# Hopf algebras and their finite dual.

The subject of this talk will be Hopf algebras and their dual theory. We will mostly focus on a particular class of Hopf algebras: noetherian Hopf algebras that are finitely-generated modules over some commutative normal Hopf subalgebra. Some properties and examples of these Hopf algebras will be mentioned. Furthermore, we will see some results on the dual of this class of Hopf algebras, some of its properties, decompositions e maybe some interesting Hopf subalgebras.

# The structure of split regular BiHom-Lie algebras

After recall classical results in order to place our work, we introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. By making use of connection techniques, we focus our attention on the study of the structure of such algebras and, under certain conditions, the simplicity is characterized.

# Quasi Euclidean Rings

For a natural number k, Cooke introduced k-stage euclidean rings as a generalization of classical Euclidean rings.  His setting was entirely commutative. Later Leutbecher introduced them in a noncommutative settings but his aim was also studying commutative rings.

# Krull-Schmidt-Remak Theorem, Direct-Sum Decompositions, and G-groups

We will begin by presenting some history of the Krull-Schmidt-Remak Theorem. From groups, we will pass to modules over a ring R, introducing some direct-sum decompositions that follow a special pattern. We will consider invariants that also appear in factorisation of polynomials. Then we will go back from (right) R-modules to groups. Here the category that appears in a natural way is that of G-groups, which substitutes the category of right R-modules. In this category, Remak's result has a natural interpretation.

# Hopf Algebras and Ore Extensions

Ore extensions provide a way of constructing new algebras from preexisting ones, by adding a new indeterminate subject to commutation relations. A recent generalization of this concept is that of double Ore extensions. On the other hand, Hopf algebras are algebras which possess a certain additional dual structure. The problem of extending a Hopf algebra structure through an Ore extension has been discussed in a recent paper by Brown, O'Hagan, Zhang and Zhuang, of which we present the main result.

# Hochschild (co)homology of down-up algebras

Let $K$ be a fixed field. Given parameters $(\alpha,\beta,\gamma) \in K^{3}$, the associated down-up algebra $A(\alpha,\beta,\gamma)$ is defined as the quotient of the free associative algebra $K\cl{u,d}$ by the ideal generated by the relations

\begin{split}
d^{2} u - (\alpha d u d + \beta u d^{2} + \gamma d),\\
d u^{2} - (\alpha u d u + \beta u^{2} d + \gamma u).
\end{split}

This family of algebras was introduced by G. Benkart and T. Roby.

# A non-abelian tensor product of Hom-Lie algebras

A non-abelian tensor product of Hom-Lie algebras is constructed and studied. This tensor product is used to describe universal (α-)central extensions of Hom-Lie algebras and to establish a relation between cyclic and Milnor cyclic homologies of Hom-associative algebras satisfying certain additional condition.

# A Poisson Gel'fand-Kirillov problem in positive characteristic.

We study a problem of birational equivalence for polynomial Poisson algebras over a field of arbitrary characteristic. More precisely, the quadratic Gel'fand-Kirillov problem asks whether the field of fractions of a given polynomial Poisson algebra is isomorphic (as Poisson algebra) to a Poisson affine field, that is the field of fractions of a polynomial algebra (in several variables) where the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively this question for a large class of polynomial Poisson algebras and their Poisson prime quotients.

# Enveloping skew fields of some super Lie algebras

In a previous paper, joint with François Dumas, we had studied a family of skew fields called "Mixed Weyl skew fields". Recently we realized that for certain particular values of the parameter these appear as enveloping skew fields of certain super Lie algebras, opening the way to a super analog of the Gelfand-Kirilllov Hypothesis.

# Universal central extensions of Hom-Leibniz algebras

The main goal of this talk is to present the generalization of   classical results that characterize universal central extensions of Leibniz (Lie) algebras to the framework of Hom-Leinbiz (Hom-Lie) algebras.

In the category of Hom-Leibniz (Hom-Lie) algebras we introduce the notion of Hom-co-representation (Hom-L-module) as  adequate coefficients to construct the chain complex  from which we compute the Leibniz (Lie) homology of Hom-Leibniz (Hom-Lie) algebras.