Uniqueness of the non-commutative divergence cocycle

2025/11/10 by Pauline Baudat, Baudat, Pauline
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2511.06903

openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

We show that, for n ≥ 3 , 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra T(A_n) with values in | T(A_n) | ⊗ | T(A_n) | are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, | T(A_n) | 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 \mathfrakL_2n, and prove the uniqueness of the Enomoto-Satoh trace.

Citations

• The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in\n higher genera
• New series in the Johnson cokernels of the mapping class groups of surfaces
• Infinitesimal presentations of the Torelli groups

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