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Quadri-algebras. (English) Zbl 1097.17002

The motivation of the paper under review comes from the theory of dendriform algebras. For a dendriform algebra \(D\) with operations \(\prec\) and \(\succ\) one can introduce a new operation by \(x\ast y= x\prec y + y\succ x\) which is associative. Hence \(D\) may be regarded as an associative algebra with operation \(\ast\) decomposed as the sum of two coherent operations. The authors introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. The paper contains several examples of quadri-algebras: the algebra of permutations, the shuffle algebra, tensor products of dendriform algebras. A pair of commuting Baxter operators on an associative algebra also gives rise to a canonical quadri-algebra structure on the underlying vector space. The main example in this direction is provided by the algebra \(\text{End}(A)\) of linear endomorphisms of an infinitesimal bialgebra \(A\) because this algebra carries a canonical pair of commuting Baxter operators. The authors also discuss commutative quadri-algebras and state some conjectures on the free quadri-algebra.

MSC:

17A30 Nonassociative algebras satisfying other identities
18D50 Operads (MSC2010)

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References:

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