INTRODUCTION TO OPERADS JON EIVIND VATNE This is the lecture notes after a short course in the theory of algebraic operads, held at the University of Bergen in the autumn of 2004. The main point has been to give an elementary introduction to operads, with a strong focus on examples. The particular examples in mind are operads that appear when trying to generalize cer- tain aspects of the classical theory of algebras, especially associative, commutative and Lie algebras. The first lecture does not mention operads at all, but show an example of a gen- eralization of two results from the classical theory: the Milnor-Moore theorem and the Poincar´e-Birkhoff-Witt theorems. These theorems are about Hopf algebra struc- tures, where the costructure is cofree cocommutative. These notions are introduced, and an analogue where the multiplicative structure is changed from being associative to being a dendriform algebra. In the classical case, the primitive elements of a Hopf algebra form a Lie algebra, but in this case, they form a brace algebra. The second lecture contains the definition of an operad. We consider composi- tion in the spaces Hom(V⊗n,V) carefully, and extract the axioms of an operad from this special case. Also, algebras over a given operad are introduced. We examplify through the classical structures, defining the operads ss, om and ie, whose al- A C L gebras are associative algebras, associative commutative algebras, and Lie algebras, respectively. We also introduce the exponential generating series for operads, and note some compatibilities among them in the classical cases. In the third lecture, we consider more examples. We see how the classical defini- tions of associative, commutative and Lie algebras can be read off the operadic struc- ture, and introduce a couple of extra examples. These new structures are magmatic algebras, dendriform algebras (already encountered in Lecture 1) and dialgebras. The fourth lecture introduces a convenient way of defining specific operads; in terms of generators and relations. The main ingredients here are a free operad (in terms of planar trees) to get relations, and an ideal in an operad, to get relations. Also, when these notions are introduced, we get a notion of duality for quadratic binary operads, which is modelled on the quadratic or Koszul duality for associative algebras. It is thus known as quadratic or Koszul duality for operads. In the ap- pendix to this section, we introduce a slightly different viewpoint on operads, that is more easily adapted to the definition of a free operad. 1 2 JON EIVINDVATNE In the fifth lecture, the Koszulity condition is introduced. It is the condition that insures that theKoszul duality hasthe expected properties. There isagainadifferent viewpoint on operads that allow us to define Koszulity in terms of the Koszul com- plex. As for algebras, there are two ways of defining Koszulity, by the bar complex or the Koszul complex. The sixth lecture introduces the operad uad, governing quadri-algebras. These Q are based on dendriform algebras, and to study them we need to scrutinize the den- driform condition. Along the way, we actually get a proof that the operad ss is A Koszul. Then, based on the known Koszulity of end, we prove the Koszulity of D uad. The idea is to find a suitable subcomplex of the bar complex, which encap- Q sulates the structure, and which behaves nicely with respect to the process which produces uad from end (the square product (cid:3)). This subcomplex is a copy of Q D cell complexes of the associahedron. The seventh lecture is devoted to