Related papers: The monoid of families of quiver representations
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…
This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The…
The (ordinary) quiver of an algebra $A$ is a graph that contains information about the algebra's representations. We give a description of the quiver of $\mathbb{C}PT_{n}$, the algebra of the monoid of all partial functions on $n$ elements.…
We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term…
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify…
An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is…
This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.
We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study…
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…
The classical construction of representations of quivers enables us to consider linear maps between several vector spaces. The mixed representations of quivers helps us to work with linear maps as well as bilinear forms on several vector…
The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion…
In this work, we introduce {\em topological representations of a quiver} as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological…
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…
In this paper, we study moduli spaces of representations of certain quivers with relations. For quivers without relations and other categories of homological dimension one, a lot of information is known about the cohomology of their moduli…
In this paper we study the representation theory of three monoids of partial functions on an $n$-set. The monoid of all order-preserving functions (i.e., functions satisfying $f(x)\leq f(y)$ if $x\leq y$) the monoid of all order-decreasing…
We introduce the notion of (twisted) quiver representations in abelian categories and study the category of such representations. We construct standard resolutions and coresolutions of quiver representations and study basic homological…
We consider quiver representations respecting a quiver automorphism and show that the dimension vectors of the indecomposables are precisely the positive roots of an associated symmetrisable Kac-Moody Lie algebra. Moreover, every such Lie…
We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…
There is a unique finite group that lies inside the 2-dimensional unitary group but not in the special unitary group, and maps by the symmetric square to an irreducible subgroup of the 3-dimensional real special orthogonal group. In an…