Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.

Abstract. A bilinear form f on a nonassociative algebra A is said to be invariant iff f(ab, c) = f(a, bc) for all a, b, c ∈ A. Finite-dimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal structure of A if f is nondegenerate and introduce the notion of T ∗-extension of an arbitrary algebra B (i.e. by its dual space B ∗ ) where the natural pairing gives rise to a nondegenerate invariant symmetric bilinear form on A: = B ⊕ B ∗. The T ∗-extension involves the third scalar cohomology H3 (B, K) if B is Lie and the second cyclic cohomology HC 2 (B) if B is associative in a natural way. Moreover, we show that every nilpotent finitedimensional algebra A over an algebraically closed field carrying a nondegenerate invariant symmetric bilinear form is a suitable T ∗-extension. As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel’d but not always isomorphic to a Manin triple. Examples involving the Heisenberg and filiform Lie algebras (whose third scalar cohomology is computed) are discussed. 1.

Abstract. Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established. The factorization method (see [1]) is used to integrate a system of ODEs of the form qt = [q+, q], q(0) = q0. (0.1) Here q(t) belongs to a Lie algebra G, which is decomposed (as a vector space) into

Abstract. Let G be a compact, connected Lie group, acting smoothly on a manifold M. In their 1998 paper, Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant cohomology of M, quasi-isomorphic to the standard (large) Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small Cartan model into the large Cartan model, intertwining the (Sg ∗)inv-module structures and inducing an isomorphism in cohomology. The construction involves the solution of a remarkable inhomogeneous Maurer-Cartan equation. This solution has further applications to the theory of transgression in the Weil algebra, and to the Chevalley-Koszul theory of the

Abstract. We show that many noetherian Hopf algebras A have a rigid dualising complex R with R ∼ = ν A 1 [d]. Here, d is the injective dimension of the algebra and ν is a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call ν the Nakayama automorphism of A. We prove that ν = S 2 ξ, where S is the antipode of A and ξ is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non-zero in the top dimension d, when A is an Artin-Schelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincaré duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating ν for A using also the opposite coalgebra structure, we determine a formula for S 4 generalising a 1976 formula of Radford for A finite dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.

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Let g be a reductive algebraic Lie algebra over an algebraically closed field k of characteristic zero and G is the corresponding connected and simply connected group. The symmetric algebra of a (finite-dimensional) g-module V is the algebra of polynomial

Abstract. We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras- nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e1,..., en, [ei, ej] = cijei+j (N-graded Lie algebras). In particular we describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list. It is proved that a symplectic filiform Lie algebra g is a filtered deformation of some N-graded symplectic filiform Lie algebra g0. But this condition is not sufficient. A spectral sequence is constructed in order to answer the question whether a given deformation of a N-graded symplectic filiform Lie algebra g0 admits a symplectic structure or not. Other applications and examples are discussed.

Abstract. We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov– Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic is motivated by statements concerning “monopole bundles ” over matrix algebras in the literature of theoretical high-energy physics. The purpose of this paper is to make precise the vague idea that if two compact metric spaces are close together then there should be a relationship between the vector bundles on the two spaces. I was led to examine this idea by statements in the theoretical high-energy physics

Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and A∞-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic