Related papers: Associated Graded Algebras and Coalgebras

This paper is a sequel to arXiv:2307.13358 and arXiv:2308.16090. A construction associating a semialgebra with an algebra, subalgebra, and a coalgebra dual to the subalgebra played a central role in the author's book arXiv:0708.3398. In…

We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a…

We consider central simple $K$-algebras which happen to bedifferential graded $K$-algebras. Two such algebras $A$ and $B$are considered equivalent if there are bounded complexes of finite dimensional$K$-vector spaces $C_A$ and $C_B$ such…

We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose…

A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with…

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…

We study a coderivation from a cobimodule into a coalgebra. Vector cofields are defined by the action of a codual bicomodule on a coalgebra. This action is induced by a codifferential. A construction of a codual object in the category of…

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators…

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of…

We present a new method, involving monads and comonads from category theory, to help establish a certain type of equivalence of subcategories. As a case study we consider the category of topological gradings of $C^*$-algebras over a fixed…

A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…

We develop a generalised gauge theory in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory,…

Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of…

In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras…

We introduce M\"obius strip diagram algebras (and their monoid and categorical versions) as subalgebras of a partition-style diagram calculus in which strands may carry handles and M\"obius strip features. We identify the resulting diagram…

We consider algebras and Frobenius algebras, internal to a monoidal category, that are graded over a finite abelian group. For the case that A is a twisted group algebra in a linear abelian monoidal category we obtain a graded…

We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in…

We show that two finite-dimensional Hopf algebras are gauge equivalent if and only if their bounded derived categories are monoidal triangulated equivalent. More generally, a monoidal derived equivalence between locally finite tensor…

We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent…

An algebra is said to be a unary Leibniz algebra if every one-generated subalgebra is a Leibniz algebra. An algebra is said to be a binary Leibniz algebra if every two-generated subalgebra is a Leibniz algebra. We give characterizations of…