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Abstract. 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 rate of intramolecular charge transfer from biphenyl to naphthalene was determined for the radical anions and radical cations of molecules with the general structure: (2-naphthyl)-(steroid spacer)-(4-biphenylyl). Varied degrees of unsaturation (one double bond, NSenB; two double bonds, NSen(2)B; and the b-ring completely aromatized, NSarB) were incorporated into the steroid spacer to examine the effect it would have on the charge transfer rate. The charge transfer rate, as inferred from the decay of the biphenyl radical ion absorption, increased in all cases relative to the completely saturated 3-(2-naphthyl)-16-(4-biphenylyl)-5alpha-androstane (NSB) reference molecule. For the anion charge transfer, the decay rates increased by factors of 1.4, 4.2, and 5.1, respectively, and for the cation, the decay rates increased by factors of 5, 276, and 470. To explain the results, the charge-transfer process was viewed as a combination of two independent mechanisms: a single-step, superexchange mechanism, and a two-step, sequential charge transfer. Using a low level of theory, simple models of the superexchange and two-step mechanisms were developed to elucidate the nature and differences between the two mechanisms. The critical variable for this analysis is the free energy of formation (DeltaG(I) degrees ) of the intermediate state: (2-naphthyl)-[spacer](1)+/--(4-biphenylyl). The conclusion from this treatment is that superexchange is the dominant mechanism when DeltaG(I) degrees is large, but at small DeltaG(I) degrees , the sequential mechanism will dominate. This is because the superexchange rate is shown to have a weak dependence on DeltaG(I) degrees , changing 10-fold for a change in DeltaG(I) degrees of 2 eV, compared to the sequential mechanism in which the rate can change over 10(3) for 0.5 V.

Table of Contents 1. Introduction 2. Continuous Hecke algebras 3. Continuous symplectic reflection algebras and Cherednik algebras 4. Infinitesimal Hecke algebras 5. Representation theory of continuous Cherednik algebras 6. Case of wreath-productsThe theory of PBW properties of quadratic algebras, to which this paper aims to be a modest contribution, originates from the pioneering work of Drinfeld (see [Dr1]). In particular, as we learned after publication of [EG] (to the embarrassment of two of us!), symplectic reflection algebras, as well as PBW theorems for them, were discovered by Drinfeld in the classical paper [Dr2] 15 years before [EG] (namely, they are a special case of degenerate affine Hecke algebras for a finite group G introduced in [Dr2], Section 4).It is our great pleasure to dedicate this paper to Vladimir Drinfeld on the occasion of his 50-th Birthday. Continuous Hecke algebras2.1 Algebraic distributions. Let X be an affine scheme of finite type over C. We shall denote by O(X) the algebra of regular functions on X. An algebraic distribution on X is an element in the dual space O(X) * of O(X). For c ∈ O(X) * and f ∈ O(X), we will denote the value c(f ) by (c, f ).The space O(X) * is naturally equipped with the weak (inverse limit) topology. Note also that O(X) * is a module over O(X): for any f ∈ O(X) and µ ∈ O(X) * we can define the elementLet Z be a closed subscheme of X, and write I(Z) for its defining ideal in O(X). We say that an algebraic distribution µ on X is supported on the scheme Z if µ annihilates I(Z). Clearly, the space of algebraic distributions on X supported on Z is naturally isomorphic to the space of algebraic distributions on Z.Now assume that Z is reduced. We say that µ ∈ O(X) * is scheme-theoretically (respectively, set-theoretically) supported on the set Z if µ annihilates I(Z) (respectively, some power of I(Z)).Example 2.1. For each point a ∈ X, the delta function δ a ∈ O(X) * is defined by δ a (f ) := f (a) where f ∈ O(X). It is scheme-theoretically supported at the point a, and its derivatives are set-theoretically supported at this point.Let G be a reductive algebraic group. Since O(G) is a coalgebra, its dual space O(G) * is an algebra under convolution. The unit of this algebra is the delta function δ 1 of the identity element 1 ∈ G.Note that a continuous representation of the algebra O(G) * is the same thing as a locally finite G-module (i.e. a G-module which is a direct sum of finite dimensional algebraic representations of G).Suppose that G acts on X. Then G acts also on O(X) and O(X) * . We have O(X) = V M V ⊗V , where V runs over the irreducible representations of G, and M V are multiplicity spaces. Thus, O(X) * = V M * V ⊗V * . In particular, O(G) * = V V ⊗V * as a G × G-module.

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 hamiltonian 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 also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.

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