Higher order Koszul brackets Hovhannes Khudaverdian University of - PowerPoint PPT Presentation

Higher order Koszul brackets Higher order Koszul brackets Hovhannes Khudaverdian University of Manchester, Manchester, UK XXXY WORKSHOP ON GEOMETRIC METHODS IN PHYSICS 26 June-2 July, Bialoweza, Poland The talk is based on the work with Ted Voronov

Higher order Koszul brackets Contents Abstracts Poisson manifold and.... Higher brackets

Higher order Koszul brackets Papers that talk is based on are [1] H.M.Khudaverdian, Th. Voronov Higher Poisson brackets and differential forms , 2008a In: Geometric Methods in Physics. AIP Conference Proceedings 1079, American Institute of Physics, Melville, New York, 2008, 203-215., arXiv: 0808.3406 [2] Th. Voronov, Nonlinear pullback on functions and a formal category extending the category of supermanifolds] , arXiv: 1409.6475 [3] Th. Voronov, Microformal geometry , arXiv: 1411.6720

Higher order Koszul brackets Abstracts Abstract... For an arbitrary manifold M , we consider supermanifolds Π TM and Π T ∗ M , where Π is the parity reversion functor. The space Π T ∗ M possesses canonical odd Schouten bracket and space Π TM posseses canonical de Rham differential d . An arbitrary even function P on Π T ∗ M such that [ P , P ] = 0 induces a homotopy Poisson bracket on M , a differential, d P on Π T ∗ M , and higher Koszul brackets on Π TM . (If P is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of Q -manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on P . Then using just recently invented theory of thick morphisms we construct a non-linear map between the L ∞ algebra of functions on Π TM with higher Koszul brackets and the Lie algebra of functions on Π T ∗ M with the canonical odd Schouten bracket.

Higher order Koszul brackets Abstracts

Higher order Koszul brackets Poisson manifold and.... Poisson manifold Let M be Poisson manifold with Poisson tensor P = P ab ∂ b ∧ ∂ a { f , g } = { f , g } P = ∂ f ∂ x a P ab ∂ g ∂ x b . {{ f , g } , h } + {{ g , h } , f } + {{ h , f } , g } = 0 , � P ar ∂ r P bc + P br ∂ r P ca + P cr ∂ r P ab = 0 . If P is non-degenerate, then ω = ( P − 1 ) ab dx a ∧ dx b is closed non-degenerate form defining symplectic structure on M .

Higher order Koszul brackets Poisson manifold and.... Differentials d —de Rham differential, d : Ω k ( M ) → Ω k + 1 ( M ) , d 2 = 0 , df = ∂ f ∂ x a dx a , d ( ω ∧ ρ ) = d ω ∧ ρ +( − 1 ) p ( ω ) ω ∧ d ρ d P —Lichnerowicz- Poisson differential, d P : A k ( M ) → A k + 1 ( M ) , P = 0 , d P f = ∂ f ∂ x b P ba ∂ d 2 ∂ x a d P P = 0 ↔ Jacobi identity for Poisson bracket { , }

Higher order Koszul brackets Poisson manifold and.... Differential forms and multivector fields A ∗ space multivector fields on M , Ω ∗ space of differential forms on M , d P A k ( M ) A k + 1 ( M ) − → ↑ ↑ d Ω k ( M ) Ω k + 1 ( M ) − →

Higher order Koszul brackets Poisson manifold and.... Differential forms and multivector fields A ∗ — multivector fields on M = functions on Π T ∗ M Ω ∗ — differential forms on M = functions on Π TM , d P d P A k ( M ) A k + 1 ( M ) C (Π T ∗ M ) C (Π T ∗ M ) − → − → ↑ ↑ ↑ ↑ d d Ω k ( M ) Ω k + 1 ( M ) − → C (Π TM ) − → C (Π TM ) d ω ( x , ξ ) = ξ a ∂ ∂ x a ω ( x , ξ ) , d P F ( x , θ ) = [ P , F ] , [ P , F ] -canonical odd Poisson bracket on Π T ∗ M .

