Title Promoter Affiliations Abstract
"Cohomological invariants of structurable algebras" "Tom De Medts" "Department of Mathematics: Algebra and Geometry" "A common theme in algebra is to understand algebraic structures over arbitrary fields by first studying them over their algebraic closure and then investigating the possible ways to ""descend"" to the base field again. A typical example occurs in the theory of quadratic forms over an arbitrary field. In order to decide when two given quadratic forms are non-isometric, a useful tool is to define invariants for quadratic forms; typical (easy) examples of such invariants are the discriminant and the Clifford algebra. These are examples of cohomological invariants (of degree 1 and 2, respectively).Our goal is to study cohomological invariants for non-associative algebras and their related linear algebraic groups. Examples that have been well studied are octonion algebras (certain 8-dimensional algebras) and Albert algebras (certain 27-dimensional Jordan algebras); these are connected to groups of type G2 and F4, respectively.We will study invariants of structurable algebras, a class of algebras with involution, simultaneously generalizing Jordan algebras and associative algebras with involution. We will mainly be interested in the exceptional structurable algebras: the 35-dimensional Smirnov algebras; tensor products of two composition algebras (of dimension 16, 32, 64); structurable algebras of skew-dimension one arising from hermitian cubic norm structures (of dimension 8, 20, 32, 56). These examples are related to groups of type 3D4, E6, E7 and E8."
"Structurable algebras, representation theory and related point-line geometries" "Tom De Medts" "Department of Mathematics: Algebra and Geometry" "The goal of the project is to investigate connections between algebraic structures (linear algebraic groups, Jordan pairs, structurable algebras, Lie algebras) and geometric structures (especially point-line geometries, but also so-called root filtration spaces). We will often rely on the representation theory of the underlying groups."
"Quantum symmetric spaces, operator algebras and quantum cluster algebras" "Kenny De Commer" "Mathematics, Topological Algebra, Functional Analysis and Category Theory" "The mathematician's notion of space has continuously evolved throughout the history of the subject. For the ancient Greeks, space was seen as a background continuum in which certain structures such as lines, triangles, circles and planes can be housed and studied. Their synthetic approach was subsequently supplemented by the Cartesian method in which algebraic methods were used to study for example curves in a plane. Gradually, it was realized that there is such a thing as an abstract space, which can have many different forms and inherent structure. A highlight in this development is the work of B. Riemann, which proved to provide the appropriate mathematicalframework in which to develop the physical theory of general relativity. At the same time, the theory of quantum mechanics has shown the need for a notion of quantum space, to be understood by the methods of abstract non-commutative algebra.The goal of this project is to study in depth a particular class of such quantum spaces. In the realm of Riemannian spaces, there is a class with a very high degree of symmetry, known as symmetric spaces. These were studied and classified by E. Cartan at the beginning of the 20th century. Only recently, particularly through the work of G. Letzter, has it become apparent that one has a corresponding notion of quantum symmetric space. We will investigate these quantum symmetric spaces both with analytic methods (operator algebras) and algebraic methods (quantum cluster algebras)."
"Dual pairs and Dirac cohomology for deformations of Weyl algebras" "Joris Van der Jeugt" "Department of Electronics and information systems, Department of Applied Mathematics, Computer Science and Statistics" "The goals of this project are to determine novel algebraic structures, classify their representations and study the connection with Dirac cohomology.By studying the symmetries of deformations of the Dirac operator interesting algebraic structures arise, which do not appear in the undeformed setting. More specifically, we work inside a deformation of the Weyl algebra of linear differential operators with polynomial coefficients, being either a rational Cherednik algebra associated with a reflection group, or an abstract generalization thereof. Within the tensor product with a Clifford algebra, we determine the subalgebra of all elements (anti)commuting with a known subalgebra containing the Dirac operator. This concept of two commuting algebraic structures is reminiscent of reductive dual pairs in Howe duality, of which Schur-Weyl duality and the decomposition of polynomials in terms of spherical harmonics are special cases.In this project, on the one hand, we will extend the framework of dual pairs and symmetries to deformations of the abstract Dirac elements appearing in Dirac cohomology. On the other hand, for the symmetry algebras obtained in this way, we will examine their action on the space of polynomials and exploit the dual pair structure to identify and classify their representations. By combining these results, we aim to develop a notion of Dirac cohomology for the abstract generalization of the rational Cherednik algebras."
