The Veronese square of the dendriform operad
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911508004339712 |
|---|---|
| author | Bremner, Murray R. |
| author_facet | Bremner, Murray R. |
| contents | Veronese powers of operads were introduced in 2020 By Dotsenko, Markl, and Remm \cite{DMR}. The $m$-th Veronese power of a weight-graded operad $\mathcal{V}$ is the suboperad $\mathcal{V}^{[m]}$ generated by the operations of weight $m$. If $\mathcal{V}$ is generated by binary operations and governs the variety $\mathbf{V}$ of algebras, this gives a natural definition of the concept of $(m{+}1)$-ary $\mathbf{V}$-algebras. In particular, the Veronese square ($m=2$) corresponds to ternary algebras. We choose five generating operations for the Veronese square of the dendriform operad. We represent the dendriform operad as a suboperad of the Rota-Baxter operad, and express the quadratic relations satisfied by the generating operations as the kernel of a rewriting map. We use combinatorics of monomials and computational linear algebra to determine the kernel. We obtain 33 linearly independent quadratic relations satisfied by the Veronese square. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Veronese square of the dendriform operad Bremner, Murray R. Rings and Algebras Category Theory Primary 17A40. Secondary 17-08, 17A30, 17A50, 17B38, 18M65, 68W30 Veronese powers of operads were introduced in 2020 By Dotsenko, Markl, and Remm \cite{DMR}. The $m$-th Veronese power of a weight-graded operad $\mathcal{V}$ is the suboperad $\mathcal{V}^{[m]}$ generated by the operations of weight $m$. If $\mathcal{V}$ is generated by binary operations and governs the variety $\mathbf{V}$ of algebras, this gives a natural definition of the concept of $(m{+}1)$-ary $\mathbf{V}$-algebras. In particular, the Veronese square ($m=2$) corresponds to ternary algebras. We choose five generating operations for the Veronese square of the dendriform operad. We represent the dendriform operad as a suboperad of the Rota-Baxter operad, and express the quadratic relations satisfied by the generating operations as the kernel of a rewriting map. We use combinatorics of monomials and computational linear algebra to determine the kernel. We obtain 33 linearly independent quadratic relations satisfied by the Veronese square. |
| title | The Veronese square of the dendriform operad |
| topic | Rings and Algebras Category Theory Primary 17A40. Secondary 17-08, 17A30, 17A50, 17B38, 18M65, 68W30 |
| url | https://arxiv.org/abs/2512.11703 |