A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with Uq(sl2). We study its representation theory, and in particular, its category O.
The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamilto-nian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We… (More)
We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of… (More)
We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various algebraic structures using operads is partly due to the theory of… (More)
It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.
We prove that the graph complex is a strong homotopy Lie (super) bialgebra.
We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's renormalization Hopf algebra, as well as to pre-Lie versions. These results are direct analogues of Turaev's results in 2004, by… (More)
We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply… (More)