Algebraic Operads

An operad is an algebraic device which encodes a type of algebras. Instead of study-

ing 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 alge-

bras and Lie algebras, give the 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 types of algebras, when written in the operadic language, can be applied to

othertypesofalgebras.Second,theoperadiclanguagesimplifiesboththestatements

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 geom-

etry, C ∗ -algebras, symplectic geometry, deformation theory, quantum field theory,

string topology, renormalization theory, combinatorial algebra, category theory, uni-

versal algebra and computer science.

Historically, the theoretical study of compositions of operations appeared in the

1950s in the work of Michel Lazard as “analyseurs”. Operad theory emerged as

an efficient tool in algebraic topology in the 1960s in the work of Frank Ada