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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.12093 |
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| _version_ | 1866909959550140416 |
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| author | Abdallah, Mohsen Ben Ennaceur, Marwa |
| author_facet | Abdallah, Mohsen Ben Ennaceur, Marwa |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12093 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogeneous Rota--Baxter Operators of Weight~0 on $B(q)$ Abdallah, Mohsen Ben Ennaceur, Marwa Rings and Algebras Mathematical Physics 17B65, 17B56, 17D25, 81R12 G.2.1; I.1.2 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. |
| title | Homogeneous Rota--Baxter Operators of Weight~0 on $B(q)$ |
| topic | Rings and Algebras Mathematical Physics 17B65, 17B56, 17D25, 81R12 G.2.1; I.1.2 |
| url | https://arxiv.org/abs/2512.12093 |