arXiv subjects

Miodrag C. Iovanov

Publications and source records attributed to Miodrag C. Iovanov.

12 recordsLinked to original sources

arXiv2018

On Incidence Algebras and their Representations

We provide a unified approach, via deformations of incidence algebras, to several important types of representations with finiteness conditions, as well as the combinatorial algebras which produce them. We show that over finite dimensional algebras, representations with finitely many orbits, or finitely many invariant subspaces, or distributive coincide, and further coincide with thin modules in the acyclic case. Incidence algebras produce examples of such modules, and we show that algebras which are locally hereditary, and whose projective are distributive, or equivalently, which have finitely many ideals, are precisely the deformations of incidence algebras, and they are the finite dimensional algebra analogue of Pr${\rm \ddot{u}}$fer rings. New characterizations of incidence algebras are obtained, such as they are exactly algebras which have a faithful thin module. A main consequence is that "every thin module comes from an incidence algebra": if $V$ is either a thin module over a finite dimensional algebra $A$, or $V$ is distributive and $A$ is acyclic, then $A/{\rm ann}(V)$ is an incidence algebra and $V$ can be presented as its defining representation. We classify thin/distributive modules, and respectively deformations, of incidence algebras in terms of first and second cohomology of the simplicial realization of the poset. As a main application we obtain a complete classification of thin modules over any finite dimensional algebra. Their moduli spaces are multilinear varieties, and we show that any multilinear variety can be obtained in this way. A few other applications, to Grothendieck rings of combinatorial algebras, to graphs and their incidence matrices, to linear algebra (tori actions on matrices), and to a positive answer to the "no-gap conjecture" of Ringel and Bongartz, in the distributive case, are given. Other results in the literature are re-derived.

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arXiv2016

Infinite-dimensional diagonalization and semisimplicity

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the "finite topology") on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn-Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.

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arXiv2015

Semiartinian profinite algebras have nilpotent Jacobson radical

We give a method to study the finiteness of the coradical filtration of a coalgebra; as a consequence, we show that a left semiartinian profinite algebra has nilpotent Jacobson radical and is right semiartinian too. Equivalently, we show that a for a semilocal profinite algebra, T-nilpotence implies nilpotence for the Jacobson radical. This answers two open questions from \cite{INT}.

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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 $k$.

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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.

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arXiv2011

Semiprimary selfinjective algebras with at most countable dimensional Jacobson quotient are QF

We give a positive answer to a conjecture of Faith stating that a self-injective semiprimary ring is QF, for algebras which are at most countable dimensional modulo their Jacobson radical. As a consequence of the method used, we also give short proofs of several other known positive cases of this conjecture.

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arXiv2011

On Extensions of Rational Modules

We investigate when the categories of all rational $A$-modules and of finite dimensional rational modules are closed under extensions inside the category of $C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We give a complete characterization of these two properties, in terms of a topological and a homological condition. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes of coalgebras for which rational modules are closed under extensions as well as various examples where this is not true.

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arXiv2011

Abstract integrals in algebra: coalgebras, Hopf algebras and compact groups

We generalize the results on existence and uniqueness of integrals from compact groups and Hopf algebras in a pure (co)algebraic setting, and find a series of new results on (quasi)-co-Frobenius and semiperfect coalgebras. For a coalgebra $C$, we introduce the generalized space of integrals $\int_M=\Hom^C(C,M)$ associated to a right $C$-comodule $M$ and study connections between "uniqueness of integrals" $\dim(\int_M)\leq \dim(M)$ and "existence of integrals" $\dim(\int_M)\geq \dim(M)$ for all $M$ and representation theoretic properties of $C$: (quasi)-co-Frobenius, semiperfect. We show that a coalgebra is co-Frobenius if and only if existence and uniqueness of integrals holds for any finite dimensional $M$. We give the interpretation for $\int_M$ for the coalgebra of representative functions of a compact group - they will be "quantum"-invariant vector integrals. As applications, new proofs of well known characterizations of co-Frobenius coalgebras and Hopf algebras are obtained, as well as the uniqueness of integrals in Hopf algebras. We also give the consequences for the representation theory of infinite dimensional algebras. We give an extensive class of examples which show that the results of the paper are the best possible. These examples are then used to give all the previously unknown connections between the various important classes of coalgebras appearing in literature.

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arXiv2010

Generalized Frobenius Algebras and the Theory of Hopf Algebras

"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or product powers of these objects are isomorphic. These show that QcF coalgebras can be viewed as generalizations of both co-Frobenius coalgebras and Frobenius algebras. Surprisingly, these turn out to have many applications to fundamental results of Hopf algebras. The equivalent characterizations of Hopf algebras with left (or right) nonzero integrals as left (or right) co-Frobenius, or QcF, or semiperfect or with nonzero rational dual all follow immediately from these results. Also, the celebrated uniqueness of integrals follows at the same time as just another equivalent statement. Moreover, as a by-product of our methods, we observe a short proof for the bijectivity of the antipode of a Hopf algebra with nonzero integral. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras.

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arXiv2010

The Bijectivity of the Antipode Revisited

We provide a very short approach to several fundamental results for Hopf algebras with nonzero integrals. Besides being short, our approach is the first to prove the bijectivity of the antipode without using the uniqueness of the integrals of Hopf algebras and to obtain the uniqueness of integrals as a corollary in a way similar to the classical theory of the Haar measure on compact groups.

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arXiv2009

When weak Hopf algebras are Frobenius

We investigate when a weak Hopf algebra H is Frobenius; we show this is not always true, but it is true if the semisimple base algebra A has all its matrix blocks of the same dimension. However, if A is a semisimple algebra not having this property, there is a weak Hopf algebra H with base A which is not Frobenius (and consequently, it is not Frobenius "over" A either). We give, moreover, a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of weak Hopf algebra.

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arXiv2006

Frobenius Extensions of Corings

Let $\Cc$ and $\Dd$ be two corings over a ring $A$ and $\Cc\stackrelλ{\longrightarrow}\Dd$ be a morphism of corings. We investigate the situation when the associated induced ("corestriction of scalars") functor $\Mm^\Cc\longrightarrow \Mm^\Dd$ is a Frobenius functor, and call these morphisms Frobenius extensions of corings. The characterization theorem generalizes notions such as Frobenius corings and is applied to several situations; in particular, provided some (general enough) flatness conditions hold, the notion proves to be dual to that of Frobenius extensions of rings (algebras). Several finiteness theorems are given for each case we consider; these theorems extend existing results from Frobenius extensions of rings or from Frobenius corings, showing that a certain finiteness property almost always occur for many instances of Frobenius functors.

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