On the coalgebra description of OCHA (Q418931)

The term OCHA refers to an open-closed homotopy algebra structure. It consists in a pair of vector spaces, the ``closed'' one \(\mathcal H_c\) being an \(L_\infty\)-algebra and the ``open'' one \(\mathcal H_o\) being an \(A_\infty\)-algebra. Extra structure appearing in the form of multilinear operations intertwines these two structures together with compatibility conditions.NEWLINENEWLINE\textit{H. Kajiura} and \textit{J. Stasheff} found a global compatibility condition for lifts of the multilinear maps to coderivations on the coalgebra \(\Lambda^* \mathcal H_c \otimes T^* \mathcal H_o\), see their survey article [``Homotopy algebra of open-closed strings'', Geom. Topol. Monogr. 13, 229--259 (2008; Zbl 1137.18306)]. In this article the space of coderivations is identified with the space of certain OCHA constrained maps, thereby singling out the OCHA structures as square zero derivations of degree one.NEWLINENEWLINEIn the last part, OCHA structures are constructed from \(A_\infty\)-extensions of the closed part by the open one. This allows the author to introduce the universal enveloping \(A_\infty\)-algebra of an OCHA as such an extension of the universal enveloping \(A_\infty\)-algebra of the \(L_\infty\)-part by the \(A_\infty\)-part.