Jacobi-Lie T-plurality

We propose a Leibniz algebra, to be called DD$^+$, which is generalization of the Drinfel'd double. find that there one-to-one correspondence between DD$^+$ and Jacobi--Lie bialgebra, extending known Lie bialgebra then construct generalized frame fields $E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+$ satisfying algebra $\mathcal{L}_{E_A}E_B = - X_{AB}{}^C\,E_C\,$, where $X_{AB}{}^C$ are structure constants $\mathcal{L}$ derivative in double field theory. Using fields, we Jacobi-Lie T-plurality show it symmetry present several examples with or without Ramond-Ramond spectator fields.