arXiv subjects

Constantin Nastasescu

Publications and source records attributed to Constantin Nastasescu.

9 recordsLinked to original sources

arXiv2025

A linear algebra approach to graded Frobenius algebras

If $A$ is a finite-dimensional algebra graded by a group $G$, and $σ\in G$, we define a variant of paratrophic matrix associated with $A$ and $σ$, and we use it to characterize the $σ$-graded Frobenius property for $A$. We discuss the invertibility of such paratrophic matrices, and then use them to check whether certain graded algebras are $σ$-graded Frobenius or (graded) symmetric. As an application, we uncover (graded) Frobenius and symmetric properties of Koszul duals of quantum polynomial algebras. We derive a structure result for $σ$-graded Frobenius algebras by only using linear algebra methods.

math.RA↗

arXiv2025

Graded Frobenius algebras from tensor algebras of bimodules

We consider certain quotient algebras of tensor algebras of bimodules $M$ over a finite-dimensional algebra $R$, and we investigate Frobenius type properties of such algebras. Our main interest is in the case where $M=R^*$, the linear dual of $R$. We obtain a large class of Frobenius or symmetric algebras, which are also equipped with a finite grading.

math.RA↗

arXiv2024

Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras

Our initial aim was to answer the question: does the Frobenius (symmetric) property transfers from a strongly graded algebra to its homogeneous component of trivial degree? Related to it, we investigate invertible bimodules and the Picard group of a finite dimensional quasi-Frobenius algebra $R$. We compute the Picard group, the automorphism group and the group of outer automorphisms of a $9$-dimensional quasi-Frobenius algebra which is not Frobenius, constructed by Nakayama. Using these results and a semitrivial extension construction, we give an example of a symmetric strongly graded algebra whose trivial homogeneous component is not even Frobenius. We investigate associativity of isomorphisms $R^*\ot_RR^*\simeq R$ for quasi-Frobenius algebras $R$, and we determine the order of the class of the invertible bimodule $H^*$ in the Picard group of a finite dimensional Hopf algebra $H$. As an application, we construct new examples of symmetric algebras.

math.RA↗

arXiv2022

Graded Frobenius Rings

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.

math.RA↗

arXiv2017

Graded semisimple algebras are symmetric

We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.

math.RA↗

arXiv2016

Symmetric algebras of corepresentations and smash products

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra $H$; for the symmetric property $H$ is assumed to be cosovereign. If $H$ is finite dimensional and $A$ is an $H$-comodule algebra, we uncover the connection between $A$ and the smash product $A\# H^*$ with respect to the Frobenius and symmetric properties.

math.RA↗

arXiv2013

Frobenius algebras of corepresentations: gradings

We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.

math.QA↗

arXiv2012

Path subcoalgebras, finiteness properties and quantum groups

We study subcoalgebras of path coalgebras that are spanned by paths (called path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a unifying approach for these classes. We discuss the left quasi-co-Frobenius and the left co-Frobenius properties for these coalgebras. We classify the left co-Frobenius path subcoalgebras, showing that they are direct sums of certain path subcoalgebras arising from the infinite line quiver or from cyclic quivers. We also discuss the coreflexive property for the considered classes of coalgebras. Finally, we investigate which of the co-Frobenius path subcoalgebras can be endowed with Hopf algebra structures, in order to produce some quantum groups with non-zero integrals, and classify all these structures over a field with primitive roots of unity of any order. These turn out to be liftings of quantum lines over certain not necessarily abelian groups.

math.QA↗

arXiv2011

The Dickson Subcategory Splitting Conjecture for Pseudocompact Algebras

Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^*$ where $C$ is a coalgebra. We show that the if the semiartinian part (the "Dickson" part) of every $A$-module $M$ splits off in $M$, then $A$ is semiartinian, also giving a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith.

math.CT↗