Year: 2026
Type: preprint
Abstract: Let k be a field, let V=k^n with n>=3, and let L_d be the homogeneous divergence-free polynomial vector fields of coefficient degree d. If E_d is the exact fibre under contraction with a volume form, then [L_2,L_{d-1}]=E_d for every d>=3. The proof is weightwise: after reducing the integer weight coordinates along Z->k, the bracket map becomes contraction by the resulting covector. Consequently the degree-d derived algebra is E_d for d>=3. In characteristic p>0, with G=GL(V), the quotient L_d/E_d is nonzero precisely for d=pq-(n-1), q>=n-1, and relative Cartier descent gives (L_d/E_d) tensor det(V*) isomorphic to F^*_{G/k}(Sym^{q-n+1}((V^{(1)})*) tensor Lambda^{n-1}((V^{(1)})*)). Equivalently, at d_r=(p-1)(n-1)+pr, one has L_{d_r}/E_{d_r} isomorphic to (det V*)^{tensor(p-1)} tensor W_r^{(1)}, where W_r^{(1)}=F^*_{G/k}(Sym^r((V^{(1)})*) tensor V^{(1)}). For p>=5, the exact fibres also satisfy the restricted-source recursion E_d=[L_2,E_{d-1}] for every d>=3.
Source: arXiv (Cornell University)
Author: -
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171302322