NOTES ON OPERADS HSUAN-YI LIAO Abstract. This note is for a talk on - PDF document

NOTES ON OPERADS HSUAN-YI LIAO Abstract. This note is for a talk on operads. The main reference is [1]. The books [2, 3] are also useful. Contents 1. Operad 1 1.1. Tree 1 1.2. Operad and cooperad 2 2. Convolution Lie algebra 3 2.1. Example: cooperad of cocommutative coalgebras 4 2.2. Cobar construction 5 References 5 1. Operad 1.1. Tree. Definition 1.1. A graph Γ = ( V Γ , E Γ ) is a pair of sets where E Γ is contained in the power set 2 V Γ (the set of subsets in V Γ ). A directed graph is a graph Γ = ( V Γ , E Γ ) with source map and target map s, t : E Γ → V Γ such that e = { s ( e ) , t ( e ) } for any e ∈ E Γ . An isomorphism Φ : Γ → ˜ Γ of graphs from Γ = ( V Γ , E Γ ) to ˜ Γ = ( V ˜ Γ , E ˜ Γ ) consists of bijections Φ V : V Γ → V ˜ Γ and Φ E : E Γ → E ˜ Γ such that Φ E ( { v, w } ) = { Φ V ( v ) , Φ V ( w ) } for any { v, w } ∈ E Γ . An isomorphism of directed graphs is an isomorphism of graphs which is compatible with the source and target maps. Let v ∈ V Γ . We denote A ( v ) := { e ∈ E Γ | v ∈ e } . The number | A ( v ) | is called the valency of v . An edge e ∈ E Γ is called a cycle if | e | = 1 . Definition 1.2. A tree T = ( v o , V T , E T ) is a connected graph without cycles which has a special vertex v o ∈ V T , called root vertex , such that | A ( v o ) | = 1 . The edge adjacent to v o is called the root edge , denoted e o . Non-root vertexes of valency 1 are called leaves . The set of leaves of T is denoted L ( T ) . A vertex is called internal if it is neither a root nor a leaf. Remark 1.3. A tree, with the direction towards the root, is naturally a directed graph. Definition 1.4. A tree T is called planar if for every internal vertex of T , the set t − 1 ( v ) carries a total order. An n -labeled planar tree is a planar tree equipped with an injective map l : { 1 , · · · , n } → L ( T ) . (The map l is not required to be monotone.) A vertex v of an n -labeled planar tree T is called nodal if v ∈ N T := V T \ { v o } \ im l . Let S, T be n -labeled planar trees. A (non-planar) morphism Φ : S → T is a pair of bijections Φ V : V S → V T and Φ E : E S → E T which are compatible with source and target maps, and Φ V ◦ l S = l T . The category 1

2 HSUAN-YI LIAO of n -labeled planar trees is denoted Tree( n ) . The full subcategory of n -labeled planar trees with k nodal vertexes is denoted Tree k ( n ) . Remark 1.5. There is a natural left S n -action on the objects of Tree( n ) . 