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Uniqueness of the non-commutative divergence cocycle

Pauline Baudat

TL;DR

The paper establishes a rigidity result for degree-zero 1-cocycles on the derivation Lie algebra of the free associative algebra, showing that for $n\ge 3$ every such cocycle with values in $|T(A_n)|\otimes|T(A_n)|$ is a linear combination of the non-commutative divergence $Div$ and its switch $\sigma\circ Div$ on finite-degree quotients. It leverages a Morita–Sakasai–Suzuki type generator description to propagate lower-degree information to all higher degrees, and confirms that no nontrivial cocycle valued in $|T(A_n)|$ exists when $n\ge 3$. In the symplectic setting, the paper proves the uniqueness (up to scalar) of the Enomoto-Satoh trace as a degree-zero cocycle on $\operatorname{Der}_{Sp}(\mathfrak{L}_{2n})$, using Hain’s generation result by $\wedge^3 H$ and representation-theoretic arguments. Together, these results clarify the non-commutative divergence’s role in governing degree-zero cohomology and connect to fundamental trace-type invariants in free Lie algebras. The work also analyzes the special commutative case $n=1$ and discusses obstructions for $n=2$.

Abstract

We show that, for $n \geq 3 $, 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra $T(A_n)$ with values in $ \rvert T(A_n) \rvert \otimes \rvert T(A_n) \rvert $ are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, $ \rvert T(A_n) \rvert $ denotes the space of cyclic words. Furthermore, we study 1-cocycles of degree zero on the Lie algebra of symplectic derivations of the free Lie algebra $ \mathfrak{L_{2n}}$, and prove the uniqueness of the Enomoto-Satoh trace.

Uniqueness of the non-commutative divergence cocycle

TL;DR

The paper establishes a rigidity result for degree-zero 1-cocycles on the derivation Lie algebra of the free associative algebra, showing that for every such cocycle with values in is a linear combination of the non-commutative divergence and its switch on finite-degree quotients. It leverages a Morita–Sakasai–Suzuki type generator description to propagate lower-degree information to all higher degrees, and confirms that no nontrivial cocycle valued in exists when . In the symplectic setting, the paper proves the uniqueness (up to scalar) of the Enomoto-Satoh trace as a degree-zero cocycle on , using Hain’s generation result by and representation-theoretic arguments. Together, these results clarify the non-commutative divergence’s role in governing degree-zero cohomology and connect to fundamental trace-type invariants in free Lie algebras. The work also analyzes the special commutative case and discusses obstructions for .

Abstract

We show that, for , 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra with values in are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, denotes the space of cyclic words. Furthermore, we study 1-cocycles of degree zero on the Lie algebra of symplectic derivations of the free Lie algebra , and prove the uniqueness of the Enomoto-Satoh trace.

Paper Structure

This paper contains 15 sections, 7 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie algebra
  4. Derivation of a Lie algebra
  5. 1-cocycles and non-commutative calculus
  6. Main result
  7. The generators of $\operatorname{Der}(T(A_n))$
  8. Proof of the main result
  9. Case $n=1$
  10. Symplectic derivation Lie algebra of the free algebra
  11. Definitions
  12. Free Lie algebra and important results
  13. Enomoto-Satoh trace
  14. Proof of uniqueness
  15. Appendix

Key Result

Theorem 1

For all $n \ge 3$, let be a 1-cocycle of degree zero. Then, $\forall k \leq n$, its restriction to the quotient by $\operatorname{Der}_{\geq k }(T(A_n))$$\quad$ is a linear combination of the non-commutative divergence $\operatorname{Div}$ and $\sigma \circ \operatorname{Div}$.

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Definition 1
  • Example
  • Example
  • Example
  • Remark
  • Remark
  • Definition 2
  • ...and 20 more