On algebraic description of the Goldman-Turaev Lie bialgebra Yusuke - PowerPoint PPT Presentation

On algebraic description of the Goldman-Turaev Lie bialgebra Yusuke Kuno Tsuda College 7 March 2016 (joint work with Nariya Kawazumi (University of Tokyo))

Contents Introduction 1 Goldman bracket 2 Turaev cobracket 3 Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 2 / 27

Introduction The Goldman-Turaev Lie bialgebra Σ: a compact oriented surface π (Σ) := π 1 (Σ) / conjugacy ∼ = Map ( S 1 , Σ) / homotopy π = ˆ ˆ Two operations to loops on Σ 1 Goldman bracket (‘86) [ , ]: ( Q ˆ π/ Q 1 ) ⊗ ( Q ˆ π/ Q 1 ) → Q ˆ π/ Q 1 , α ⊗ β �→ [ α, β ] 1 ∈ ˆ π : the class of a constant loop 2 Turaev cobracket (‘91) δ : Q ˆ π/ Q 1 → ( Q ˆ π/ Q 1 ) ⊗ ( Q ˆ π/ Q 1 ) Theorem ( Goldman (bracket) +Turaev (cobracket, Lie bialgebra)+Chas (involutivity)) The triple ( Q ˆ π/ Q 1 , [ , ] , δ ) is an involutive Lie bialgebra. Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 3 / 27

Introduction Lie bialgebra The operation [ , ] is defined by using the intersection of two loops, while the operation δ by using the self-intersection of a loop. Theorem (bis) The triple ( Q ˆ π/ Q 1 , [ , ] , δ ) is an involutive Lie bialgebra. Definition A triple ( g , [ , ] , δ ) is a Lie bialgebra if 1 the pair ( g , [ , ]) is a Lie algebra, 2 the pair ( g , δ ) is a Lie coalgebra, and 3 the maps [ , ] and δ satisfy a comatibility condition: ∀ α, β ∈ g , δ [ α, β ] = α · δ ( β ) − β · δ ( α ) . Moreover, if [ , ] ◦ δ = 0 then ( g , [ , ] , δ ) is called involutive. Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 4 / 27

Introduction Fundamental group and tensor algebra We have a binary operation [ , ] and a unary operation δ on Q ˆ π/ Q 1 . The goal is to express them algebraically, i.e., by using tensors. Assume ∂ Σ ̸ = ∅ (e.g., Σ = Σ g , 1 , Σ = Σ 0 , n +1 ). Then any “group-like” Magnus expansion θ gives an isomorphism (of complete Hopf algebras) � ∼ = → � Q π 1 (Σ) − θ : T ( H ) onto the complete tensor algebra generated by H := H 1 (Σ; Q ). Moreover, we have an isomorphism (of Q -vector spaces) ∼ θ : � = → � T ( H ) c yc . Q ˆ π − Here, 1 the source � Q ˆ π is a certain completion of Q ˆ π , c yc means taking the space of cyclic invariant tensors. 2 Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 5 / 27

Introduction Algebraic description of the Goldman bracket We can define [ , ] θ by the commutativity of the following diagram. [ , ] Q ˆ π ⊗ Q ˆ Q ˆ π − − − − → π     � � θ θ ⊗ θ [ , ] θ T ( H ) c yc � � ⊗ � → � T ( H ) c yc T ( H ) c yc − − − − Theorem(Kawazumi-K., Massuyeau-Turaev), stated roughly For some choice of θ , [ , ] θ has a simple, θ -independent expression. 1 For Σ = Σ g , 1 , it equals the associative version of the Lie algebra of symplectic derivations introduced by Kontsevich. 2 For Σ = Σ 0 , n +1 , it equals the Lie algebra of special derivations in the sense of Alekseev-Torossian (c.f. the work of Ihara). Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 6 / 27

