We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provi…
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19 results for “self-distributivity”
This research classifies deformations of Yang-Baxter operators using cohomology of -Lie algebras.
problem Classifying deformations of Yang-Baxter operators via cohomology of -Lie algebras.
method Introducing a cohomology theory for -ary self-distributive objects, showing natural injections and isomorphisms, and constructing deformation theories.
result The self-distributive deformations classify the Yang-Baxter operator deformations, with nontrivial examples provided.
Study higher arity self-distributive operations and their cohomology.
problem Understanding cohomology groups of higher arity operations.
method Introduced mutually distributive n-ary operations and defined a cohomology theory.
result Geometric interpretation of cohomology in terms of framed links.
New cohomology theories for heaps and ternary operations linked to group cohomology.
problem Defining and studying cohomology theories for heaps and ternary operations.
method Introduced para-associative and heap cohomology theories, and ternary self-distributive cohomology with abelian heap coefficients.
result Heap cohomology is related to group cohomology via a long exact sequence, and injects into ternary self-distributive cohomology.
New homology theory for semi-groups with specific properties.
problem Developing a homology theory for semi-groups with self-distributivity or idempotency.
method Constructing a new homology theory and comparing it with existing theories.
result Comparison and connections with rack homology and knot theory.
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self-distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study ass…
S2D efficiently trains models to estimate uncertainty without increasing resource costs.
problem Efficiently estimating uncertainty in deep learning models for safety-critical applications.
method Self-distribution distillation (S2D) approach to train a single model for uncertainty estimation.
result S2D models outperform standard models and Monte-Carlo dropout in uncertainty estimation.
The paper computes rack homology for a specific family of quandles.
problem Computing rack homology for graphic quandles.
method Review of rack homology, computation of second rack homology groups for graphic quandles.
result Computed second rack homology groups for a large family of graphic quandles.
Heap theory applied to framed links yields new invariants.
problem Developing invariants for framed links using heap theory.
method Introducing fundamental heap, defining cocycle invariant using ternary cohomology.
result Found cocycles and computed invariants for specific link families.
This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to…
This paper investigates Lie Quandles and Leibniz Racks, extending Noether's first theorem.
problem Classifying and understanding Lie Quandles and Leibniz Racks.
method Investigates linear/nonlinear correspondences and classifies generalizations.
result Describes a nonlinear analogue of Noether's first theorem.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each …
The paper constructs Yang-Baxter solutions using categorical augmented racks.
problem Solutions to the Yang-Baxter equation in knot theory.
method Interpreting augmented racks in tensor categories and constructing solutions using quantum heaps and Hopf algebra modules.
result Explicit constructions and infinite families of Yang-Baxter solutions are provided.
While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied …
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
Study YB operators and their deformations, finding integrable and nontrivial cases.
problem Understanding deformations of Yang-Baxter operators and their integrability.
method Relating deformations to Lie algebra deformations, analyzing cohomology groups.
result Existence of integrable YB deformations and nontrivial cases not arising from SD deformations.
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
problem Characterizing extensions of augmented racks and constructing invariants for surfaces.
method Characterization of rack extensions through fibrant and additive cohomology, construction of invariants using cocycles.
result Characterization of extensions of augmented racks and construction of surface invariants.
The paper constructs new algebraic structures from Lie algebras and ternary Nambu-Lie algebras, leading to Yang-Baxter operators.
problem Constructing new algebraic structures from Lie algebras and ternary Nambu-Lie algebras.
method Using compositions of binary Lie algebras, 3-Lie algebras, and ternary Nambu-Lie algebras, the paper constructs ternary self-distributive objects and Yang-Baxter operators.
result The constructed Yang-Baxter operators are not gauge equivalent to the transposition operator and can be deformed to new solutions.