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On some anticyclic operads. (English) Zbl 1060.18004
An operad \(\mathcal P\) is a collection of modules \({\mathcal P}(n)\) over the symmetric group of degree \(n\) equipped with composition maps satisfying some axioms modelled after the composition of multilinear maps. Similar structures without the actions of symmetric groups are called non-symmetric operads. A non-symmetric operad with action of the cyclic group of order \(n+1\) on \({\mathcal P}(n)\), with some axioms, is called anticyclic. The characteristic function of the operad \({\mathcal P}\) is defined as the infinite sum of the symmetric functions for the modules \({\mathcal P}(n)\).
The most classical operads are the three operads describing commutative-associative, associative and Lie algebras. They fit the following diagram: \(\text{Comm}\leftarrow \text{Assoc} \leftarrow \text{Lie}\).
In the paper under review the author shows that most of the properties of this classical sequence of operads hold also for two other diagrams involving some binary quadratic operads, namely \[ \text{Perm}\leftarrow \text{Dias} \leftarrow \text{Leib},\quad \text{Zinb}\leftarrow \text{Dend} \leftarrow \text{PreLie}. \] Here Leibniz algebras and pre-Lie algebras satisfy the identities \([x,[y,z]]=[[x,y],z]-[[x,z],y]\) and \((xy)z-x(yz)=(xz)y-x(zy)\), respectively. Diassociative algebras have two binary multiplications \(\dashv\) and \(\vdash\) subject to some associativity conditions. Dendriform algebras also have two operations \(\prec\) and \(\succ\) such that the product made of the sum \(x\prec y+y\succ x\) is associative. The Perm operad is a quotient of the diassociative operad Dias by the ideal generated by \(x\dashv y - y\vdash x\) (and the Leibniz operad Leib is a suboperad of Dias generated by the same element). Similarly, the Zinbiel operad is a quotient of the dendriform operad modulo the ideal generated by \(x\prec y - y\succ x\) (and PreLie is realized as a suboperad of Dend generated by \(x\succ y-y\prec x\)).
First the author proves that Dias and Dend can be endowed in a unique way with non-symmetric anticyclic structures. Then he establishes that Perm and Leib, respectively Zinb and PreLie inherit the anticyclic structures of Dias, respectively Dend. The author also calculates the characteristic functions of Dias, Dend and Perm and conjectures a formula for the characteristic function of PreLie.

MSC:
18D50 Operads (MSC2010)
05E05 Symmetric functions and generalizations
17A30 Nonassociative algebras satisfying other identities
17A32 Leibniz algebras
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