"Rigidity and structural results in von Neumann algebras and Ergodic Theory" "Stefaan Vaes" Analysis " In their pioneering work, Murray and von Neumann found a natural way to associate a von Neumann algebra to every countable group G and to any of its measure preserving actions. The classification of these von Neumann algebras is in general a hard problem and it is driven by the following fundamental question: what aspects of the group G and of an action of G are remembered by the associated von Neumann algebras? In the amenable case, no information can be recovered excepts the amenability of the group. On the other hand, the non-amenable case revealed a beautiful rigidity theory. Various aspects of the groups and their actions are recovered by their von Neumann algebras. In this proposal we aim to obtain new structural results for von Neumann algebras associated to lattices in higher rank Lie groups and to classify all the tensor product decompositions of von Neumann algebras arising from their actions.Orbit equivalence (OE) theory has seen an explosion of activity in the last two decades and it was triggered in part by the success of Popa's deformation/rigidity approach in the classification of von Neumann algebras. In this project, we plan to find new classes of actions that are OE superrigid (i.e. the OE relation completely remembers the underlying action) and obtain new structural and rigidity results in the OE theory. In particular, we aim to find new classes of groups G for which any Bernoulli or profinite action of G is cocycle and OE superrigid. "
"Algebraic structure and representations of concrete classes of algebras, groups and semigroups" "Eric Jespers" Mathematics "In this project we will treat fundamental structural issues of important algebraic constructions for which the interest still grows. The motivation comes from one side of classical algebraic constructions, like group rings, polynome rings and their generalizations, and from another side of the still growing interest and need for new algebraic methods in fast developing areas, such as quantum groups and their representations."
"Almost inner derivations of Lie algebras" "Karel Dekimpe" "Mathematics, Kulak Kortrijk Campus" "A classical problem in spectral geometry was to determine whether or not isospectral manifolds are necessarily isometric. It turns out that the answer to this question is negative and several counterexamples have been given. For the construction of continuous families of isospectral and non-isometric manifolds, class preserving automorphisms of nilpotent Lie groups were crucial.An automorphism of a group is called class preserving if and only if every element is conjugate to its image. So this condition is related to, but less strict than the one for an inner automorphism. A nilpotent Lie group admitting a discrete and cocompact subgroup and a class preserving automorphism which is not inner can be used to construct a continuous family of isospectral but non-isometric nilmanifolds. Class preserving automorphisms of a Lie group are very closely related to almost inner derivations of the corresponding Lie algebra. These are derivations for which each element is mapped to the Lie bracket of itself with some other element. The set of all almost inner derivations forms a Lie subalgebra of the derivation algebra and contains the inner derivations.Up till now, almost inner derivations of Lie algebras have not been studied in detail yet. They have almost only been considered from a differential geometric point of view, where the focus was on constructing some examples. The goal of this thesis is to study this notion in a purely algebraic way. Although the motivation from spectral geometry only makes sense for nilpotent Lie algebras, from an algebraic point of view, there is no reason to restrict to this class only. Hence, we study almost inner derivations of Lie algebras more generally. We also consider non-nilpotent Lie algebras and Lie algebras over an arbitrary field, so not only real and complex Lie algebras. This dissertation consists of three main parts and one appendix. The first part is an introduction, which provides all preliminaries to understand the rest. In Chapter 2, we define Lie algebras and develop the necessary notions which will be important in the study of almost inner derivations. Chapter 3 contains more information about the geometric motivation from spectral theory. We also present some properties of the related notion of class preserving automorphisms of groups. Finally, in Chapter 4, we describe some interesting techniques for doing computations on the almost inner derivations of (a class of) Lie algebras.In the second part, we will focus on the fact that the dimension of the set of almost inner derivations depends on the field over which the Lie algebra is defined. Chapter 5 contains an elaborated example, where a Lie algebra is given by means of the structure constants. The distinction for various fields has to do with a different factorisation of polynomials. In Chapter 6, we show a procedure to construct new almost inner derivations by using finite field extensions. In particular, this gives a way to set up a Lie algebra for which the dimension of the set of almost inner derivations is distinct when we consider different fields. Chapter 7 focuses on Lie algebras related to finite groups. We explain the connection with class preserving automorphisms of finite groups and compare the results we have for the two notions.In the last part, we will use the observations from the two other parts to compute the set of almost inner derivations for different classes of Lie algebras. In Chapter 8, we give an overview of almost inner derivations for low-dimensional Lie algebras. The appendix contains tables where the non-vanishing Lie brackets for a lot of low-dimensional Lie algebras are collected. Each time, we also provide tables with results, such as the dimension of some subalgebras of the derivation algebra. The next three chapters are devoted to other classes of nilpotent Lie algebras. Two-step nilpotent Lie algebras are studied in Chapter 9. Further, we also consider filiform Lie algebras and free nilpotent Lie algebras (in Chapter 10 respectively Chapter 11). The last chapter contains results for some other classes of (not only nilpotent) Lie algebras."