examples. Many of the operads we have intro- duced are related by morphisms, or functors between the categories of algebras. The simplest examples are the inclusion of commmutative associative algebras in associa- tive algebras, andtheassociatedLiealgebra toanassociative algebra. These functors have left adjoints; the abelianization of an associative algebra and the universal en- veloping algebra of a Lie algebra, respectively. The other functors we introduce also have left adjoints, so we include a short introduction to the theory of adjoints. Later we will also have need for free and cofree algebras, which are left and right adjoints to forgetful functors. For operads there is a notion of coalgebra, generalizing the classical definition. In the eighth lecture, we introduce these to get homology and cohomology of algebras over an operad. We also need derivations and coderivations, and in the end we get our homology theories. Now the Koszul condition can be formulated in terms of vanishing of homology for the free algebras. In the three classical cases, we can prove Koszulity by identifying the homology theory created as Hochschild homology, Harri- son homology and Chevalley-Eilenberg homology for ss, om and ie, respectively. A C L The classical theory is (very) briefly reviewed in the appendix. The ninth lecture returns to the example of the first lecture, considering algebra structure, coalgebra structures and primitive elements governed by three operads. We think a bit about what happens in the classical case, and then examine a new case of classical interest, namely the case where both the operation and cooperation is governed by the operad ss. A The tenth lecture is about infinity structures, or strong homotopy algebras. The classical example is the class of -algebras, which are exactly infinity algebras ∞ A over ss. The definition can be given an operadic generalization, in terms of a A (conjectured) closed model structure on the category of operads. The easy case is the case of an operad which is Koszul. The bar complex can be augmented to the INTRODUCTION TO OPERADS 3 original operad, and this augmentation map is a quasi-isomorphism in the Koszul case. Now the bar complex determines a minimal model (in particular, a cofibrant model), and algebras over this operad are the infinity algebras over the original operad. By using the coderivations from Lecture 8, we get another description of these algebras. In the appendix, we take a look at the definition of a closed model category. 4 JON EIVINDVATNE 1. Lecture 1: Introduction and first examples In this first lecture we consider examples of relations between different algebraic structures. There are similarities between the relations, pointing towards a richer theory underlying the examples. We work over k = C. 1.1. The triple Ass,Com,Lie. Recall that an asociative algebra A/k consists of a vector space with a fixed mapping µ : A A A, the multiplication, satisfying the ⊗ → associativity condition: µ(a,µ(b,c)) = µ(µ(a,b),c) The diagrammatic representation of this relation is perhaps more appealing (I is the identity operator on A): I⊗µ A A A // A A ⊗ ⊗ ⊗ µ⊗I µ (cid:15)(cid:15) µ (cid:15)(cid:15) A A // A ⊗ is a commutative diagram. If we want a unit u : k A, the axiom should be that → the the diagram below commutes: I⊗u u⊗I A k // A A oo k A ⊗ GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG⊗(cid:15)(cid:15) µwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww ⊗ A The dual notion is that of a coalgebra C. We then have a comultiplication ∆ : C C C satisfying → ⊗ ∆⊗I C C C oo C C ⊗ OO ⊗ ⊗OO I⊗∆ ∆ C C oo C ⊗ ∆ is a commutative diagram. If we want a counit ǫ : C k, the axiom should be → that the the diagram below commutes: INTRODUCTION TO OPERADS 5 I⊗ǫ ǫǫ⊗⊗II C k oo C C oo // k C ⊗ GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG∆⊗OO wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww ⊗ C These items often appear in combination, on the same vector space, with an extra piece of information called the antipode, an anti-homomorphism S : H H. The → diagram for the antipode is S⊗I H H // H H ⊗OO ⊗ ∆ µ ǫ u (cid:15)(cid:15) H // k // H We also demand that ∆ is a morphism of algebras, and that µ is a morphism of coalgebras. To make sense of the latter condition, we need a coalgebra structure on a tensor product of two coalgebras: Definition 1.1 (Tensor product of coalgebras). Let (C,∆ ,ǫ ) and (D,∆ ,ǫ ) C C D D be two coalgebras. We define their tensor product (C D,∆,ǫ) by tensoring the ⊗ underlying vector spaces, ∆ = (I τ I) ∆ ∆ (τ switches the second and the C D ⊗ ⊗ ◦ ⊗ third tensor factors), and ǫ = φ (ǫ ǫ ), where φ : k k k is the canonical C D ◦ ⊗ ⊗ → isomorphism. This determines a coalgebra. Definition 1.2 (Hopf algebra). The data (H,µ,u,∆,ǫ,S) satisfying the require- ments above is known as a Hopf algebra. Example 1.3. Let G be a finite group, k[G] = k g the group algebra. The g∈G · multiplicationµisinducedbythemultiplicationinLG: µ(g g ) = g g ,andextended 1 2 1 2 ⊗ linearly. The comultiplication is defined by ∆(g) = g g, and extended linearly. The ⊗ antipode is defined by taking the inverse in the group. Since (g g )−1 = g−1g−1, this 1 2 2 1 is an anti-homomorphism. Definition 1.4 (Grouplike). InanycoalgebraC, anelement g such that∆(g) = g g ⊗ is called grouplike. The example of the group algebra is special: it is a cocommutative Hopf algebra. For any vector space V, there is a morphism τ : V V V V such that ⊗ → ⊗ τ(v v ) = v v . The diagram for an associative algebra A to be commutative is 1 2 2 1 ⊗ ⊗ 6 JON EIVINDVATNE τ A A // A A ⊗ FFFFFµFFFF## {{xxxµxxxxxx ⊗ A Dually, a coalgebra C is cocommutative if τ C C // C C ⊗ ccGGGGG∆GGGG www∆wwwwww;; ⊗ C is commutative. For the group algebra, ∆(g) = g g = τ(g g) = (τ ∆)(g) ⊗ ⊗ ◦ Now g is a Lie algebra if there is a morphism [,] : g g g such that [x,x] = 0 ⊗ → and [x,[y,z]]+[y,[z,x]]+[z,[x,y]] = 0 Especially, [x,y] = [y,x]. − Any associative algebra give rise to a Lie algebra on the same vector space, by setting [X,Y] = XY YX. − Let Tg be the set of non-commutative polynomials on g∗; the free associative algebra on the vector space g. It is a graded algebra, with degree n being the n th − tensor power of g. If g has a basis g , ,g , a basis for Tg is given by 1 r ··· g g n N, g g , ,g { i1··· in| ∈ ij ∈ 1 ··· r} This T is a functor defined on vector spaces. To take the structure of Lie algebra into account, consider the quotient algebra Ug = Tg/(x y y x [x,y]) ⊗ − ⊗ − This algebra is called the universal enveloping algebra of g. It has a structure of Hopf algebra: the associative structure is given by the qoutient structure, whereas the comultiplication is given by ∆(x) = x 1+1 x ⊗ ⊗ for all x g, and extended to all of Ug by demanding that ∆ is a map of algebras. ∈ Definition 1.5 (Primitive elements). In any bialgebra H, an element X H such ∈ that ∆(X) = X 1+1 X ⊗ ⊗ INTRODUCTION TO OPERADS 7 is called primitive. The set PrimH of primitive elements form a Lie algebra: ∆([X,Y]) = ∆(X)∆(Y) ∆(Y)∆(X) − = (X 1+1 X)(Y 1+1 Y) (Y 1+1 Y)(X 1+1 X) ⊗ ⊗ ⊗ ⊗ − ⊗ ⊗ ⊗ ⊗ = XY 1+X Y +Y X +1 XY ⊗ ⊗ ⊗ ⊗ YX 1 Y X X Y 1 YX − ⊗ − ⊗ − ⊗ − ⊗ = [X,Y] 1+1 [X,Y] ⊗ ⊗ Now the enveloping algebra Ug is cocommutative. We have the following theorem: Theorem 1.6 (Milnor-Moore and Poincar´e-Birkhoff-Witt). The following are equiv- alent: 1 H is connected and cocommutative. 