A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations of canonical form, and reduce the problem of classifying Eu...

24 LLE Review, Volume 69 The ponderomotive force associated with a light wave of variable amplitude1–10 drives many phenomena that occur in inertial confinement fusion11 and particle acceleration12 experiments. The existing formula for the ponderomotive force was derived under the assumption that the quiver speed of electrons oscillating in the applied electric field is much less than the speed...

We show that the bases of irreducible integrable highest weight module of a non-symmetric Kac-Moody algebra, which is associated to a quiver with a nontrivial admissible automorphism, can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of certain varieties associated with the quiver defined by Nakajima. In the case of non-symmetric affine or finite K...

We show that the string spectrum in the pp-wave limit of AdS5 × S/ZM (orbifolded pp-wave) is reproduced from the N = 2 quiver gauge theory by quantizing the Green-Schwarz string theory on the orbifolded pp-wave in light cone gauge. We find that the twisted boundary condition on the world-sheet is naturally interpreted from the viewpoint of the quiver gauge theory. The correction of order g2 YM ...

We study quiver gauge theories on D7-branes wrapped over vanishing holomorphic 4-cycles. We investigate how to incorporate O7-planes and/or flavor D7branes, which are necessary to cancel anomalies. These theories are chiral, preserve four supercharges and exhibit very rich infrared dynamics. Geometric transitions and duality in the presence of O-planes are analyzed. We study the Higgs branch of...

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution o...

In [24] and [26] Jørgensen introduced the Auslander-Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form ZA∞ and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d−1 such components. In this thesis we show that this is the only case where finitely many components appear. More precisely, we construct families of modules, ...

We construct the entire generalized Kac-Moody Lie algebra as a quotient of the positive part of another generalized Kac-Moody Lie algebra. The positive part of a generalized Kac-Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac-Moody Lie algebra. This idea arises from the...

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the ”truncated simple reflections” on the set of almost positive roots Φ≥−1 associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we ...

We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a cl...