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Structure theorems for braided Hopf algebras

Craig Westerland

TL;DR

The paper extends the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems to braided Hopf algebras, introducing braided primitives through the operad BrPrim and a Woronowicz framework that links primitives to Nichols algebras. It develops profinite and monadic methods via profinite completions of braided operads, defines enveloping algebras $U_{ ext{C}}(L)$ for suboperads of $\\widehat{\text{BrAss}}$, and proves CM-Moore-type equivalences for operads with a perfect structure theory, as well as PBW-type filtrations in the braided setting. The results show that primitively generated, finitely braided Hopf algebras $A$ are isomorphic to enveloping algebras of their primitive pieces, $A \cong U_{\\mathfrak{W}_*}(P_{\\mathfrak{W}_*}(A))$, and that iterated primitive filtrations lead to braided Nichols algebras governing the associated graded structures. The framework unifies and extends classical results to braided monoidal categories, enabling a robust operadic approach to primitives, enveloping constructions, and their filtrations. These insights advance the understanding of how braiding affects Lie-type primitives and their enveloping constructions in a broad categorical setting.

Abstract

We develop versions of the Poincaré-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.

Structure theorems for braided Hopf algebras

TL;DR

The paper extends the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems to braided Hopf algebras, introducing braided primitives through the operad BrPrim and a Woronowicz framework that links primitives to Nichols algebras. It develops profinite and monadic methods via profinite completions of braided operads, defines enveloping algebras for suboperads of , and proves CM-Moore-type equivalences for operads with a perfect structure theory, as well as PBW-type filtrations in the braided setting. The results show that primitively generated, finitely braided Hopf algebras are isomorphic to enveloping algebras of their primitive pieces, , and that iterated primitive filtrations lead to braided Nichols algebras governing the associated graded structures. The framework unifies and extends classical results to braided monoidal categories, enabling a robust operadic approach to primitives, enveloping constructions, and their filtrations. These insights advance the understanding of how braiding affects Lie-type primitives and their enveloping constructions in a broad categorical setting.

Abstract

We develop versions of the Poincaré-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.
Paper Structure (47 sections, 71 theorems, 222 equations, 1 figure)

This paper contains 47 sections, 71 theorems, 222 equations, 1 figure.

Table of Contents

  1. Background and technical lemmas
  2. Braid groups and the Artin representation
  3. Braided vector spaces and braided monoidal categories
  4. Finiteness conditions
  5. Finitely braided vector spaces detect the braid group
  6. Pro-objects
  7. Braided Hopf algebras
  8. Definitions
  9. The tensor algebra and its dual
  10. Nichols algebras
  11. Braided operads
  12. Definition and examples
  13. Algebras for braided operads
  14. Modules for algebras over an operad
  15. Presentations of algebras
  16. ...and 32 more sections

Key Result

Theorem 1

If $A$ is a primitively generated, finitely braided Hopf algebra over a field $k$ of characteristic zero, there is an algebra isomorphism

Figures (1)

  • Figure 1: Cabling a 2-strand braid onto the second strand (with strands read from the top) in a 3-strand braid.

Theorems & Definitions (191)

  • Theorem 1
  • Theorem 2
  • Proposition 1.1
  • proof
  • Example 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 1.5
  • Example 1.6
  • Definition 1.7
  • ...and 181 more