The following courses and events will run during the two-week research school:
Professor Jesús A. Laliena Clemente
Every Lie algebra L can be embeded in an associative algebra A in which the associative product has been changed by the bracket product [a,b] = ab – ba. The new algebra built in this way is denoted by A–. I.N. Herstein and W. Baxter noticed this feature and, in the 1970’s, they researched the relation among the ideals of the associative algebra A and the ideals of the Lie algebra A–.
In this course, we will study the relationship between the set of ideals of an associative algebra A, and the set of ideals of the Lie algebra obtained from A by replacing the associative product ab in A by [a,b] = ab – ba. In addition, if A has an involution, we will study the relationship between the ideals of A and the ideals of the Lie algebra consisting of the skew symmetric elements of A.
An introduction to Lie algebras
Professor Mª del Pilar Benito Clavijo
This is a introductory course to the theory of Lie algebras, including a treatment of nilpotent, soluble and semisimple Lie algebras. In addition, we briefly talk about representation of Lie algebras and the relation between Lie algebras and other non-associative algebraic structures.
A constructive approach to the structure theory for the group algebra of Sn
Professor Murray Bremner
We discuss applications of representation theory of symmetric groups Sn to polynomial identities for associative and nonassociative algebras:
- Section 1 presents complete proofs of the classical structure theory for the group algebra FSn over a field F of characteristic 0 (or p > n). We obtain a constructive version of the Wedderburn decomposition which gives an isomorphism ψ from the group algebra to the direct sum of simple two-sided ideals isomorphic to full matrix algebras. Alfred Young showed how to compute ψ; to compute ψ−1, we use an efficient algorithm for representation matrices discovered by Clifton.
- Section 2 discusses constructive methods which allow us to analyze the polynomial identities satisfied by a specific (non)associative algebra: the fill and reduce algorithm, the module generators algorithm, and Bondari’s algorithm for finite dimensional algebras.
- Section 3 applies these methods to the study of multilinear identities satisfied by octonion algebras over a field of characteristic 0.
An exceptional Lie algebra: G2
Dr Cristina Draper Fontanals
The Killing-Cartan classification of finite-dimensional complex simple Lie algebras was one of the great milestones of 19th century mathematics. According to it, there are four infinite families of classical simple Lie algebras (special linear, orthogonal and symplectic) and five isolated exceptional examples, G2, F4, E6, E7 and E8, of dimensions 14, 52, 78, 133, and 248 respectively.
In this brief course, we would like to speak about the smallest of the exceptional algebras, G2, as well as its relationship with another relevant nonassociative algebra, the octonion algebra, for which G2 is the derivation algebra. We will use this example to illustrate the structure theory of simple Lie algebras over ℂ, while giving some hints about the classification over the reals. Hopefully we speak about the relevance of G2 to Geometry or Physics.
Graded ring theory with applications to the study of Leavitt path algebras
Dr Roozbeh Hazrat
In ring theory, in many instances there is a possibility of partitioning the structure of the ring and then rearranging the partitions. These rings are called graded rings. This adds an extra layer of structure (and complexity) to the theory. We will discuss the natural grading of Leavitt path algebras, and consider the category of graded modules. We study the information that comes out of this grading. We recall the concept of the graded Grothendieck group and summarise the results on the classification of Leavitt path algebras via graded K-theory.
Ample groupoid algebras
Dr Lisa Orloff Clark
A groupoid is a generalisation of a group in which the operation is only partially defined. Groupoids are very general objects that appear in a variety of different mathematical settings. In this course, we will begin with an introduction to ample groupoids and their associated algebras. Then we will show how Leavitt path algebras can be realised as ample groupoid algebras. Finally, we will demonstrate how the groupoid model can give valuable insight by looking at the ideal structure of Leavitt path algebras.
Leavitt path algebras
Dr Dolores Martín Barquero and Professor Cándido Martín González
This is an introductory course on Leavitt path algebras. After giving the definition and examples, we focus on the so called “Reduction Theorem” and its application to the description of simple LPAs. Then, we review some of the more important algebra properties, which can be characterized by a graphical property (primeness, primitivity, existence of a socle, chain conditions, etc). We will also review the use of computational techniques implemented under Magma and Mathematica which have turned out to be of utility in our studies of LPAs.
Professor Mercedes Siles Molina and Dr Yolanda Cabrera Casado
Poster exhibition organised by students
All participants should contribute a poster on a research topic of their own choice. There will be prizes for the best posters.