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Algebras with one operation including Poisson and other Lie-admissible algebras. (English) Zbl 1101.18004

The paper studies when algebras with one operation form a monoidal category and analyzes Koszulness, cyclicity and (here introduced) dihedrality of the corresponding operads. The main tool, the polarization, permits to represent algebras with one operation without any symmetry (such as associative algebras) as structures with one commutative and one anticommutative operation (such as Poisson algebras).

MSC:

18D50 Operads (MSC2010)
17D25 Lie-admissible algebras
17A99 General nonassociative rings
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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