Eliezer Batista

• PhD
• Professor (Full) at Universidade Federal de Santa Catarina

In this work, we introduce the notion of a partial action of a group on a strict monoidal category. We propose, in the context of Monoidal categories, new constructions analogous to those existing for partial group actions over an algebra such as the globalization, the subalgebra of partial invariants, and the partial smash product.

Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$, and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$. To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. I...

In this work, the notion of a quantum inverse semigroup is introduced as a linearized generalization of inverse semigroups. Beyond the algebra of an inverse semigroup, which is the natural example of a quantum inverse semigroup, several other examples of this new structure are presented in different contexts; those are related to Hopf algebras, wea...

In this work, the notion of a quantum inverse semigroup is introduced as a linearized generalization of inverse semigroups. Beyond the algebra of an inverse semigroup, which is the natural example of a quantum inverse semigroup, several other examples of this new structure are presented in different contexts, those are related to Hopf algebras, wea...

We show that the category of partial comodules over a Hopf algebra $H$ is comonadic over ${\sf Vect}_k$ and provide an explicit construction of this comonad using topological vector spaces. The case when $H$ is finite dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connect...

We introduce the notion of a partial corepresentation of a given Hopf algebra H over a coalgebra C and the closely related concept of a partial H-comodule. We prove that there exists a universal coalgebra Hpar, associated to the original Hopf algebra H, such that the category of regular partial H-comodules is isomorphic to the category of Hpar-como...

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra RcoA̲ with a certain subalgebra of the smash product R#Â. Besides that, we present the notion of partial Galois coaction, which is closely related to this Mori...

We introduce the notion of a partial corepresentation of a given Hopf algebra $H$ over a coalgebra $C$ and the closely related concept of a partial $H$-comodule. We prove that there exists a universal coalgebra $H^{par}$, associated to the original Hopf algebra $H$, such that the category of regular partial $H$-comodules is isomorphic to the catego...

We introduce the notion of a dilation for a partial representation (i.e. a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (i.e.a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of parti...

We introduce the notion of a dilation for a partial representation (i.e. a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (i.e.a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of parti...

In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming fro...

In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming fro...

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.

In this work, we give a survey of recent developments in the theory of partial actions of groups and Hopf algebras.

In this work, we give a survey of recent developments in the theory of partial actions of groups and Hopf algebras.

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

As árvores com raiz, ou rooted trees, são grafos conexos, sem ciclos e com um vértice especial, chamado raiz. Um dos primeiros estudos a respeito foi feito em 1857, por Arthur Cayley. Mais tarde, em 1998, no contexto de renormalização em Teoria Quântica de Campos, Alain Connes e Dirk Kreimer construiram uma álgebra (de Hopf) usando as árvores como...

The duality between partial actions and co-actions of a Hopf algebra are fully explored in this work. The good properties of Hopf algebras with respect to duality are enlightened, giving rise to new constructions, like partial $H$ module coalgebras and partial $H$ comodule coalgebras. The inter relation between partial coactions of commutative Hopf...

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which has the universal property that each partial representation of $H$ can be factorized by an algebra morphism from...

The effectiveness of the aplication of constructions in $G$-graded $k$-categories to the computation of the fundamental group of a finite dimensional $k$-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a $k$-linear category, motivated th...

In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebra...

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to prove the existence of enveloping actions, i.e., every partial Hopf action on a algebra A is induced by a Hopf action on a algebra B that contains A as a right ideal....

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this artic...

Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construc...

This work is a short review on recent results about the Hopf algebraic approach to noncommutative differential geometry for a non specialist audience. This approach is different from the spectral triple formulation because it does not need an extra element, such as the Dirac operator, in order to construct a differential calculus. We show how the d...

We study the standard angular momentum algebra [x(i), x(j)] = tlambdaepsilon(ijk)x(k) as a noncommutative manifold R-lambda(3). There is a natural 4D differential calculus and it is possible to obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory. The space R-lambda(...

We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for...

It has been known for some time that topological geons in quantum gravity may lead to a complete violation of the canonical spin-statistics relation: There may be no connection between spin and statistics for a pair of geons. We present an algebraic description of quantum gravity in a (2 + 1)D manifold of the form Σ × ℝ, based on the first-order ca...

It is well-known that is spite of sharing some properties with conventional particles, topological geons in general violate the spin-statistics theorem. On the other hand, it is generally believed that in quantum gravity theories allowing for topology change, using pair creation and annihilation of geons, one should be able to recover this theorem....

The topology of orientable (2 + 1)d spacetimes can be captured by certain lumps of non-trivial topology called topological geons. They are the topological analogues of conventional solitons. We give a description of topological geons where the degrees of freedom related to topology are separated from the complete theory that contains metric (dynami...

In this paper we employ the construction of the Dirac bracket for the remaining current of 0305-4470/31/29/001/img6 deformed Kac - Moody algebra when constraints similar to those connecting the sl(2)-Wess - Zumino - Witten model and the Liouville theory are imposed to show that it satisfies the q-Virasoro algebra proposed by Frenkel and Reshetikhin...

The construction of a q-deformed N=2 superconformal algebra is proposed in terms of level 1 currents of U q (su(2)) quantum affine Lie algebra and a single real Fermi field. In particular, it suggests the expression for the q-deformed Energy-Momentum tensor in the Sugawara form. Its constituents generate two isomorphic quadratic algebraic structure...