Related papers: Quiver Grassmannians and their Euler characteristi…
This overview paper reviews several results relating the representation theory of quivers to algebraic geometry and quantum group theory. (Potential) applications to the study of the representation theory of wild quivers are discussed. To…
This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (19-24 January 2020). It introduces a geometric orbifold model for the bounded derived category of a skew-gentle…
In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second-named author to introduce a variety called transverse quiver Grassmannian and the first-named and third-named authors to…
This paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type. We first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized Gelfand-MacPherson…
We give a new characterization of tilted algebras by the existence of certain special subquivers in their Auslander-Reiten quiver. This result includes the existent characterizations of this kind and yields a way to obtain more tilted…
In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…
In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of…
For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping…
We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the…
We use Seiberg--Witten-like relations in the topological recursion framework to obtain virtual Euler characteristics for uni- and multicellular maps for ensembles of classic orthogonal polynomials and for ensembles related to nonorientable…
The idea is to identify certain path algebra elements with symmetric functions. We propose such a morphism by solving the quiver relations, which describe the Plucker-type embedding for quiver grassmannians.
Some algebraic, geometric and geometroalgebraic characteristics of pairs of operators are discussed.
We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example…
This informal introduction is an extended version of a three hours lecture given at the 6th Peyresq meeting ``Integrable systems and quantum field theory''. In this lecture, we make an overview of some of the mathematical results which…
Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant K-homology of M. The…
We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $\tt A$ flag varieties, their linear degenerations, finite dimensional…
This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent…
We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and…
This is an expanded version of the talk I gave at Oberwolfach, MIT and several other universities.
This is an expository article, closely following the author's lecture at the 2014 Journal Differential Geometry conference.