Weak Lie 2-bialgebra

TL;DR

This paper introduces the concept of weak Lie 2-bialgebras, extending Lie bialgebra theory to a higher categorical setting, and establishes a correspondence with crossed modules of Lie bialgebras.

Contribution

It defines weak Lie 2-bialgebras, describes their compatibility via the big bracket, and links strict cases to crossed modules of Lie bialgebras.

Findings

Weak Lie 2-bialgebras are introduced as compatible pairs of 2-term L-infinity algebras.

A correspondence between strict Lie 2-bialgebras and crossed modules of Lie bialgebras is established.

The compatibility condition is formulated using the big bracket.

Abstract

We introduce the notion of weak Lie 2-bialgebra. Roughly, a weak Lie 2-bialgebra is a pair of compatible 2-term $L_\infty$-algebra structures on a vector space and its dual. The compatibility condition is described in terms of the big bracket. We prove that (strict) Lie 2-bialgebras are in one-one correspondence with crossed modules of Lie bialgebras.