arXiv subjects

Sorin Dascalescu

Publications and source records attributed to Sorin Dascalescu.

12 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↗

arXiv2018

Structural matrix algebras, generalized flags and gradings

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 classify the gradings obtained in this way as the orbits of the action of a double semidirect product on a certain set. Under some conditions on the associated graph, all good gradings on $A$ are of this type. As a bi-product, we obtain a new approach to compute the automorphism group of a structural matrix algebra.

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↗

arXiv2015

Frobenius Structural Matrix Algebras

We discuss when the incidence coalgebra of a locally finite preordered set is right co-Frobenius. As a consequence, we obtain that a structural matrix algebra over a field $k$ is Frobenius if and only if it consists, up to a permutation of rows and columns, of diagonal blocks which are full matrix algebras over $k$.

math.RT↗

arXiv2015

Semiperfect and coreflexive coalgebras

We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we introduce a new notion of left coreflexivity for counital coalgebras, namely, a coalgebra is left coreflexive if $C$ is isomorphic canonically to the finite dual of its left rational dual $Rat(_{C^*}C^*)$. We show that right semiperfectness for coalgebras is in fact essentially equivalent to this left reflexivity condition, and we give the connection to usual coreflexivity. As application, we give a generalization of some recent results connecting dual objects such as quiver or incidence algebras and coalgebras, and show that Hopf algebras with non-zero integrals (compact quantum groups) are coreflexive.

math.RT↗

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↗

arXiv2001

Co-Frobenius Hopf algebras and the coradical filtration

We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius, i. e. there is a non-zero integral on it. As a consequence, we show that algebras of functions on quantum groups at roots of one are co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. This characterization is in the framework of the lifting method due to H.-J. Schneider and the first-named author. Here is our main result. Let H be a Hopf algebra whose coradical is a Hopf subalgebra. Let gr H be the associated graded Hopf algebra and let R be the diagram of H. Then the following are equivalent: (1) H is co-Frobenius, (2) gr H is co-Frobenius, (3) R is finite dimensional, (4) the coradical filtration of H is finite. This Theorem allows to construct many new examples of co-Frobenius Hopf algebras and opens the way to the classification of ample classes of such Hopf algebras.

math.QA↗