Related papers: Quivers and Three-Dimensional Lie Algebras

We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension $\mu(\Lg)$ of a…

We study the representation theory of quantizations of Gieseker moduli spaces. Namely, we prove the localization theorems for these algebras, describe their finite dimensional representations and two-sided ideals as well as their categories…

We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras $\mathfrak{sl}(n^\infty)$, $\mathfrak o(n^\infty)$, and $\mathfrak{sp}(n^\infty)$. Let $\mathfrak g_\infty$ be one of…

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore…

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…

We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of…

We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as…

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…

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 the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge…

We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of infinite type A. We…

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $\Lambda$ are explored, with regard to both their geometry and the structure of the modules they encode. Provided…

Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph $A_{\infty, \infty}$ is Krull-Schmidt, as long as the arrows in the quiver eventually point outward.

By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure ${{\bar \BF}}_q$ of the finite field $\BF_q$ of $q$ elements, we establish a relation between the representation theory of $A$ over ${{\bar…

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study…

We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful…

In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…

We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for…

We transpose Jones' technology and the authors' C*-algebraic techniques to study representations of the Leavitt path algebra L (over an arbitrary row-finite graph) by using its quiver algebra A. We establish an equivalence of categories…

For a natural number $m$, a Lie algebra $L$ over a field $k$ is said to be of breadth type $(0, m)$ if the co-dimension of the centralizer of every non-central element is of dimension $m$. In this article, we classify finite dimensional…