"Calabi-Yau property of Hopf algebras" "Yinhuo ZHANG" Algebra "Calabi-Yau categories came from Mathematical Physics and Algebraic Geometry. The Serre functor in the bounded derived category of coherent sheaves on a Calabi-Yau variety is an iteration of the shift functor. Triangulated categories with this property are called Calabi-Yau categories. An algebra is a Calabi-Yau algebra, if the associated bounded derived category of all finite dimensional modules is a Calabi-Yau category. From this definition, a finite dimensional Calabi-Yau algebra must be semisimple. However, for a finite dimensional selfinjective algebra, its stable module category is a triangulated category. A selfinjective algebra is called a stably Calabi-Yau algebra if its stable category is Calabi-Yau. Calabi-Yau categories and (stably) Calabi-Yau algebras are now popular in many braches of mathematics, such as representation theory, algebraic geometry, mathematical physics and so on. Hopf algebras, as a special kind of algebras endowed with coalgebra structure have been extensively studied since they were introduced in 1940s. In this project we study the Calabi-Yau property of Hopf algebras. This project will deal with the following problems."
"Representation rings of pointed Hopf algebras of finite type" "Yinhuo ZHANG" Algebra "The computation of the representation ring (or the Green ring) of a Hopf algebra (or a group) is one of the most difficult topics in the Representation Theory of Hopf algebras (or groups). However, the resent research progress made by UHasselt, UAntwerpen and Yangzhou University sheds light on the development on this topic. This proposal forms one part of the ongoing joint research project between the Algebra group of UHasselt and the algebra group of Yangzhou University. The proposed project aims to study the representation rings of finite dimensional pointed Hopf algebras of finite type in the next two years. In the first year of his Ph.D. research, the candidate has successfully computed the representation rings of pointed Hopf algebras of rank one, an important class of pointed Hopf algebras of finite type. The results in his preprint [1] generalize the results of Li and Zhang [2] and the results of Chen, Van Oystaeyen and Zhang [3]. In this proposal, the candidate will compute the representations rings of the other classes of pointed Hopf algebras of finite type, and study the properties of the representation rings such as the semisimplicity, commutativity, semiprime, etc. It is expected that the ring properties of the representation ring of a Hopf algebra H will reflect the structure of the monoidal category of representation over H. The results from this work will make great contribution to the knowledge of the Representation Theory of Hopf algebras and quantum groups. [1] Z. Wang, Representation rings of pointed Hopf algebras of rank one of nilpotent type, preprint. [2] L. Li and Y. Zhang, The Green rings of the generalized Taft algebras, to appear in Contemporary Mathematics, AMS. [3] H.Chen, F. Van Oystaeyen and Y.Zhang, The Green rings of Taft algebras, to appear in Proc. AMS."
"Von Neumann algebras and discrete groups." "Stefaan Vaes" Analysis "Von Neumann algebras, and more specifically II_1 factors, arise naturally in the study of countable groups and their actions on measure spaces. A central, but extremely hard problem is the classification of these von Neumann algebras in terms of their group/action data. In 2001 Sorin Popa initiated his breakthrough deformation/rigidity theory leading to the solution of many long standing open problems. Since then overwhelming progress has been made in the understanding of group measure space II_1 factors associated with measure preserving group actions. The aim of this project is to obtain structural results of similar strength about group von Neumann algebras L(G). This includes the construction of a wide class of group von Neumann algebras L(G) that entirely remember the group G. We will also continue the study of group measure space II_1 factors and make a systematic study of (non-)uniqueness of Cartan subalgebras."