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

TL;DR

This paper classifies all homogeneous weight 0 Rota--Baxter operators on the Block-type Witt algebra, revealing that solutions are either constant, delta, or finite-support in the non-resonant case, and fully flexible in the resonant case.

Contribution

It provides a complete classification of homogeneous weight 0 Rota--Baxter operators on B(q), correcting previous misconceptions and exploring their implications for Lie algebra deformations.

Findings

Nonlinear functional equation admits only specific solutions in the non-resonant case.

In the resonant case, any profile function is admissible.

Classification is cohomologically exhaustive for generic 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 $(k,k') \in \mathbb{Z}^2$. A key correction is established in the non+resonant regime $q \ne k'$ 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 $q = k'$ 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.