On the coalgebra description of OCHA

TL;DR

This paper explores the intrinsic coalgebraic structure of OCHA, a homotopy algebra for open-closed strings, and introduces a universal enveloping A-infinity algebra framework.

Contribution

It demonstrates the coalgebraic nature of OCHA and constructs its universal enveloping A-infinity algebra from A-infinity extensions.

Findings

OCHA structure is intrinsic to tensor products of coalgebras.

OCHA can be derived from A-infinity extensions.

Defined the universal enveloping A-infinity algebra of an OCHA.

Abstract

OCHA is the homotopy algebra of open-closed strings. It can be defined as a sequence of multilinear operations on a pair of DG spaces satisfying certain relations which include the $L_\infty$ relations in one space and the $A_\infty$ relations in the other. In this paper we show that the OCHA structure is intrinsic to the tensor product of the symmetric and tensor coalgebras. We also show how an OCHA can be obtained from $A_\infty$-extesions and define the {\it universal enveloping} $A_\infty$-algebra of an OCHA as an $A_\infty$-extension of the universal enveloping of its $L_\infty$ part by its $A_\infty$ part.