Abstract

We will start by defining quivers, which are directed graphs, and their representations. Afterwards we will motivate why quiver representations are natural to study in representation theory and algebraic geometry. We will then state Gabriel’s theorem about representations of quivers of ADE type, which gives us a connection to the theory of Lie algebras. Afterwards we will explore different algebras associated to a quiver. We will touch on a selection of the following topics: The Ringel-Hall algebra of a quiver and its relation to the positive part of the quantum group, definition of the cohomological Hall algebra (CoHA) of a quiver in terms of equivariant cohomology, concrete description of the multiplication of the CoHA and first examples. This talk should not have too many prerequisites, in particular, no knowledge about quivers is assumed. Some knowledge about Lie algebras and algebraic geometry is useful to understand some of the motivations.