Higher order Koszul brackets Poisson manifold and.... x a = ( x 1 ,..., x n ) — coordinates on M ( x a , ξ b ) = ( x 1 ,..., x n ; ξ 1 ,..., ξ n ) , —coordinates on Π TM p ( ξ a ) = p ( x a )+ 1 , x a ′ = x a ′ ( x a ) → ξ a ′ = ξ a ∂ x a ′ ( dx a ↔ ξ a ) . ∂ x a . ‘ Respectively ( x a , θ b ) = ( x 1 ,..., x n ; θ 1 ,..., θ n ) , —coordinates on Π T ∗ M ∂ x a p ( θ a ) = p ( x a )+ 1 , x a ′ = x a ′ ( x a ) → θ a ′ = θ a ( ∂ a ↔ θ a ) . ∂ x a ′ . ‘ Example Ω ∗ ∋ ω = l a dx a + r ab dx a ∧ dx b ↔ ω ( x , ξ ) = l a ξ a + r ab ξ a ξ b ∈ C (Π TM ) A ∗ ∋ F = X a ∂ a + M ab ∂ a ∧ ∂ b ↔ F ( x , θ ) = X a θ a + M ab θ a θ b ∈ C (Π T ∗ M ) .

Higher order Koszul brackets Poisson manifold and.... Canonical odd Poisson bracket F , G functions on Π T ∗ M F , G multivector fields [ F , G ] Schouten commutator , [ F , G ] odd Poisson bracket , X = X a ∂ a , [ X , F ] = L X F [ X , F ] = [ X a θ a , F ( x , θ )] P = P ab ∂ a ∧ ∂ b , [ P , F ] = d P F , d P F = ( P , F ) = [ P ab θ a θ b , F ( x , θ )] [ F ( x , θ ) , G ( x , θ )] = ∂ F ( x , θ ) ∂ G ( x , θ ) +( − 1 ) p ( F ) ∂ F ( x , θ ) ∂ G ( x , θ ) ∂ x a ∂θ a ∂θ a ∂ x a . odd Poisson bracket Schouten bracket Names are Buttin bracket anti-bracket

Higher order Koszul brackets Poisson manifold and.... Koszul bracket on differential forms C (Π T ∗ M ) ξ a = P ab θ b or dx a = P b ∂ b ϕ ∗ ↑ P : C (Π TM ) From { , } on functions to Koszul bracket on differential forms P ) − 1 ([ ϕ ∗ [ ω , σ ] P = ( ϕ ∗ P ( ω ) , ϕ ∗ P ( σ )]) . [ f , g ] P = 0 , [ f , dg ] P = ( − 1 ) p ( f ) { f , g } P , [ df , dg ] P = ( − 1 ) p ( f ) d ( { f , g } P ) This formula survives the limit if P is degenerate.

Higher order Koszul brackets Poisson manifold and.... Lie algebroid E → M —vector bundle, [[ , ]] —commutator on sections, ρ : E → TM —-anchor � � � [[ s 1 ( x ) , f ( x ) s 2 ( x )]] = f ( x )[[ s 1 ( x ) , s 2 ( x )]]+ ρ ( s 1 ( x )) f ( x ) s 2 ( x ) , Jacobi identity: [[[[ s 1 , s 2 ]] , s 3 ]]+ cyclic permutations = 0 . ik ( x ) e m ( x ) , ρ ( e i ) = ρ µ s ( x ) = s i ( x ) e i ( x ) , [[ e i ( x ) , e k ( x )]] = c m i ∂ µ , � � 1 ρ µ 2 ρ µ s i 1 s k 2 c m ik + s i i ∂ µ s m 2 ( x ) − s i i ∂ µ s m [[ s 1 ( x ) , s 2 ( x )]] = 1 ( x ) e m

