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 Koszul duality for operads; see eg. [K] o...

Let V be an n-dimensional vector space. We give a direct construction of an exact sequence that gives a GL(V)-equivariant " resolution " of each symmetric power S t V in terms of direct sums of tensor products of the form ∧ i 1 V ⊗ · · · ⊗ ∧ ip V. This exact sequence corresponds to inverting the relation in the representation ring of GL(V) that is described by the Koszul complex, and has appear...

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. We prove a Koszul duality theory for PROPs generalizing the one for associative algebras and for operads.

We study the cohomology H(A) = Ext∗A(k, k) of a locally finite, connected, cocommutative Hopf algebra A over k = Fp. Specifically, we are interested in those algebras A for which H∗(A) is generated as an algebra by H(A) and H(A). We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras F → A → B with F monogenic and B semiKoszul, we use the Cartan-Eilenberg spectral s...

Braverman, Gaitsgory, Polishchuk, and Positselski gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincaré-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypot...