Homogeneous Rota--Baxter Operators of Weight~0 on $B(q)$
Mohsen Ben Abdallah, Marwa Ennaceur
TL;DR
This work provides a complete and corrected classification of homogeneous weight-0 Rota--Baxter operators on Block-type Witt algebras $B(q)$ with integral degree $(k,k')$. By deriving the correct shifted RB equation and a universal resonance constraint, the authors establish a sharp rigidity–flexibility dichotomy: in the non-resonant regime ($q\neq k'$) with $k\neq0$, RB profiles are forced to live on the line $m=-k$ and must satisfy a stringent nonlinear equation whose solutions are only constants, Kronecker deltas, or finite-support profiles; in the resonant regime ($q=k'$) any profile on $m=-k$ is allowed, with no two-line superpositions. The results are cohomologically complete for generic $q$ (where $H^1(B(q),B(q))=0$), and they yield explicit induced pre-Lie structures and Lie algebra deformations, clarifying rigidity in the purely even setting and contrasting with flexible behavior in deformed superalgebras. This corrects prior claims in the literature and provides a rigorous foundation for RB cohomology in Block-type algebras.
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.