Intersection cohomology of coisotropic submanifolds Work in progress - PowerPoint PPT Presentation

347 Grundlehren der mathematischen Wissenschaften 347 Grundlehren der mathematischen Wissenschaften Vanhaecke Laurent-Gengoux · Pichereau A Series of Comprehensive Studies in Mathematics A Series of Comprehensive Studies in Mathematics Camille Laurent-Gengoux · Anne Pichereau · Pol Vanhaecke Poisson Structures Camille Laurent-Gengoux Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and diff erential geometry to abstract algebra, algebraic Anne Pichereau geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are de- Pol Vanhaecke cisive for the solution to the problem in nearly all cases. Poisson Structures is the fi rst book that off ers a comprehensive introduction to the theory, as well as an overview of the diff erent aspects of Poisson structures. T e fi rst part covers solid foundations, the central part consists of a detailed exposition of the diff erent known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the fi nal Poisson Structures part is devoted to the two main applications of Poisson structures (integrable systems 1 and deformation quantization). T e clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures. Poisson Structures Mathematics ISSN 0072-7830 ISBN 978-3-642-31089-8 9 7 8 3 6 4 2 3 1 0 8 9 8

Intersection cohomology of coisotropic submanifolds Work in progress Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 1 / 1

Gerstenhaber algebra, Gerstenhaber module. A Gerstenhaber algebra is a graded vector space E , endowed with a graded commutative product ∧ on E 1 a graded Lie algebra structure [ ., . ] on E [ − 1 ] , 2 + compatibility: [ Q ∧ R , P ] = [ Q , P ] ∧ R + ( − 1 ) q ( p − 1 ) Q ∧ [ R , P ] for 3 all P , Q , R of degrees p , q , r . Relaxing Jacobi for [ ., . ] leads to pre-Gerstenhaber algebra . A Gerstenhaber algebra module is a graded vector space F endowed with: a structure ı of module for the graded algebra ( E , ∧ ), 1 a structure L of module for the graded Lie algebra ( E [ − 1 ] , [ ., . ]) ), 2 + compatibility: L P ◦ ı Q α − ( − 1 ) q ( p − 1 ) ı Q ◦ L P α = ı [ P , Q ] α . 3 Corresponding notion of a pre-Gerstenhaber module . Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 2 / 1

(Pre)-Lie algebroid Definition Let A → M be a vector bundle. A pre-Lie algebroid structure on A is a graded derivation of degree + 1: D : Γ( ∧ • A ∗ ) → Γ( ∧ • + 1 A ∗ ) , said to be a Lie algebroid when it squares to 0. The bracket of P ∈ Γ( ∧ p A ) with Q ∈ Γ( ∧ q A ) is the unique element R := [ P , Q ] D ∈ Γ( ∧ p + q − 1 A ) s.t. [ L P , ı Q ] = ı R . In the previous, L P := ı P ◦ D − ( − 1 ) p D ◦ ı P is the Lie derivative . Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 3 / 1

Gerstenhaber algebra and Gerstenhaber module. Proposition Let A → M be a vector bundle. For every Lie algebroid structure D , the triple (Γ( ∧ • A ) , ∧ , [ · , · ] D ) is a Gerstenhaber algebra, and Γ( ∧ • A ∗ ) , equipped with: the action of (Γ( ∧ • A ) , ∧ ) by contractions, 1 the action of (Γ( ∧ • A ) , [ ., . ]) by Lie derivatives, 2 is a module over this Gerstenhaber algebra. This can be weakened as follows. Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 4 / 1

Gerstenhaber algebra and Gerstenhaber module. Proposition Let A → M be a vector bundle. For every pre -Lie algebroid structure D , the triple (Γ( ∧ • A ) , ∧ , [ · , · ] D ) is a pre -Gerstenhaber algebra, and Γ( ∧ • A ∗ ) , equipped with: the action of (Γ( ∧ • A ) , ∧ ) by contractions, 1 the action of (Γ( ∧ • A ) , [ ., . ] D ) by Lie derivatives, 2 is a module over this pre -Gerstenhaber algebra. Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 5 / 1

Koszul complex Let A → M be a vector bundle. Choose a section ϕ ∈ Γ( A ∗ ) . Let ı ϕ : Γ( ∧ • A ) → Γ( ∧ •− 1 A ) be the contraction by ϕ 1 Let m ϕ : Γ( ∧ • A ∗ ) → Γ( ∧ • + 1 A ∗ ) be the (left) mutiplication by ϕ . 2 Both ı ϕ and m ϕ square to zero, hence define a homology H ∗ ( ϕ ) and a cohomology H ∗ ( ϕ ) , said to be attached to ϕ . The graded algebra structure ∧ and module structure ı go down: Proposition The homology H ∗ ( ϕ ) is a graded commutative algebra. 1 The cohomology H ∗ ( ϕ ) is a module over the algebra H ∗ ( ϕ ) . 2 Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 6 / 1

