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Gabriela Olteanu

Publications and source records attributed to Gabriela Olteanu.

13 recordsLinked to original sources
arXiv2024

Rational group algebras of generalized strongly monomial groups: primitive idempotents and units

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra $\mathbb{Q}G$ for $G$ a finite generalized strongly monomial group. For the same groups with no exceptional simple components in $\mathbb{Q}G$, we describe a subgroup of finite index in the group of units $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$ that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.

math.RA↗
arXiv2018

Strongly Rickart objects in abelian categories

We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.

math.CT↗
arXiv2018

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.

math.CT↗
arXiv2017

Rickart and dual Rickart objects in abelian categories: transfer via functors

We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.

math.CT↗
arXiv2014

On idempotents and the number of simple components of semisimple group algebra

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly monomial groups. Further, for this class of group algebras, we describe when the number of simple components agrees with the number of simple components of the rational group algebra. Finally, we give a formula for the rank of the central units of the group ring over the ring of integers of a number field for a strongly monomial group.

math.RT↗
arXiv2014

Construction of minimal non-abelian left group codes

Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra FG for a large class of groups G. As an illustration of our methods, alternative constructions to some best linear codes over F_2 and F_3 are given.

math.RT↗
arXiv2013

Group rings of finite strongly monomial groups: central units and primitive idempotents

We compute the rank of the group of central units in the integral group ring $\Z G$ of a finite strongly monomial group $G$. The formula obtained is in terms of the strong Shoda pairs of $G$. Next we construct a virtual basis of the group of central units of $\Z G$ for a class of groups $G$ properly contained in the finite strongly monomial groups. Furthermore, for another class of groups $G$ inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of $\Q G$. Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of $\Z G$, this for metacyclic groups $G$ of the form $G=C_{q^m}\rtimes C_{p^n}$ with $p$ and $q$ different primes and the cyclic group $C_{p^n}$ of order $p^n$ acting faithfully on the cyclic group $C_{q^m}$ of order $q^m$.

math.RA↗
arXiv2012

Central units of integral group rings

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\Z G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal{Z} (\U (\Z G))$ of the unit group $\U (\Z G)$ in case $G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal{Z}(\U(\Z G))$ for all finite strongly monomial groups $G$. We call these units generalized Bass units. Finally, we show that the commutator group $\U(\Z G)/\U(\Z G)'$ and $\mathcal{Z}(\U(\Z G))$ have the same rank if $G$ is a finite group such that $\Q G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra, or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $\Q$. This allows us to prove that in this case the natural images of the Bass units of $\Z G$ generate a subgroup of finite index in $\U(\Z G)/\U(\Z G)'$.

math.RA↗
arXiv2012

Correspondences of coclosed submodules

We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if $M$ is a finitely generated quasi-projective left $R$-module with $S=End_R(M)$ and $N$ is an $M$-generated left $R$-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left $R$-submodules of $N$ and coclosed left $S$-submodules of $Hom_R(M,N)$.

math.RA↗
arXiv2007

The Schur group of an abelian number field

We characterize the maximum $r$-local index of a Schur algebra over an abelian number field $K$ in terms of global information determined by the field $K$, for $r$ an arbitrary rational prime. This completes and unifies previous results of Janusz and Pendergrass.

math.RT↗
arXiv2007

Group algebras of Kleinian type and groups of units

The algebras of Kleinian type are finite dimensional semisimple rational algebras $A$ such that the group of units of an order in $A$ is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings $RG$, with $R$ an order in a number field and $G$ a finite group, such that $RG^*$ is virtually a direct product of free-by-free groups.

math.RT↗