The aim of this paper is to further explore an idea from J.-L. Loday brieflyexposed in [5]. We impose a natural and simple symmetry on a unit action overthe most general quadratic relation which can be written. This leads us to twofamilies of binary, quadratic and regular operads whose free objects, as wellas their duals in the sense of Ginzburg and Kapranov are computed. Roughlyspeaking, free objects found here are in relation to $m$-ary trees, triangularnumbers and more generally $m$-tetrahedral numbers, homogeneous polynomials on$m$ commutative indeterminates over a field $K$ and polygonal numbers.Involutive connected P-Hopf algebras are constructed and a link to genomics isdiscussed. We also propose in conclusion some open questions.
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机译：本文的目的是进一步探索J.-L.洛迪（Loday）在[5]中作了简要介绍。我们在可以写的最一般的二次关系上，对单元动作施加自然而简单的对称性。这导致我们得到二元，二次和正则操作符的两个家族，它们的自由对象以及在Ginzburg和Kapranov意义上的对偶被计算出来。粗略地说，这里发现的自由物体与$ m $的一元树，三角数以及更普遍的$ m $四面体数有关，$ m $交换性的齐次多项式在字段$ K $和多边形上不确定。构造霍夫代数，并讨论与基因组学的联系。最后，我们还提出了一些未解决的问题。
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