Shifted $\text{L}_\infty$-Bialgebras


A Lie bialgebra is a pair consisting of a Lie algebra and a Lie coalgebra structure on a space $\mathfrak{g}$, such that the cobracket is a 1-cocycle for the adjoint representation of $\mathfrak{g}$ on $\bigwedge\mathfrak{g}$. When $\mathfrak{g}$ is finite-dimensional, such Lie bialgebras are in one-to-one correspondence with Manin triples $\mathfrak{d} = \mathfrak{g} \oplus \mathfrak{g}^*$. A third equivalent definition is as Maurer–Cartan elements of $\mathcal{S}(\mathfrak{g}^*)\otimes\mathcal{S}(\mathfrak{g})$ w.r.t. Kosmann-Schwarzbach’s big bracket construction.

We introduce a generalization of Kosmann-Schwarzbach’s construction to chain complexes with a certain shift in degree, i.e. we extend the big bracket to $\mathcal{B}_k(L) = \mathcal{S}(L^*)\otimes\mathcal{S}(L[k])$. We use this construction to give definitions of Lie $k$-bialgebras and Manin $k$-triples on underlying $k$-term complexes $L$ in such a way, that $L$ becomes a Lie $k$-algebra and $L[1-k]$ becomes a Lie $k$-coalgebra resp. $L’=L[1-k]^*$ a Lie $k$-algebra. It is shown that Lie $k$-bialgebras and Manin $k$-triples correspond to Maurer-Cartan elements in certain sub-DGLAs of $\mathcal{B}_k$.