2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126026We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.24 pages, accepted by Functional Analysis and its Applications, a few typos correctedQuantum AlgebraRepresentation Theory18D50, 20C30 (Primary); 05E25, 06A11, 37K10 (Secondary)text