We propose a Leibniz algebra, to be called
DD^++,
which is a generalization of the Drinfel’d double. We find that there is
a one-to-one correspondence between a DD^++
and a Jacobi–Lie bialgebra, extending the known correspondence between a
Lie bialgebra and a Drinfel’d double. We then construct generalized
frame fields E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+EAM∈O(D,D)×ℝ+
satisfying the algebra \hat{\pounds}_{E_A}E_B = - X_{AB}{}^C\,E_C£̂EAEB=−XABCEC,
where X_{AB}{}^CXABC
are the structure constants of the DD^++
and \hat{\pounds}£̂
is the generalized Lie derivative in double field theory. Using the
generalized frame fields, we propose the Jacobi–Lie
TT-plurality
and show that it is a symmetry of double field theory. We present
several examples of the Jacobi–Lie TT-plurality
with or without Ramond–Ramond fields and the spectator fields.
Finite-dimensional Leibniz algebra representations of $\mathfrak{sl}_2$
