# Utumi Modules

## Speaker:

Mohamed F. Yousif

## Date:

Thursday, 28 June, 2018 - 11:00

## Venue:

Room 1.08, Mathematics building, FCUP

A right R-module M is called a Utumi Module (U-module) if, whenever A and B are isomorphic submodules of M with A ∩ B = 0, there exist two summands K and T of M such that A is an essential submodule of K, B is an essential submodule of T and K ⊕ T is a direct summand of M . The class of U -modules is a simultaneous and strict generalization of three fundamental classes of modules; namely the quasi-continuous, the square-free and the automorphism-invariant modules.

# On denominator vectors of Cluster Algebras

Pin Liu

## Date:

Wednesday, 23 May, 2018 - 14:00

## Venue:

Room 1.09, Mathematics building, FCUP

At the beginning of this century, Fomin and Zelevinsky invented a new class of algebras called cluster algebras motivated by total positivity in algebraic groups and canonical bases in quantum groups. Since their introduction, cluster algebras have found application in a diverse variety of settings which include Poisson geometry, Teichmüller theory, tropical geometry, algebraic combinatorics and last not least the representation theory of quivers and finite dimensional algebras.

# Long cycles in Hamiltonian graphs

António Girão

## Date:

Friday, 20 April, 2018 - 15:30

## Venue:

Room 1.22, Mathematics building, FCUP

In 1975, Sheehan conjectured that every d-regular Hamiltonian graph contains a second Hamiltonian cycle. This conjecture has been verified for all d greater than 22. In the light of Sheehan’s conjecture, it is natural to ask if regularity is genuinely necessary to force the existence of a second Hamiltonian cycle, or if a minimum degree condition is enough.

# Hopf algebras and their finite dual.

Miguel Couto

## Date:

Friday, 24 November, 2017 (All day)

## Venue:

Room FC1.122, DMat-FCUP, 15h30 - 16h30

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

José Mª Sánchez

## Date:

Monday, 11 September, 2017 (All day)

## Venue:

11h, room 004, FC1 (Maths Bldg)

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

André Leroy

## Date:

Wednesday, 26 April, 2017 (All day)

## Venue:

FCUP, Maths building FC1, room 1.22 at 11:30

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

Alberto Facchini

## Date:

Friday, 24 March, 2017 (All day)

## Venue:

Room 1.22, Mathematics building, FCUP

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

## Speaker:

Manuel José Ribeiro de Castro Silva Martins

## Date:

Tuesday, 29 November, 2016 - 15:00

## Venue:

FCUP, Maths building FC1, room 0.06

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

Andrea Solotar

## Date:

Friday, 17 June, 2016 - 10:00

## Venue:

Room 004 (FC1-Maths Building)

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

## Speaker:

J.M. Casas (University of Vigo)

## Date:

Friday, 21 November, 2014 - 11:30

## Venue:

Room 0.29, Mathematics building, FCUP

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.