1.2. Operad and cooperad. Let C be the category of cochain complexes. Definition 1.6. A S -module is a sequence { P ( n ) } n ≥ 0 of objects in C such that for each n ∈ N 0 , the object P ( n ) is equipped with a left S n -action. Let T ∈ Tree( n ) . Define � P ( | t − 1 ( v ) | ) P ( T ) := v ∈ N T where the tensor product is done in the order induced by T . Definition 1.7. A (dg) operad is an S -module { P ( n ) } n ≥ 0 equipped with “composition maps” µ T : P ( T ) → P ( n ) for any T ∈ Tree( n ) , and equipped with a unit u : k → P (1) which satisfies a list of axioms (“associativity,” “ S -equivalent,” “unit”). Proposition 1.8. Let V be a cochain complex. The direct sum ∞ � P ( n ) ⊗ V ⊗ n � � P ( V ) := S n n =0 with the natural P -algebra structure is the free P -algebra generated by V . Consider the S -module s 1 − n sign n , � n ≥ 1; Λ( n ) := 0 , n = 0 , where sign n = k with the S n -action σ · 1 := ( − 1) σ · 1 . The compositions are defined by 1 m ◦ i 1 n := ( − 1) (1 − n )( i − 1) 1 n + m − 1 . Remark 1.9. The sign assignment of insertion is different from [1] . It is not clear to the author how the sign convention was chosen in [1] . Let V be a cochain complex, and let Φ : Λ → End V be a morphism of dg operads. Let ˜ Φ : Com → End V [1] be the map � n j =1 ( n − j ) | v j | s − 1 ◦ Φ(1 n )( sv 1 , · · · , sv n ) . ˜ Φ(˜ 1 n )( v 1 , · · · , v n ) := ( − 1) Proposition 1.10. The assignment Λ - Alg → Com - Alg 1 : Φ �→ ˜ Φ is a bijection, where V ∈ Com - Alg 1 iff V [1] ∈ Com - Alg . Proof. We prove ˜ Φ is a morphism of operads. The other parts of proof should be easy. Since Φ is a morphism, we have Φ(1 n )( v σ (1) , · · · , v σ ( n ) ) = ǫ ( σ, v )( − 1) σ Φ(1 n )( v 1 , · · · , v n )

NOTES ON OPERADS 3 Then � n j =1 ( n − j ) | v j | s − 1 ◦ Φ( σ · 1 n )( sv 1 , · · · , sv n ) Φ( σ ⋆ ˜ ˜ 1 n )( v 1 , · · · , v n ) = ( − 1) σ ( − 1) � n j =1 ( n − j ) | v j | s − 1 ◦ Φ(1 n )( sv σ (1) , · · · , sv σ ( n ) ) = ( − 1) σ ǫ ( σ, sv )( − 1) � n j =1 ( n − j ) | v j | s − 1 ◦ Φ(1 n )( sv σ (1) , · · · , sv σ ( n ) ) = ǫ ( σ, v )( − 1) = ˜ Φ(˜ � � 1 n ) σ ⋆ ( v 1 ⊗ · · · v n ) . and ˜ Φ(˜ ◦ i ˜ 1 m ¯ 1 n )( v 1 , · · · , v m + n − 1 ) � m + n − 1 ( m + n − 1 − j ) | v j | s − 1 ◦ Φ(1 m + n − 1 )( sv 1 , · · · , sv m + n − 1 ) = ( − 1) j =1 � m + n − 1 ( m + n − 1 − j ) | v j | s − 1 ◦ Φ(1 m ◦ i 1 n )( sv 1 , · · · , sv m + n − 1 ) = ( − 1) (1 − n )( i − 1) ( − 1) j =1 � m + n − 1 ( m + n − 1 − j ) | v j | ( − 1) | Φ(1 n ) | ( i − 1+ � i − 1 = ( − 1) (1 − n )( i − 1) ( − 1) j =1 | v j | ) j =1 · s − 1 ◦ Φ(1 m ) sv 1 , · · · , sv i − 1 , ss − 1 Φ(1 n )( sv i , · · · , sv i + n − 1 ) , sv i + n , · · · , sv m + n − 1 � � � m + n − 1 ( m + n − 1 − j ) | v j | ( − 1) (1 − n )( i − 1+ � i − 1 = ( − 1) (1 − n )( i − 1) ( − 1) j =1 | v j | ) j =1 � i + n − 1 � i − 1 � m + n − 1 � i + n − 1 ( n + i − 1 − j ) | v j | ( − 1) j =1 ( m − j ) | v j | ( − 1) j = i + n ( m + n − 1 − j ) | v j | ( − 1) ( m − i ) | v j | · ( − 1) j = i j = i · ˜ Φ(˜ v 1 , · · · , v i − 1 , ˜ Φ(˜ � � 1 m ) 1 n )( v i , · · · , v i + n − 1 ) , v