Introduction Algebraic description of the Turaev cobracket Similarly, we can define δ θ by the commutativity of the following diagram. δ Q ˆ π/ Q 1 → ( Q ˆ π/ Q 1 ) ⊗ ( Q ˆ π/ Q 1 ) − − − −     � � θ ⊗ θ θ δ θ T ( H ) c yc ⊗ � � � T ( H ) c yc T ( H ) c yc − − − − → Question Can we have a simple expression for δ θ ? Our motivation: the Johnson homomorphism I (Σ): the Torelli group of Σ h (Σ): Morita’s Lie algebra (Kontsevich’s “lie”) τ Kawazumi-K δ � → � ⊗ � π � Q ˆ Q ˆ Q ˆ I (Σ) ֒ → h (Σ) ֒ → π − π. Then Im ( τ ) ⊂ Ker ( δ ). For instance, the Morita trace factors through δ . Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 7 / 27

Goldman bracket Introduction 1 Goldman bracket 2 Turaev cobracket 3 Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 8 / 27

Goldman bracket Definition of the Goldman bracket π (Σ) = Map ( S 1 , Σ) / homotopy. Recall: ˆ π = ˆ Definition (Goldman) α, β ∈ ˆ π : represented by free loops in general position ∑ ε p ( α, β ) α p β p ∈ Q ˆ [ α, β ] := π. p ∈ α ∩ β Here, ε p ( α, β ) = ± 1 is the local intersection number of α and β at p , and α p is the loop α based at p . This formula induces a Lie bracket on Q ˆ π , and 1 ∈ ˆ π is centeral. Background Study of the Poisson structures on Hom ( π 1 (Σ) , G ) / G . Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 9 / 27

Goldman bracket The action σ For ∗ 0 , ∗ 1 ∈ ∂ Σ, ΠΣ( ∗ 0 , ∗ 1 ) := Map (([0 , 1] , 0 , 1) , (Σ , ∗ 0 , ∗ 1 )) / homotopy . Definition (Kawazumi-K.) For α ∈ ˆ π and β ∈ ΠΣ( ∗ 0 , ∗ 1 ), ∑ ε p ( α, β ) β ∗ 0 p α p β p ∗ 1 ∈ Q ΠΣ( ∗ 0 , ∗ 1 ) . σ ( α ) β := p ∈ α ∩ β This formula induces a Q -linear map σ = σ ∗ 0 , ∗ 1 : Q ˆ π → End ( Q ΠΣ( ∗ 0 , ∗ 1 )) . The Leibniz rule holds: for β 1 ∈ ΠΣ( ∗ 0 , ∗ 1 ) and β 2 ∈ ΠΣ( ∗ 1 , ∗ 2 ), σ ( α )( β 1 β 2 ) = ( σ ( α ) β 1 ) β 2 + β 1 ( σ ( α ) β 2 ) . Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 10 / 27

Goldman bracket The action σ (continued) Write ∂ Σ = ⊔ i ∂ i Σ with ∂ i Σ ∼ = S 1 . For each i , choose ∗ i ∈ ∂ i Σ. The small category Q ΠΣ Objects: {∗ i } i Morphisms: Q ΠΣ( ∗ i , ∗ j ) Consider the Lie algebra Der ( Q ΠΣ) := { ( D i , j ) i , j | D i , j ∈ End ( Q ΠΣ( ∗ i , ∗ j )) , D i , j satisfy the Leibniz rule . } Then the collection ( σ ∗ i , ∗ j ) i , j defines a Lie algebra homomorphism σ : Q ˆ π → Der ∂ ( Q ΠΣ) . Example If ∂ Σ = S 1 , we have σ : Q ˆ π → Der ∂ ( Q π 1 (Σ)). Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 11 / 27