Higher order Koszul brackets Poisson manifold and.... Trivial examples of Lie algebroid G G −− Lie algebra , ↓ , where [[ , ]] — usual commutator , ∗ TM ↓ tangent bundle where [[ , ]] — commutator of vector fields , M For TM anchor is identity map

Higher order Koszul brackets Poisson manifold and.... Poisson algebroid ( M , P ) Poisson manifold, ( P = P ab ∂ b ∧ ∂ a , { f , g } = ∂ a fP ab ∂ b g ) T ∗ M ∂ [[ df , dg ]] = d { f , g } , anchor ρ : ρ ( ω a dx a ) = D ω = P ab ω b ↓ , ∂ x a , M � 1 � 2 ω a σ b ∂ c P ab + P ab ω b ∂ a σ c − ( ω ↔ σ ) [[ ω a dx a , σ b dx b ]] = dx x (This is Koszul bracket [ , ] P on 1-forms).

Higher order Koszul brackets Poisson manifold and.... Anchor—morphism of algebroids     T ∗ M TM  → Anchor ρ : ↓ ↓    , M M morphism of algebroid T ∗ M to tangent algebroid. ρ [[ ω , σ ]] = [ ρ ( ω ) , ρ ( σ )] .

Higher order Koszul brackets Poisson manifold and.... One very useful object— Q manifold Definition A pair ( M , Q ) where M is (super)manifold, and Q is odd vector field on it such that Q 2 = 1 2 [ Q , Q ] = 0 is called Q -manifold. Q is called homological vector field.

Higher order Koszul brackets Poisson manifold and.... Lie algebroid and its neighbours Algebroid has diffferent manifestations Π E E ↓ ↓ M M , Π E is Q manifold with E → M is Lie algebroid with ∂ξ m + ξ i ρ µ ∂ ∂ ik , ρ ( e i ) = ρ µ Q = ξ k ξ i c m ∂ [[ e i , e k ]] = c m ik i ∂ x µ ∂ x µ i E ∗ Π E ∗ ↓ ↓ —(even, odd)Poisson manifolds , M M Lie–Poisson bracket: ik u m , { x µ , u i } = ρ µ i , { x µ , x ν } = 0 . { u i , u k } = c m

Higher order Koszul brackets Poisson manifold and.... Neighbours of G → ∗ Π G G ∗ G ↓ ↓ ↓ ∗ ∗ ∗ ∂ , , Q = ξ i ξ k c m [ e i , e k ] = c m { u i , u k } = c m ik e m ik u m ik ∂ξ m � �� � � �� � � �� � structure constants Lie-Poisson bracket homological vector field

Higher order Koszul brackets Poisson manifold and.... Neighbours of tangent algebroid TM → M Π TM ↓ M ∂ Q = ξ m , ∂ x m � �� � homological vector field—de Rham differential d (functions on Π TM ) —differential forms on M ) T ∗ M Π T ∗ M ↓ ↓ M M , canonical symplectic structure canonical odd sympletic structure

Higher order Koszul brackets Poisson manifold and.... Neighbours of Poisson algebroid T ∗ M → M ( M , P ) —Poisson manifold, { x a , x b } = P ab Π T ∗ M T ∗ M ↓ ↓ M M ∂ P ba ∂ + θ a P ab ∂ , Q = θ a θ b Poisson algebroid ∂ x c ∂ x b ∂θ c [[ dx a , dx b ]] = dP ab , ρ ( dx a ) = P ab ∂ b � �� � homological vector field Π TM ↓ M { , } = [ , ] P is Koszul bracket on Π TM .

Higher order Koszul brackets Poisson manifold and.... Π T ∗ M TM ↓ ↓ is in the neighbourhood of tangent algebroid M M T ∗ M Π TM ↓ is in the neighbourhood of Poisson algebroid ↓ M M Π T ∗ M → Π TM � �� � � �� � Odd canonical Poisson bracket Odd Koszul bracket i Linear map ξ a = 1 ∂ P ( x , θ ) ( dx a = P ab ∂ b = P ab θ b , ) 2 ∂θ a