Jordan (super)coalgebras and Lie (super)coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Köecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra $(J,\Delta)$ with a dual algebra $J^*$, there exists a Lie supercoalgebra $(L^c(J),\Delta_L)$ whose dual algebra $(L^c(J))^*$ is the Lie $KKT$-superalgebra for the Jordan superalgebra $J^*$. It is well known that some Jordan coalgebra $J^0$ can be constructed from an arbitrary Jordan algebra $J$. We find necessary and sufficient conditions for the coalgebra $(L^c(J^0),\Delta_L)$ to be isomorphic to the coalgebra $(\operatorname{Loc}(L_{\textup{in}}(J)^0),\Delta^0_L)$, where $L_{\textup{in}}(J)$ is the adjoint Lie $KKT$-algebra for the Jordan algebra $J$.