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Ana González

Publications and source records attributed to Ana González.

11 recordsLinked to original sources
arXiv2024

An Imprimitivity Theorem for finite algebraic quantum groups

Let $\mathcal{G}$ be an algebraic quantum group and $\mathcal{U}$ a compact quantum subgroup. Given a left $\hat{\mathcal{U}}$-module algebra A with unit, we can endow $A\otimes\mathcal{G}$ with a structure of a right $\hat{\mathcal{U}}$-module algebra. The algebra of invariants for this action $(A\otimes\mathcal{G})^{\hat{\mathcal{U}}}$ has a left action of $\hat{\mathcal{G}}$. We prove that for finite $\mathcal{G}$, $(A\otimes\mathcal{G})^{\hat{\mathcal{U}}}\#\hat{\mathcal{G}}$ is Morita equivalent to $A\#\hat{\mathcal{U}}$.

math.QA↗
arXiv2024

Primitive elements in infinitesimal bialgebras

For any set S, the free magmatic algebra spanned by card(S) binary products is the vector space spanned by the set of all planar rooted binary trees with the internal nodes colored by the elements of S, graded by the number of leaves of a tree. We show that it has a unique structure of coassociative coalgebra such that the coproduct satisfies the unital infinitesimal condition with each magmatic product, and prove an analog of Aguiar-Sottile formula in this context, describing the coproduct in terms of the Moebius basis for the Tamari order. The last result allows us to compute the subspace of primitive elements of any unital infinitesimal S-magmatic bialgebra. As an example, we construct a set of generators of the dual of Pilaud and Pons bialgebra of integer relations and compute an explicit basis of its subspace of primitive elements.

math.CO↗
arXiv2023

Handle element on nearly Frobenius algebras

In this article the concept of handle element of Frobenius algebras, as in [16], will be extended to nearly Frobenius algebras. The main properties of this element will be analyzed in this case and many examples will be constructed. Also the Casimir and Schur elements of symmetric algebras will be considered ([8]) and generalized for Frobenius and nearly Frobenius algebras showing which results still hold in this framework and which ones fail.

math.RA↗
arXiv2020

Nearly Frobenius dimension on Frobenius algebras

This article is divided into two parts. In the first part we work over a field $\mathbb{k}$ and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. In particular, we prove that the Frobenius dimension coincides with the dimension of the algebra. In the second part we work with a commutative ring $k$. We introduce the concept of nearly Frobenius algebras in this context and construct solutions of the Yang-Baxter equation starting from elements in the Frobenius space. Also, we give a list of equivalent characterizations of nearly Frobenius algebras.

math.RT↗
arXiv2019

Nearly Frobenius theory and semisimplicity of bimodules

In the first part of this article we prove that one of the conditions required in the original definition of nearly Frobenius algebra, the coassociativity, is redundant. Also, we determine the Frobenius dimension of the product and tensor product of two nearly Frobenius algebras from the Frobenius dimension of each of them. We apply these results to semisimple algebras. In the second part we introduce the notion of normalized nearly Frobenius algebra. We prove a series of equivalences: the concept of normalized nearly Frobenius algebra is equivalent to the concept of separable algebra, equivalent to the fact that the algebra is projective as a bimodule on itself and, finally, equivalent to the category of bimodules is semisimple. Also, we relate these concepts with the property of semisimplicity of the category of modules over the algebra.

math.RA↗
arXiv2019

Nearly Frobenius Algebras

In this introductory paper we study nearly Frobenius algebras which are generalizations of the concept of a Frobenius algebra which appear naturally in topology: nearly Frobenius algebras have no traces (co-units). We survey the most basic foundational results and some of the applications they encounter in geometry, topology and representation theory.

math.RA↗
arXiv2019

Nearly Frobenius structures in some families of algebras

In this article we continue with the study started in [1] of nearly Frobenius structures in some representative families of finite dimensional algebras, as the radical square zero algebras, string algebras and the toupie algebras. We prove that the radical square zero algebras with at least one path of length two are nearly Frobenius. As for the string algebras, in the ones that are not gentle, we can afirm that there is at least one non-trivial nearly Frobenius structure. Finally, in the case of the toupie algebras, we prove that the existence of monomial relations is a suficient condition to have non-trivial nearly Frobenius structure. Using the technics developed for the previous families of algebras we prove suficient conditions for the existence of non-trivial Frobenius structures in quotients of path algebras in general.

math.RA↗
arXiv2013

Equivariant Topological Quantum Field Theory in Dimension 2

For $G$ a finite group, we prove in dimension 2 that there is a monoidal equivalence between the category of $G$-equivariant topological quantum field theories and the category of $G$-Frobenius algebras, this was proved by G. Moore and G. Segal. This work consists to give, in more detail, a proof of this result.

math.AT↗
arXiv2013

Constructing nearly Frobenius algebras

In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver $(Q,I)$ with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver is $\overrightarrow{\mb{A}_n}$ and I=0. We also present an algorithm that determines the number of independent nearly Frobenius structures for Gentle algebras without oriented cycles.

math.RA↗
arXiv2011

The Lie bialgebra structure of the vector space of cyclic words

In this paper we give a combinatory proof of the Lie bialgebra structure presented in the vector space of reduced cyclic words. This structure was introduce by M. Chas in Combinatorial lie bialgebras of curves on surfaces, where the proof of the existence of this Lie bialgebra structure is based on the existence of an isomorphism between the space of reduced cyclic word and the space of curves on a surface.

math.AT↗