scholarly journals f-Racks, f-quandles, their extensions and cohomology

2017 ◽  
Vol 16 (11) ◽  
pp. 1750215 ◽  
Author(s):  
Indu R. U. Churchill ◽  
Mohamed Elhamdadi ◽  
Matthew Green ◽  
Abdenacer Makhlouf
Keyword(s):  
Cohomology Theory ◽  
Low Order

The purpose of this paper is to introduce and study the notions of [Formula: see text]-rack and [Formula: see text]-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order [Formula: see text]-quandles. Moreover, we define modules over [Formula: see text]-racks, discuss extensions and define a cohomology theory for [Formula: see text]-quandles and give examples.

scholarly journals Ternary and n-ary f-distributive structures

2018 ◽  
Vol 16 (1) ◽  
pp. 32-45 ◽  
Author(s):  
Indu R. U. Churchill ◽  
M. Elhamdadi ◽  
M. Green ◽  
A. Makhlouf
Keyword(s):  
Cohomology Theory ◽  
Extension Theory ◽  
Low Dimensions

AbstractWe introduce and study ternary f-distributive structures, Ternary f-quandles and more generally their higher n-ary analogues. A classification of ternary f-quandles is provided in low dimensions. Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally n-ary, f-quandles. Furthermore, we give some computational examples.

scholarly journals Group Classification and Conservation Laws of a Class of Hyperbolic Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
J. C. Ndogmo
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Linear Dependence ◽  
Hyperbolic Equations ◽  
Group Classification ◽  
The Family ◽  
The Given ◽  
Low Order

Abstracts. A method for the group classification of differential equations is proposed. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing in the determining equations of symmetries of the equation. The method is simple and systematic and applied to a family of hyperbolic equations. Moreover, as the given family contains several known equations with important physical applications, low-order conservation laws of some relevant equations from the family are computed, and the results obtained are discussed with regard to the symmetry integrability of a particular class from the underlying family of hyperbolic equations.

scholarly journals ON THE CLASSIFICATION OF QUANDLES OF LOW ORDER

2012 ◽  
Vol 21 (09) ◽  
pp. 1250088 ◽  
Author(s):  
L. VENDRAMIN
Keyword(s):  
Type D ◽  
Transitive Groups ◽  
Low Order

Using the classification of transitive groups we classify indecomposable quandles of size < 36. This classification is available in Rig, a GAP package for computations related to racks and quandles. As an application, the list of all indecomposable quandles of size < 36 not of type D is computed.

scholarly journals A computer aided classification of certain groups of prime power order

1977 ◽  
Vol 17 (2) ◽  
pp. 257-274 ◽  
Author(s):  
Judith A. Ascione ◽  
George Havas ◽  
C. R. Leedham-Green
Keyword(s):  
Prime Power ◽  
Maximal Class ◽  
Prime Power Order ◽  
Computer Aided ◽  
Low Order

A classification of two-generator 3-groups of second maximal class and low order is presented. All such groups with orders up to 38 are described, and in some cases with orders up to 310. The classification is based on computer aided computations. A description of the computations and their results are presented, together with an indication of their significance.

scholarly journals Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants*

2021 ◽  
Vol 38 (12) ◽  
pp. 127101
Author(s):  
Yunqing Ouyang ◽  
Qing-Rui Wang ◽  
Zheng-Cheng Gu ◽  
Yang Qi
Keyword(s):  
Computational Cost ◽  
Discrete Symmetry ◽  
Cohomology Theory ◽  
Explicit Calculation ◽  
Topological Invariants ◽  
Symmetry Groups ◽  
Great Success ◽  
Generalized Cohomology ◽  
Symmetry Protected

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.

A 3-Dimensional Non-Abelian Cohomology of Groups With Applications to Homotopy Classification of Continuous Maps

1991 ◽  
Vol 43 (2) ◽  
pp. 265-296 ◽  
Author(s):  
Manuel Bullejos ◽  
Antonio M. Cegarra
Keyword(s):  
General Problem ◽  
Cohomology Theory ◽  
Homotopy Classification ◽  
3 Dimensional ◽  
Continuous Maps ◽  
Crossed Modules ◽  
Long Time ◽  
Cohomology Of Groups

The general problem of what should be a non-abelian cohomology, what is it supposed to do, and what should be the coefficients, form a set of interesting questions which has been around for a long time. In the particular setting of groups, a comprehensible and well motivated cohomology theory has been so far stated in dimensions ≤ 2, the coefficients for being crossed modules. The main effort to define an appropriate for groups has been done by Dedecker [16] and Van Deuren [40]; they studied the obstruction to lifting non-abelian 2-cocycles and concluded with first approach for , which requires “super crossed groups” as coefficients. However, as Dedecker said “some polishing work remains necessary” for his cohomology.

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