Related papers: Classification and structure of generalized Legend…
A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian…
We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic…
In this paper, we explore the algebraic structure of GL-racks, and demonstrate that finite GL-racks decompose canonically into permutation GL-racks and block GL-racks. As a corollary, we verify that two Legendrian knots with the same…
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the…
We study rack and quandle coverings from a universal algebraic viewpoint and we show how they can be understood using the notion of strongly abelian congruences. We provide an abstract characterization of several particular types of…
A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations…
We study generic graded contractions of Lie algebras from the perspectives of group cohomology, affine algebraic geometry and monoidal categories. We show that generic graded contractions with a fixed support are classified by a certain…
Virtual racks and virtual quandles are nonassociative algebraic structures based on the Reidemeister moves of virtual knots. In this note, we enumerate virtual dihedral quandles and several families of virtual permutation racks and virtual…
There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…
A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e., oriented compact surfaces with boundaries embedded…
In categorical realizability, it is common to construct categories of assemblies and categories of modest sets from applicative structures. These categories have structures corresponding to the structures of applicative structures. In the…
Given a symmetric Leibniz algebra $(\mathcal{L},.)$, the product is Lie-admissible and defines a Lie algebra bracket $[\;,\;]$ on $\mathcal{L}$. Let $G$ be the connected and simply-connected Lie group associated to $(\mathcal{L},[\;,\;])$.…
In this paper we consider representations of generalized $k$-linear Reedy categories $\underline{\mathscr{C}}$, a common generalization of $k$-linear Reedy categories introduced by Georgiois-\v{S}t'ov\'{\i}\v{c}ek and $k$-linearizations of…
The Lie algebra $gl(V)$ is the Lie algebra of all endomorphisms of a countable-dimensional complex vector space $V$. We define a tensor category of topological representations of the Lie algebra $gl(V)$, so that $V$, its dual and the…
We show that a structural matrix algebra $A$ is isomorphic to the endomorphism algebra of an algebraic-combinatorial object called a generalized flag. If the flag is equipped with a group grading, an algebra grading is induced on $A$. We…
The goal of this paper is to present results which are consistent with conjectures about the Leibniz (co)homology for discrete groups stated by J. L. Loday. We show that rack cohomology has properties very close to the properties expected…
We show that if $V$ is a vertex operator algebra such that all the irreducible ordinary $V$-modules are $C_1$-cofinite and all the grading-restricted generalized Verma modules for $V$ are of finite length, then the category of finite length…
In this paper, we introduce the category of Lie $n$-racks and generalize several results known on racks. In particular, we show that the tangent space of a Lie $n$-Rack at the neutral element has a Leibniz $n$-algebra structure. We also…
We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of…
The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state…