$\mathrm{QP}$-structures of degree $3$ and $\mathsf{CLWX} \: 2$-algebroids

In this paper, we give the notion of a $\mathsf{CLWX} \: 2$-algebroid and show that a $\mathrm{QP}$-structure (symplectic NQ structure) of degree $3$ gives rise to a $\mathsf{CLWX} \: 2$-algebroid. This is the higher analogue of the result that a $\mathrm{QP}$-structure of degree $2$ gives rise to a Courant algebroid. A $\mathsf{CLWX} \: 2$-algebroid can also be viewed as a categorified Courant algebroid. We show that one can obtain a Lie $3$-algebra from a $\mathsf{CLWX} \: 2$-algebroid. Furthermore, $\mathsf{CLWX} \: 2$-algebroids are constructed from split Lie $2$-algebroids and split Lie $2$-bialgebroids.

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Research supported by NSFC (11922110, 11901501), NSF of Jilin Province (20170101050JC), Nanhu Scholars Program for Young Scholars and Nanhu Scholar Development Program of XYNU.

Received 3 March 2016

Accepted 13 September 2018

Published 17 January 2020