Paper
The Veronese square of the dendriform operad
Authors
Murray R. Bremner
Abstract
Veronese powers of operads were introduced in 2020 By Dotsenko, Markl, and Remm. The -th Veronese power of a weight-graded operad is the suboperad generated by the operations of weight . If is generated by binary operations and governs the variety of algebras, this gives a natural definition of the concept of -ary -algebras. In particular, the Veronese square () 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 defining the Veronese square.