Operad

In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad ${\displaystyle O}$, one defines an algebra over ${\displaystyle O}$ to be a set together with concrete operations on this set which behave just like the abstract operations of ${\displaystyle O}$. For instance, there is a Lie operad ${\displaystyle L}$ such that the algebras over ${\displaystyle L}$ are precisely the Lie algebras; in a sense ${\displaystyle L}$ abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.