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Free quadri-algebras and dual quadri-algebras. (English) Zbl 1484.16046

Author’s abstract: We study quadri-algebras and dual quadri-algebras. We describe the free quadri-algebra on one generator as a subobject of the Hopf algebra of permutations FQSym, proving a conjecture due to Aguiar and Loday, using that the operad of quadri-algebras can be obtained from the operad of dendriform algebras by both black and white Manin products. We also give a combinatorial description of free dual quadri-algebras. A notion of quadri-bialgebra is also introduced, with the applications to the Hopf algebras FQSym and WQSym.

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
18M60 Operads (general)
16T05 Hopf algebras and their applications

Software:

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References:

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