Gerstenhaber structure on the Koszul complex. Question 1. Let A → M be a vector bundle. Given a section ϕ ∈ Γ( A ∗ ) , a Lie algebroid D : Γ( ∧ • A ∗ ) �→ Γ( ∧ • + 1 A ∗ ) , do the Gerstenhaber algebra (Γ( ∧ • A ) , ∧ , [ ., . ] D ) , and its module (Γ( ∧ • A ∗ ) , ı, L ) , go to the quotient with respect to m ϕ , ı ϕ and define structures of Gerstenhaber algebra and module on H ∗ ( ϕ ) and H ∗ ( ϕ ) respectively ? Answer: Yes, if D ( ϕ ) = 0. Proposition Let A → M be a vector bundle. For every Lie algebroid D and 1-cocyle ϕ ∈ Γ( A ∗ ) , H ∗ ( ϕ ) and H ∗ ( ϕ ) are Gerstenhaber algebras and modules respectively. Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 7 / 1

Gerstenhaber structure on the Koszul complex. Question 1. Let A → M be a vector bundle. Given a section ϕ ∈ Γ( A ∗ ) , a pre -Lie algebroid D : Γ( ∧ • A ∗ ) �→ Γ( ∧ • + 1 A ∗ ) , do the pre -Gerstenhaber algebra (Γ( ∧ • A ) , ∧ , [ ., . ] D ) , and its module (Γ( ∧ • A ∗ ) , ı, L ) , go to the quotient with respect to m ϕ , ı ϕ and define structures of pre -Gerstenhaber algebra and module on H ∗ ( ϕ ) and H ∗ ( ϕ ) respectively ? Answer: Yes, if and only if D ( ϕ ) = 0. Proposition Let A → M be a vector bundle. For every pre -Lie algebroid D and 1-cocyle ϕ ∈ Γ( A ∗ ) , H ∗ ( ϕ ) and H ∗ ( ϕ ) are pre -Gerstenhaber algebras and modules respectively. Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 8 / 1

Induced structures on Koszul complex-2. Question 2. Could it be that D be only a pre -Lie algebroid but still the induced structure is a (honnest) Gerstenhaber algebra ? Answer: Theorem Let A → M be a vector bundle. Given: a section ϕ ∈ Γ( A ∗ ) 1 a pre -Lie algebroid D : Γ( ∧ • A ∗ ) → Γ( ∧ • + 1 A ∗ ) , 2 such that ♣ D ( ϕ ) = 0, ♠ D 2 = C ◦ m ϕ + m ϕ ◦ C for some operator C (i.e. D 2 homotopic to zero - recall that m ϕ ( α ) = ϕ ∧ α computes H ∗ ( ϕ ) ), then D induces a structure of Gerstenhaber algebra on the homology H ∗ ( ϕ ) and a structure of Gerstenhaber module on the cohomology H ∗ ( ϕ ) . Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 9 / 1

P ∞ -algebras Question 3. Where do pre-Lie algebroid structures D and sections ϕ satisfying ♣ (i.e. D ( ϕ ) = 0) and ♠ (i.e. D 2 = C ◦ m ϕ + m ϕ ◦ C ) can arise from? Answer. From a Maurer-Cartan element in a P ∞ algebras on ∧ A ∗ . Definition Let A → M be a vector bundle. A P ∞ -algebra structure on ∧ A ∗ a sequence of n -ary “brackets” : [Γ( ∧ a 1 A ∗ ) , . . . , Γ( ∧ a n A ∗ )] n ⊂ Γ( ∧ a 1 + ··· + a n − n + 2 A ∗ ) for all n ∈ N ∗ , such that [ · · · ] n skew-symmetric and is a derivation in each variable, 1 the “higher” Jacobi identities hold: 2 � � ( − 1 ) σ, x [[ x σ ( 1 ) , . . . , x σ ( i ) ] i , x σ ( i + 1 ) , . . . , x σ ( n + 1 ) ] j = 0 . i + j = n + 1 σ ∈ Σ i , j Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 10 / 1

Precisions on the degrees. By construction: [Γ( A ∗ ) , Γ( A ∗ )] 2 ⊂ Γ( ∧ 2 A ∗ ) (so sections of A ∗ do not come equipped with a bracket !) Remark. For every ϕ ∈ Γ( A ∗ ) and every sequence ( a n ) n ≥ 1 in R , the operator � D : α �→ a n [ ϕ, . . . , ϕ, α ] n n ≥ 1 is a pre-Lie algebroid, provided that it converges. Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 11 / 1

Let A → M be a vector bundle, and ([ . . . ] n ) n ∈ N ∗ be a P ∞ -structure on ∧ A ∗ . Definition A section ϕ ∈ Γ( A ∗ ) is said to be Maurer-Cartan when: � [ ϕ, . . . , ϕ ] n = 0 . n ! n ∈ N ∗ By construction: � [ ϕ, . . . , ϕ, α ] n D ϕ ( α ) := n ! n ∈ N ∗ is a pre-Lie algebroid that satisfies ♣ (i.e. D ϕ ( ϕ ) ). Poisson 2012 (C. Laurent-Gengoux) Intersection cohomology of coisotropic submanifolds Utrecht, August 2012 12 / 1