Non-commutative algebra and geometry

NCAG Seminar

Research outline

Our main research interests are the study of different structural properties of algebraic quantum groups and topics related to algebraic K-theory, using tools of non-commutative geometry, Hopf algebras, Lie theory, algebraic geometry, representation theory, tensor categores and triangulated categories.

The current research topics consist mainly in the following:

- Clasification of complex pointed Hopf algebras over finite simple groups of Lie type
- Hopf algebras without the (dual) Chevalley property
- Multiparametric quantum groups
- Quantum subgroups of simple quantum groups
- Quantum determinants and Nichols algebras
- Representations of generalized small quantum groups
- The Baum-Connes conjecture in the context of algebraic K-theory

The theory of quantum groups had its origin in the study of quantum integrable systems in statistical mechanics. As objects on their own right, they were introduced in the beginning of the 80’s by Drinfeld and Jimbo, independently.

From that moment on, they are considered as important and efficient tools in diverse problems of mathematics and theoretical physics. Examples can be found from invariants in low dimensional topology to conformal field theory.

The theory of analytic quantum groups was introduced to understand the Pontrjagin duality in a non-commutative context, as a non-trivial generalization of this duality for locally compact non-abelian groups.

In a series of papers, Woronowicz developed a general theory for compact quantum groups, and after several contributions of many authors, Kustermans and Vaes established the axiomatic definition of a locally compact quantum group. It is worth noting that this is the only class of quantum groups with such a description. In spite of the lack of an axiomatic general definition, experts in the field coincide that the category of quantum groups should correspond to the opposite category of (C*-) Hopf algebras.

In general, quantum groups might be presented as multiparametric deformations of universal enveloping algebras of semisimple Lie algebras or function algebras over reductive affine algebraic (or compact) groups.

The theory of quantum groups provides an infinite source of non-trivial examples of Hopf algebras and operator algebras. In this sense, one may consider them as objects that codify certain symmetries in non-commutative geometric spaces.

Members