Homogeneous Rota--Baxter Operators of Weight~0 on $B(q)$

Abstract

We give a complete and rigorous classification of homogeneous weight $0$ Rota--Baxter operators on the Block-type Witt algebra $B(q)$, assuming the operator has integral degree $(k,k^{\prime })\in {\mathbb{Z}}^2$. A key correction is established in the non+resonant regime $q\ne k^{\prime }$ with $k\ne 0$: the profile function $g(i)=f(-k,i)$ must satisfy the nonlinear functional equation \[ (i - j)g(i)g(j) = g(i+j+k')\big[(i + k' + q)g(i) - (j + k' + q)g(j)\big], \] which admits only constant, Kronecker-delta, or finite-support solutions. This excludes previously and erroneously claimed families such as non-constant polynomials, exponentials, or nontrivial periodic functions. In contrast, the resonant case $q=k^{\prime }$ exhibits full flexibility: any profile $g$ is admissible, provided the operator is supported on the single line $m=-k$. The classification is cohomologically exhaustive for generic $q$ (i.e., when $H^1(B(q),B(q))=0$), and is applied to derive all homogeneous post-Lie structures and associated Lie algebra deformations.