2 H = UPrimH. ∼ 3 H is cofree as a connected cocommutative coalgebra. The first part (1 implies 2) is the Milnor-Moore theorem, and the second (2 implies 3) is the Poincar´e-Birkhoff-Witt theorem. That 3 implies 1 is of course a tautology. It is this theorem that explains the interest in the triple of associative, commuta- tive and Lie algebras: We have a special class of associative algebras H, where the comultiplication is cocommutative and the primitive elements form a Lie algebra. And most importantly, these three structures come together in a very precise form. The notionsfreeandcofreewillbeexplained indetaillater. Connectivity isdefined for graded and filtered structures by demanding that the zeroeth part is the ground field k, and that it is generated by its elements in degree one (in a suitable sense). The terminology comes from algebraic geometry, where an affine scheme whose ring satisfies this condition is connected. There are other and more well known formulations of the Poincar´e-Birkhoff-Witt theorem. 1.2. Algebraic structure on binary trees. We will form a vector space with ba- sis corresponding to binary trees, introduce multiplication and comultiplication, and explain the primitive elements. Then we have a theorem exactly as above, but for a different triple (in fact, dendriform, associative and brace algebras). The material here is from work of Loday and Ronco, see [9], [11], [17] and [18]. We start with the binary trees, which we think of as a prescription where each vertex takes two inputs and gives one output, and the final output is called the root (we often suppress the edge leaving the root, and then refer to the vertex as the root). There is one special tree, without any internal vertices, which we write just as 8 JON EIVINDVATNE a vertical segment. There is one tree with one internal vertex: 6 6666 (cid:8)(cid:8)(cid:8)(cid:8) (cid:8) ◦◦◦◦ There are two trees with two internal vertices: 6 6 6666 (cid:8)(cid:8)(cid:8)(cid:8) 6666 (cid:8)(cid:8)(cid:8)(cid:8) ◦◦◦(cid:8)6666 (cid:8)(cid:8)(cid:8)(cid:8) 66666 (cid:8)(cid:8)(cid:8)◦◦◦(cid:8) (cid:8) and (cid:8) ◦◦◦◦◦ ◦◦◦◦◦ In general, the number of trees with n internal vertices is given by the Catalan number C = 1 2n . Let Y be the set of such trees, k[Y ] the vector space with n n+1 n n n the trees as basis.(cid:0)Fin(cid:1)ally, we let k[Y ] be the direct sum of all these vector spaces. ∞ Given any two trees, there is a new tree constructed from the first two by grafting the roots. We write this T T′. E.g. ∨ 6 6666 (cid:8)(cid:8)(cid:8)(cid:8) 66666 (cid:8)(cid:8)(cid:8)(cid:8) 66666 (cid:17)(cid:17)(cid:17) ◦◦//(cid:8)// (cid:8)(cid:8)(cid:8)(cid:8) 66666 (cid:8)(cid:8)(cid:8)(cid:8) ◦◦◦(cid:8)6666 (cid:8)(cid:8)(cid:8)(cid:8) ◦◦◦(cid:17)6666 (cid:8)(cid:8)(cid:8)◦◦◦(cid:8) (cid:8) (cid:8) = (cid:8) ◦◦◦◦ ∨ ◦◦◦◦◦ ◦◦◦◦◦ Inversely, given any tree T, by removing the root we get two new trees, one left T and one right T (this is not well defined for the tree without vertices). These 1 2 operations are extended to k[Y ] by linearity. ∞ Using these structures, we can introduce certain operations on the space k[Y ] as ∞ follows. Definition 1.7 (Dendriform operations). There are two basic operations and , ≺ ≻ and for convenience, we introduce = + . The definitions are as follows: and ∗ ≻ ≺ |≺| are undefined. T = T,T = 0, T = 0 and T = T. So T = T = T |≻| ≺| ≻| |≺ |≻ |∗ ∗| for all T. T T′ = T (T T′) 1 