На 7 и 8 юни 2017 г. (сряда и четвъртък) от 15:00 часа в Заседателната зала на ФМИ гостът по програма Еразъм във ФМИ-СУ Esperanza López Centella от Университета в Гренада ще ще проведе два открити семинара на следните теми:

**Title**: A brief introduction to Hopf algebras and its generalizations.

**Abstract**:

The multiples applications of Hopf algebras and its generalizations makes this theory specially valuable and interesting in Mathematics and Theoretical Physics. This series of lectures is intended to give an introduction to the basic notions and some of the main results of Hopf Algebra Theory from a purely algebraic point of view, focusing on the ideas, tools and calculus techniques of this beautiful, useful and powerful theory and its more general versions.

**Outline**:

1. Notations and conventions.

2. Algebras and coalgebras.

3. Bialgebras.

4. Hopf algebras. Relation to groups.

5. Weak bialgebras and weak Hopf algebras.

6. Frobenius algebras.

7. A categorical description of weak bialgebras.

8. Extension of the classical relation between groups and Hopf algebras to groupoids and weak Hopf algebras.

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**Title**: Weak factorizations of algebras and weak quantum duplicates.

**Abstract**:

Twisted tensor products (wreath products) are the key for the (strict) factorization problem of unital algebras. As it is well-known, for a unital algebra, there is a one-to-one correspondence between the set of factorization structures admitting two given algebras as factors and the set of so-called twisting aps (distributive laws). In this lecture we recall the factorization problem answered by a weak wreath product of algebras. Motivated by arguments from Mathematical Physics, we introduce the notion of weak quantum duplicate of an algebra, a construction based on a weak wreath product of the algebra under consideration and a two-dimensional factor. We provide a characterization of weak quantum duplicates of a finite-dimensional algebra, extending that one of quantum duplicates given by C. Cibils. As an application, we explicitly describe a great part of the set of weak factorization structures (and weak distributive laws by R. Street) existing between two two-dimensional unital algebras over a field, classifying (up-to isomorphism) the weak wreath products arising from them and covering the description by Ó. Cortadellas, G. Navarro, J. López-Peña.

**Outline**:

1. Classical factorization problem of algebras.

2. Weak factorization problem of algebras.

3. Weak quantum duplicates. Characterization.

4. Application: partial classification of weak wreath products of two-dimensional algebras.

5. Examples.

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