Structure theorems for braided Hopf algebras

TL;DR

This paper extends fundamental theorems of Lie theory to braided Hopf algebras by introducing braided analogues of Lie algebras using braided operads, advancing the algebraic framework in braided monoidal categories.

Contribution

It develops braided versions of the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems, introducing new braided Lie algebra concepts via braided operads.

Findings

Established braided PBW and CMM theorems

Introduced braided Lie algebra analogues

Expanded algebraic tools in braided categories

Abstract

We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the notion of a braided operad.