Loop of formal diffeomorphisms A. Frabetti (Lyon, France) based on a work in progress with Ivan P. Shestakov (Sao Paulo, Brasil) Potsdam, 8–12 February, 2016
Motivation: from renormalization Hopf algebras to series ‚ Diffeomorphism groups fix structure in geometry and physics.
Motivation: from renormalization Hopf algebras to series ‚ Diffeomorphism groups fix structure in geometry and physics. ‚ Taylor expansion gives formal diffeomorphisms , ok for perturbations. They form proalgebraic groups , represented by commutative Hopf algebras on infinitely many generators.
Motivation: from renormalization Hopf algebras to series ‚ Diffeomorphism groups fix structure in geometry and physics. ‚ Taylor expansion gives formal diffeomorphisms , ok for perturbations. They form proalgebraic groups , represented by commutative Hopf algebras on infinitely many generators. ‚ In pQFT, ren. Hopf algebras do represent groups of formal series on the coupling constants [Connes-Kreimer 1998, Pinter 2001, Keller 2010]
Motivation: from renormalization Hopf algebras to series ‚ Diffeomorphism groups fix structure in geometry and physics. ‚ Taylor expansion gives formal diffeomorphisms , ok for perturbations. They form proalgebraic groups , represented by commutative Hopf algebras on infinitely many generators. ‚ In pQFT, ren. Hopf algebras do represent groups of formal series on the coupling constants [Connes-Kreimer 1998, Pinter 2001, Keller 2010] ‚ Renormalization Hopf algebras: - are right-sided combinatorial Hopf alg [Loday-Ronco 2008, Brouder-AF-Menous 2011] - are all related to operads and produce P -expanded series [Chapoton 2003, van der Laan 2003, AF 2008] - admit non-commutative lifts [Brouder-AF 2000, 2006, Foissy 2001]
Motivation: from renormalization Hopf algebras to series ‚ Diffeomorphism groups fix structure in geometry and physics. ‚ Taylor expansion gives formal diffeomorphisms , ok for perturbations. They form proalgebraic groups , represented by commutative Hopf algebras on infinitely many generators. ‚ In pQFT, ren. Hopf algebras do represent groups of formal series on the coupling constants [Connes-Kreimer 1998, Pinter 2001, Keller 2010] ‚ Renormalization Hopf algebras: - are right-sided combinatorial Hopf alg [Loday-Ronco 2008, Brouder-AF-Menous 2011] - are all related to operads and produce P -expanded series [Chapoton 2003, van der Laan 2003, AF 2008] - admit non-commutative lifts [Brouder-AF 2000, 2006, