TY - JOUR

T1 - Quandle cohomology, extensions and automorphisms

AU - Bardakov, Valeriy

AU - Singh, Mahender

PY - 2021/11/1

Y1 - 2021/11/1

N2 - A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles. (C) 2021 Elsevier Inc. All rights reserved.

AB - A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles. (C) 2021 Elsevier Inc. All rights reserved.

KW - Automorphism

KW - Dynamical cocycle

KW - Factor set

KW - Group extension

KW - Group cohomology

KW - Quandle module

KW - Quandle cohomology

KW - Quandle extension

KW - INVARIANTS

KW - RACKS

U2 - 10.1016/j.jalgebra.2021.06.016

DO - 10.1016/j.jalgebra.2021.06.016

M3 - Article

VL - 585

SP - 558

EP - 591

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -