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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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References:
 [1] M Aguiar, Pre-Poisson algebras, Lett. Math. Phys. 54 (2000) 263 · Zbl 1032.17038 [2] F Chapoton, Un endofoncteur de la catégorie des opérades, Lecture Notes in Math. 1763, Springer (2001) 105 · Zbl 0999.17004 [3] F Chapoton, M Livernet, Pre-Lie algebras and the rooted trees operad, Internat. Math. Res. Notices (2001) 395 · Zbl 1053.17001 [4] J Conant, K Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003) 1167 · Zbl 1063.18007 [5] E Getzler, M M Kapranov, Cyclic operads and cyclic homology, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA (1995) 167 · Zbl 0883.18013 [6] E Getzler, M M Kapranov, Modular operads, Compositio Math. 110 (1998) 65 · Zbl 0894.18005 [7] V Ginzburg, M Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994) 203 · Zbl 0855.18006 [8] S Huang, D Tamari, Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law, J. Combinatorial Theory Ser. A 13 (1972) 7 · Zbl 0248.06003 [9] M Kontsevich, Formal (non)commutative symplectic geometry, Birkhäuser (1993) 173 · Zbl 0821.58018 [10] G Labelle, Some new computational methods in the theory of species, Lecture Notes in Math. 1234, Springer (1986) 192 · Zbl 0698.05007 [11] J L Loday, Dialgebras, Lecture Notes in Math. 1763, Springer (2001) 7 · Zbl 0999.17002 [12] M Markl, Cyclic operads and homology of graph complexes, Rend. Circ. Mat. Palermo $$(2)$$ Suppl. (1999) 161 · Zbl 0970.18011
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