications

References 11 publications

Quandle cohomology is a Quillen cohomology

Quandle cohomology is a Quillen cohomology

The local integration of Leibniz algebras

Virtual Yang-Baxter cocycle invariants

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