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Название: Algebraic Operads. Version 0.99
Авторы: Loday J.-L., Vallette B.
An operad is an algebraic device, which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The classical types of algebras, that is associative algebras, commutative algebras and Lie algebras, give first examples of algebraic operads. Recently, there has been much interest in other types of algebras, to name a few: Poisson algebras, Gerstenhaber algebras, Jordan algebras, pre-Lie algebras, Batalin-Vilkovisky alge- bras, Leibniz algebras, dendriform algebras and the various types of algebras up to homotopy. The notion of operad permits us to study them conceptually and to compare them. The operadic point of view has several advantages. First, many results known for classical algebras, when written in the operadic language, allows us to apply them to other types of algebras. Second, the operadic language simplifies both the statements and the proofs. So, it clarifies the global understanding and allows one to go further. Third, even for classical algebras, the operad theory provides new results that had not been unraveled before. Operadic theorems have been applied to prove results in other fields, like the deformation-quantization of Poisson manifolds by Maxim Kontsevich and Dmitry Tamarkin for instance. Nowadays, operads appear in many different themes: algebraic topology, differential geometry, noncommutative geometry, C ∗ -algebras, symplectic geometry, deformation theory, quantum field theory, string topology, renormalization theory, combinatorial algebra, category theory, universal algebra and computer science.