2 ≺ ∨ ∗ and T T′ = (T T′) T′ ≻ ∗ 1 ∨ 2 Proposition 1.8 (Dendriform axioms). These operations satisfy the axioms (1) (T T′) T′′ = T (T′ T′′) ≺ ≺ ≺ ∗ (2) (T T′) T′′ = T (T′ T′′) ≻ ≺ ≻ ≺ (3) (T T′) T′′ = T (T′ T′′) ∗ ≻ ≻ ≻ (1)+(2)+(3) (T T′) T′′ = T (T′ T′′) ∗ ∗ ∗ ∗ INTRODUCTION TO OPERADS 9 So we have an associative operation which is split in two pieces. Proof. (T T′) T′′ = (T (T T′)) T′′ 1 2 ≺ ≺ ∨ ∗ ≻ = T ((T T′) T′′) 1 2 ∨ ∗ ∗ = T (T (T′ T′′)) 1 2 ∨ ∗ ∗ = T (T′ T′′) ≻ ∗ (T T′) T′′ = ((T T′) T′) T′′ ≻ ≺ ∗ 1 ∨ 2 ≺ = (T T′) (T′ T′′) ∗ 1 ∨ 2 ∗ = T (T′ (T′2 T′′)) ≻ 1 ∨ ∗ = T (T′ T′′) ≻ ≺ (T T′) T′′ = ((T T′) T′′) T′′ ∗ ≻ ∗ ∗ 1 ∨ 2 = (T (T′ T′′)) T′′ ∗ ∗ 1 ∨ 2 = T ((T′ T′′) T′′) ≻ ∗ 1 ∨ 2 = T (T′ T′′) ≻ ≻ (cid:3) Definition 1.9 (Dendriform algebra). A vector space with operations and as ≺ ≻ above, satisfying these axioms, is called a dendriform algebra. There is also a cooperation ∆, similarly defined inductively. Definition 1.10 (Cooperation). If T Y , T Y , T Y , and if n+m+1 1 n 2 m ∈ ∈ ∈ ∆(T ) = T T′ 1 1,i ⊗ 1,(n−i) Xi and ∆(T ) = T T′ 2 2,j ⊗ 2,(m−j) Xj then we define ∆(T) = (T T ) (T′ T′ )+T 1,i ∗ 2,j ⊗ 1,n−i ∨ 2,m−j ⊗| Xi,j Proposition 1.11. ∆ is coassociative. Proposition 1.12 (Hopf algebra of trees). k[Y ] with the operation and the co- ∞ ∗ operation ∆ is a Hopf algebra. There is also a refined notion of dendriform Hopf algebra which also applies to k[Y ]. ∞ Definition 1.13 (Dendriform Hopf algebra). Writing ∆(x) = x x , the (1) (2) ⊗ conditions on ∆ to be compatible with the dendriform structure aPre ∆(x y) = (x y x y +x y x +y x y +x x y)+y x (1) (1) (2) (2) (1) (2) (1) (2) (1) (2) ≺ ∗ ⊗ ≺ ∗ ⊗ ⊗ ≺ ⊗ ≺ ⊗ X 10 JON EIVINDVATNE and ∆(x y) = (x y x y +x y y +y x y +x x y)+x y (1) (1) (2) (2) (1) (2) (1) (2) (1) (2) ≻ ∗ ⊗ ≻ ∗ ⊗ ⊗ ≻ ⊗ ≻ ⊗ X In analogy with our examples earlier, the next natural question is the question of primitive elements. The answer is that the primitive elements form a brace algebra. Definition 1.14 (Bracealgebra). Abrace algebraAis avector space withoperations < >: A⊗n A ··· → for all n 1 (where the operation is the identity for n = 1), satisfying a series of ≥ axioms, the first of which can be written << u,v >,w > < u,< v,w >>=<< u,w >,v > < u,< w,v >> − − This is the pre-Lie axiom; in particular [u,v] :=< u,v > < v,u > defines a Lie − bracket. For the general axioms, we need a little input from thetheory of symmetric groups: first, we define an action of Σ on the r-th tensor power of a vector space V by r σ(v v ) = v v 1 r σ−1(1) σ−1(r) ⊗···⊗ ⊗···⊗ Recall that an (n,m)-shuffle in Σ is a permutation respecting the order of the n+m first n and the last m terms; σ(1) < < σ(n), σ(n+1) < < σ(m+n) ··· ··· For a finite ordered subset X of V, let X⊗ be the tensor product of the elements of X. For any (n,m)-shuffle σ, we say that a family of disjoint ordered subsets χ = X , ,X of v , ,v ,w , w is σ-admissible if 1 r 1 n 1 m { ··· } { ··· ··· } X⊗ X⊗ is equal to σ(v , ,v ,w , w ). • 1 ⊗···⊗ r 1 ··· n 1 ··· m X = i • 6 ∅ If X > 1, then X contains exactly one w, and this is the last element of X . i i i • | | Now we can write up the rest of the axioms: < v , v ,< w , ,w ,z >>= (Σ << X >, < X >,z >) 1 n 1 m χ 1 r ··· ··· ··· X σ∈Shn,m for all shuffles and all admissible families. There is a functor U from the category of brace algebras to the category of dendri- form Hopf algebras, which we again will call the universal enveloping functor. Now we can state the analogue of Theorem 1.6 Theorem 1.15 (MM and PBW for dendriform Hopf algebras). The following are equivalent for a dendriform Hopf algebra H: H is connected. • H = UPrimH. ∼ •