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In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading…
We construct a category equivalent to the category $\mathbf{Mon}$ of monoids and monoid homomorphisms, based on categories with strict factorization systems. This equivalence is then extended to the category $\mathbf{Mon_s}$ of unital…
For a Frobenius abelian category $\mathcal{A}$, we show that the category ${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$ modulo…
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that…
We develop the Morita theory of fusion 2-categories. In order to do so, we begin by proving that the relative tensor product of modules over a separable algebra in a fusion 2-category exists. We use this result to construct the Morita…
We give a classification of semisimple and separable algebras in a multi-fusion category over an arbitrary field in analogy to Wedderben-Artin theorem in classical algebras. It turns out that, if the multi-fusion category admits a…
In a triangulated symmetric monoidal closed category, there are natural dualities induced by the internal Hom. Given a monoidal functor f^* between two such catgories and adjoint couples (f^*,f_*) and (f_*,f^!), we prove the necessary…
We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…
We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give…
We study the symmetric monoidal 2-category of finite semisimple module categories over a symmetric fusion category. In particular, we study $E_n$-algebras in this 2-category and compute their $E_n$-centers for $n=0,1,2$. We also compute the…
By providing equivalent definitions of fractional Brauer configuration algebras in certain special cases, we associate to each monomial algebra some combinatorial data called a fractional Brauer configuration, from which we construct a…
From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the…
The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study extension dimensions linked by recollements of derived module categories…
To a quiver we associate a finite length monoidal abelian category which categorifies the corresponding preprojective K-theoretic Hall algebra of Varagnolo-Vasserot. The simples in this category provide a (dual) canonical basis of the Hall…
We introduce the notions of a commutative square ring $R$ and of a quadratic map between modules over $R$, called $R$-quadratic map. This notion generalizes various notions of quadratic maps between algebraic objects in the literature. We…
In this paper, we introduce the cofibrant derived category of a group algebra $kG$ and study its relation to the derived category of $kG$. We also define the cofibrant singularity category of $kG$, whose triviality characterizes the…
We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and…
For a semisimple multiring category with left duals, we prove that the unit object is simple if and only if the tensor functors by any non-zero algebra are separable (resp. faithful, resp. Maschke, resp. dual Maschke, resp. conservative).…