Homology of Moduli Spaces of Curves and Commutative Homotopy Algebras

There have been a number of mathematical results recently identifying algebras over certain operads [4, 25, 17, 28, 16, 10, 14, 11]. See [1, 26] for expository surveys of the basics of operad theory. Before citing any of these results, let us mention some trivial classical examples. Let A denote one of the three words: “commutative”, “associative” and “Lie”. In each of these cases, consider the corresponding operad O(n) = 〈 words in the free A algebra on n generators having at most one occurance of each generator 〉k, n ≥ 1, 〈 〉k meaning the linear span over the ground field k. When A is commutative, associative or Lie, we will denote the corresponding operad O by C, As and Lie, respectively. Note that C(n) = 〈x1 . . . xn〉k = k and As(n) = 〈xσ(1) . . . xσ(n) | σ ∈ Sn〉k = k[Sn]. The main feature of these three operads is that they describe algebras of the corresponding types. More precisely, algebras over an operad O, O = C, A∫ or L〉⌉, (or simply, O-algebras) are exactly A algebras. A hint of another relation between operads and algebras may be given by the formula: ⊕