1-shifted Lie bialgebras and their quantizations
Wenjun Niu, Victor Py
TL;DR
The paper extends Lie bialgebra quantization to the setting of 1-shifted structures by introducing 1-shifted metric Lie algebras and transversal Lagrangian (Manin-triple) data, establishing a one-to-one correspondence with 1-shifted Lie bialgebras and constructing their classical double with a 1-shifted r-matrix satisfying a shifted classical Yang–Baxter equation. It then develops a canonical quantization framework using curved differential graded algebras to deform the double and derive monoidal actions on quantized substructures, with a crucial assumption g_2=0 enabling the quantization of the Lagrangian halves and their Koszul duality. The approach yields DG analogues of Yangians, including the notion of DG 1-shifted Yangians for loop-group–based examples, and connects to physical interpretations in 2d bulk-boundary TQFTs and 3d HT theories. Overall, the work provides a shifted generalization of Drinfeld–Etingof–Kazhdan quantization, yielding new algebraic objects and monoidal structures with potential applications to shifted Poisson theory and HTQFTs.
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.