A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras

@misc{nsiloulou2026note,
  title = {{A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras}},
  author = {Servais Cyr Gatsé \and Basile Guy Richard Bossoto \and Irmely Gladesh Mabanza Nsiloulou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32371},
  eprint = {2609.32371},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32371},
}

TY  - UNPB
TI  - A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras
AU  - Servais Cyr Gatsé \and Basile Guy Richard Bossoto \and Irmely Gladesh Mabanza Nsiloulou
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32371
UR  - https://arxiv.org/abs/2609.32371
AB  - Let A be a commutative associative unital algebra over a field K of characteristic zero, and let (G,ρ) be a Lie--Rinehart algebra over A. We revisit the Gerstenhaber structure carried by the exterior algebra bigwedge_A G from the point of view of graded derivations. Particular attention is paid to the well-definedness of the Schouten--Nijenhuis extension over the coefficient algebra A: the Lie--Rinehart identity is shown to provide exactly the correction terms needed for the bracket to descend to the A-balanced exterior algebra. After establishing graded antisymmetry, the Leibniz rule and the graded Jacobi identity without circularity, we study the inner graded derivations adSN(P)=[P,·]_SN and prove [adSN(P),adSN(Q)]=adSN([P,Q]_SN). Finally, we give a complete proof of the converse construction: Gerstenhaber brackets on bigwedge_A G are in one-to-one correspondence with Lie--Rinehart structures on G.
ER  -