Related papers: Orbit semigroups and the representation type of qu…

Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We introduce $\mathcal{Q}^N$ quivers and construct maximal green sequences for these quivers. We prove that any finite connected full subquiver of the quivers defined by Hernandez and Leclerc, arising in monoidal categorifications of…

We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are…

In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of…

We study a monoid associated to complex semisimple Lie algebras, called the quantic monoid. Its monoid ring is shown to be isomorphic to a degenerate quantized enveloping algebra. Moreover, we provide normal forms and a straightening…

In Lie theory, a dense orbit in the unipotent radical of a parabolic group under the adjoint action is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalising the classical definition in the case of…

We study the canonical orbit category of the bounded derived category of finite dimensional representations of the quiver of type $D_{\infty}$. We prove that this orbit category is a cluster category, that is, its cluster-tilting…

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector $\mathbf{d}$ and each integral weight $\theta$ of Q, the moduli space $\mathcal{M}(Q,\mathbf{d})^{ss}_{\theta}$ of…

In this note we give an example of a one-dimensional manifold with two connected components and a complete metric whose group of isometries has an orbit which is not closed. This answers a question of S. Gao and A. S. Kechris.

We study the distribution of non-discrete orbits of geometrically finite groups in $\operatorname{SO}(n,1)$ acting on $\mathbb{R}^{n+1}$, and more generally on the quotient of $\operatorname{SO}(n,1)$ by a horospherical subgroup. Using…

The general linear group GL(n, K) over a field K contains a particularly prominent subgroup U(n, K), consisting of all the upper triangular unipotent elements. In this paper we are interested in the case when K is the finite field F_q, and…

We present several results regarding the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay…

The connection between simple Lie algebras and their Yangian algebras has a long history. In this work, we construct finite-dimensional representations of Yangian algebras $\mathsf{Y}(\mathfrak{sl}_{n})$ using the quiver approach. Starting…

We characterize all bounded orbits of two similar Collatz-type quadratic mappings of the set of non-negative integers. In one case, where cycles of all possible lengths may occur, an orbit is bounded if and only if it reaches a cycle. For…

With the help of Lusztig's canonical basis, we study local intersection cohomology of the Zariski closures of orbits of representations of a quiver of type A, D or E. In particular, we characterize the rationally smooth orbits and prove…

We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic…

We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to…

Let Q be a finite quiver without oriented cycles, and let k be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Coxeter group W_Q and the cofinite…

We continue the study of the closures of $GL(V)$-orbits in the enhanced nilpotent cone $V\times\cN$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety.…