Quantization of the Lie bialgebra of string topology

TL;DR

This paper demonstrates that the reduced equivariant homology of the free loop space of a simply-connected manifold forms a Lie bialgebra and constructs a Hopf algebra that quantizes this structure.

Contribution

It introduces a novel quantization of the Lie bialgebra structure in string topology using Poincare duality models.

Findings

Reduced equivariant homology has a Lie bialgebra structure

Constructed a Hopf algebra quantizing the Lie bialgebra

Provides new algebraic tools for string topology

Abstract

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.