Goldman bracket Completions We have a Lie algebra homomorphism σ : Q ˆ π → Der ∂ ( Q ΠΣ) . The augumentation ideal I ⊂ Q π 1 (Σ) defines a filtration { I m } of Q π 1 (Σ). We set � Q π 1 (Σ) / I m . Q π 1 (Σ) := lim ← − m Likewise, we can consider the completions of Q ˆ π and Q ΠΣ. For example, 1 the Goldman bracket induces a complete Lie bracket [ , ]: � ⊗ � π → � π � Q ˆ Q ˆ Q ˆ π , 2 we get a Lie algebra homomorphism π → Der ∂ ( � σ : � Q ˆ Q ΠΣ) . Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 12 / 27

Goldman bracket Magnus expansion Let π be a free group of finite rank. T ( H ) := ∏ ∞ Set H := π abel ⊗ Q ∼ = H 1 ( π ; Q ) and � m =0 H ⊗ m . Definition (Kawazumi) A map θ : π → � T ( H ) is called a (generalized) Magnus expansion if 1 θ ( x ) = 1 + [ x ] + (terms with deg ≥ 2), 2 θ ( xy ) = θ ( x ) θ ( y ). Definition (Massuyeau) A Magnus expansion θ is called group-like if θ ( π ) ⊂ Gr ( � T ( H )). If θ is a group-like Magnus expansion, then we have an isomorphism ∼ = θ : � → � Q π − T ( H ) of complete Hopf algebras. Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 13 / 27

Goldman bracket The case of Σ = Σ g , 1 Definition (Massuyeau) A group-like expansion θ : π 1 (Σ) → � T ( H ) is called symplectic if θ ( ∂ Σ) = exp( ω ), where ω ∈ H ⊗ 2 corresponds to 1 H ∈ Hom ( H , H ) = H ∗ ⊗ H ∼ P.d. H ⊗ H . = Fact: symplectic expansions do exist. The Lie algebra of symplectic derivations (Kontsevich): Der ω ( � T ( H )) := { D ∈ End ( � T ( H )) | D is a derivation and D ( ω ) = 0 } . The restriction map Der ω ( � T ( H )) → Hom ( H , � T ( H )) ∼ P.d. H ⊗ � T ( H ) ⊂ � = T ( H ) , D �→ D | H induces a Q -linear isomorphism Der ω ( � T ( H )) ∼ = � T ( H ) cyc . Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 14 / 27

Goldman bracket The case of Σ = Σ g , 1 : the Goldman bracket Consider the diagram [ , ] Q ˆ π ⊗ Q ˆ − − − − → Q ˆ π π     � � θ θ ⊗ θ [ , ] θ T ( H ) c yc � � ⊗ � → � T ( H ) c yc T ( H ) c yc − − − − where the vertical map θ is induced by π ∋ x �→ − ( θ ( x ) − 1) ∈ � T ( H ). Theorem (Kawazumi-K., Massuyeau-Turaev) If θ is symplectic, [ , ] θ equals the Lie bracket in � T ( H ) cyc = Der ω ( � T ( H )) . Explicit formula: for X 1 , . . . , X m , Y 1 , . . . , Y n ∈ H , [ X 1 · · · X m , Y 1 · · · Y n ] θ ∑ = ( X i · Y j ) X i +1 · · · X m X 1 · · · X i − 1 Y j +1 · · · Y n Y 1 · · · Y j − 1 . i , j Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 15 / 27

Goldman bracket The case of Σ = Σ g , 1 : the action σ Consider the diagram σ Q ˆ π ⊗ Q π 1 (Σ) → Q π 1 (Σ) − − − −     � � θ θ ⊗ θ T ( H ) c yc � � ⊗ � � T ( H ) − − − − → T ( H ) Here, the bottom horizontal arrow is the action of T ( H ) c yc = Der ω ( � � T ( H )) by derivations. Theorem (Kawazumi-K., Massuyeau-Turaev) If θ is symplectic, this diagram is commutative. Kawazumi-K.: use (co)homology theory of Hopf algebras Massuyeau-Turaev: use the notion of Fox paring (see the next page) Yusuke Kuno (Tsuda College) The Goldman-Turaev Lie bialgebra 7 March 2016 16 / 27