1-shifted Lie bialgebras and their quantizations

Abstract

In this paper, we define (cohomologically) 1-shifted Manin triples and 1-shifted Lie bialgebras, and study their properties. We derive many results that are parallel to those found in ordinary Lie bialgebras, including the double construction and the existence of a 1-shifted $r$-matrix satisfying the classical Yang-Baxter equation. Turning to quantization, we first construct a canonical quantization for each 1-shifted metric Lie algebra $\mathfrak{g}$, producing a deformation to the symmetric monoidal category of $\mathfrak{g}$ modules over a formal variable $\hbar$. This quantization is in terms of a curved differential graded algebra. Under a further technical assumption, we construct quantizations of transverse Lagrangian subalgebras of $\mathfrak{g}$, which is a pair of DG algebras connected by Koszul duality, and give rise to monoidal module categories of the quantized double. Finally, we apply this to Manin triples arising from Lie algebras of loop groups, and construct 1-shifted meromorphic $r$-matrices. The resulting quantizations are the cohomologically-shifted analogue of Yangians.