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)
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 prove that the graph complex is a strong homotopy Lie (super) bialgebra.
We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. When the quiver is of type A 0 , we recover the results for gl n. The proof of our Chevalley restriction theorem is similar to the proof for gl n ; however,… (More)
We construct a long exact sequence involving the homology of an FI-module. Using the long exact sequence, we give two methods to bound the Castelnuovo–Mumford regularity of an FI-module which is generated and related in finite degree. We also prove that for an FI-module which is generated and related in finite degree, if it has a nonzero higher homology,… (More)
We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.
We determine explicitly the center of the twisted graded Hecke algebras associated to homocyclic groups. Our results are a generalization of formulas by M. The notion of twisted graded Hecke algebras was introduced by S. Witherspoon in ; they are variants of the graded Hecke algebras of V. Drinfel'd  and G. Lusztig  (see also ) and twisted… (More)
Let M be a smooth, simply-connected, closed oriented manifold, and LM the free loop space of M. Using a Poincare duality model for M , we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.