i + n , · · · , v m + n − 1 = ˜ v 1 , · · · , v i − 1 , ˜ Φ(˜ Φ(˜ � � 1 m ) 1 n )( v i , · · · , v i + n − 1 ) , v i + n , · · · , v m + n − 1 � ˜ Φ(˜ ◦ i ˜ Φ(˜ �� � = 1 m )¯ 1 n ) v 1 , · · · , v m + n − 1 . � Definition 1.11. A (dg) cooperad is an S -module { Q ( n ) } n ≥ 0 equipped with “decomposition maps” ∆ T : Q ( n ) → Q ( T ) for any T ∈ Tree( n ) , and equipped with a counit ˜ u : Q (1) → k which satisfies a list of axioms (“coassocia- tivity,” “ S -equivalent,” “counit”). A cooperad Q is coaugmented if we have a cooperad morphism ǫ : ∗ → Q , where ∗ is the natural cooparad with ∗ (1) = k and ∗ ( n ) = 0 if n � = 1 . We denote the pseudo-cooperad coker( ǫ ) by Q o . Example 1.12. The S -module Λ also caries a cooperad structure: ∆ i : Λ m + n − 1 → Λ m ⊗ Λ n , ∆ i (1 m + n − 1 ) := ( − 1) (1 − n )( i − 1) · 1 m ⊗ 1 n . 2. C onvolution Lie algebra The notation π 0 denotes the collection of isomorphism classes in a category. Let P be a dg (pseudo-)operad, and Q be a dg (pseudo-)cooperad. Consider � Conv( Q, P ) := Hom S n ( Q ( n ) , P ( n )) n ≥ 0 with the operation • defined by the sum of the compositions ∆ T f ⊗ g µ T Q ( n ) − − → Q ( n 1 ) ⊗ Q ( n 2 ) − − → P ( n 1 ) ⊗ P ( n 2 ) − − → P ( n )

4 HSUAN-YI LIAO where T ∈ Tree 2 ( n ) , n i = | t − 1 ( v i ) | , N T = { v 1 , v 2 } . More precisely, � f • g ( x ) := µ T ◦ ( f ⊗ g ) ◦ ∆ T ( x ) T ∈ π 0 (Tree 2 ( n )) for x ∈ Q ( n ) . Lemma 2.1. The bracket [ f, g ] := f • g − ( − 1) | f || g | g • f satisfies the Jacobi identity. The differentials on P and Q induce a differential on the convolution Conv( Q, P ) . Proposition 2.2. The convolution Conv( Q, P ) is a dgla. 2.1. Example: cooperad of cocommutative coalgebras. Let coCom be the cooperad of cocommutative coassociative coalgebras. More precisely, � 0 , n = 0; coCom( n ) := k · δ n , n � = 0 , with trivial S n -action and with the cocompositions ∆ T : coCom( n ) → coCom( n 1 ) ⊗ coCom( n 2 ) : δ n �→ δ n 1 ⊗ δ n 2 for T ∈ Tree 2 ( n ) . We endow coCom with the coaugmentation ǫ : ∗ → coCom : 1 �→ δ 0 . If V is a cochain complex, then coCom( V ) ∼ = S ≥ 1 V with the differential induced from V and the natural comultiplication. Proposition 2.3. Let V be a cochain complex. Then Conv(coCom o , End V ) ∼ = coDer ′ (coCom( V )) , where coDer ′ (coCom( V )) is the set of coderivations on coCom( V ) ∼ = S ≥ 1 V which vanish on V . Proof. Note that coDer ′ (coCom( V )) ∼ = Hom( S ≥ 2 V, V ) ∞ ∼ � Hom( S n V, V ) = n =2 ∞ ∼ � k , Hom( S n V, V ) � � Hom = n =2 ∞ ∼ � coCom o ( n ) , Hom( S n V, V ) � � = Hom n =0 ∞ ∼ � coCom o ( n ) , Hom( V ⊗ n , V ) � � Hom S n = . n =0 It’s straightforward to check the isomorphisms preserve the dgla structures. � Remark 2.4. According to [1] , the above proposition is true for general coaugmented cooperads.