Steve Shnider

Quantum symmetric spaces

Let G be a semisimple Lie group, g its Lie algebra. For any symmetric space M over G we construct a new (deformed) multiplication in the space A of smooth functions on M. This multiplication is invariant under the action of the Drinfeld-Jimbo quantum group Uhg and is commutative with respect to an involutive operator S: A ⊗ A → A ⊗ A. Such a multiplication is unique. Let M be a kahlerian symmetric space with the canonical Poisson structure. Then we construct a Uhg-invariant multiplication in A which depends on two parameters and is a quantization of that structure.

Associahedra, cellular -construction and products of _{∞}-algebras

The aim of this paper is to construct a functorial tensor product of A ∞ -algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.

Operads as abstract algebras, and the Koszul property

Abstract The purpose of this work is to clarify some aspects of the theory of operads – especially of Koszul operads. We describe the free operad and give the categorical bar construction in terms of it. After giving a description of operads as abstract “algebras”, we give a self-contained proof of Theorem 4.1.13 of Ginzburg and Kapranov (1994). Our approach uses two spectral sequences associated to the categorical cobar complex of a Koszul operad.