Related papers: Minicourse on Kac-Moody groups
The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their…
Over-extended Kac-Moody algebras contain so-called gradient structures - a gl(d)-covariant level decomposition of the algebra contains strings of modules at different levels that can be interpreted as spatial gradients. We present an…
The paper is a short survey of recent developments in the area of first order descriptions of linear groups. It is aimed to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups…
We provide some necessary details to several arguments appearing in our previous paper ``Canonical bases for quantum generalized Kac-Moody algebras''. We also make the link with some other work on the same subject.
This is a brief introduction to the theories of Lie groups, algebraic groups and their discrete subgroups, which is based on a lecture series given during the Summer School held in the Banach Centre in Poland in Summer 2011.
In this study, we try to generalize Bruhat-Tits's theory to the case of a Kac-Moody group, that is to define an affine building for a Kac-Moody group over a local field. Actually, we will obtain a geometric space wich lacks some of the…
We introduce a new class of infinite-dimensional Lie algebras, which we refer to as continuum Kac-Moody algebras. Their construction is closely related to that of usual Kac-Moody algebras, but they feature a continuum root system with no…
We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.
These are expanded and revised notes for a minicourse entitled "Affine W-algebras", which took place as part of the thematic month "Quantum Symmetries" at the Centre de Recherches Mathematiques in Montreal, Canada in October 2022. The first…
The goal of the present paper is to obtain new free field realizations of affine Kac-Moody algebras motivated by geometric representation theory for generalized flag manifolds of finite-dimensional semisimple Lie groups. We provide an…
This paper is an overview of my recent work on abstract homomorphisms of algebraic groups. It is based on a talk given at the Conference on Group Actions and Applications in Geometry, Topology, and Analysis held in Kunming in July 2012.
Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of…
We investigate smooth representations of complete Kac-Moody groups. We approach representation theory via geometry, in particular, the group action on the Davis realisation of its Bruhat-Tits building. Our results include an estimate on…
We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section…
These notes give an elementary introduction to Lie groups, Lie algebras, and their representations. Designed to be accessible to graduate students in mathematics or physics, they have a minimum of prerequisites. Topics include definitions…
A tutorial introduction is given to general Hopf algebras and to general compact quantum groups. In the definition and further treatment of compact quantum groups C*-algebras are avoided. Contact with Woronowicz's compact matrix quantum…
Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups…
These are the lecture notes from my portion of a mini-course for the summer school "Building Bridges 3" that was held in Sarajevo during July 2016. My lectures covered the Katz definition of modular forms, a family of forms defined from…
These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism…
This article is based on a talk given at the Ghent Geometric Analysis Seminar in 2023. We review basic notions from the theory of weightings along submanifolds, with special emphasis on multiplicative weightings for Lie groupoids along…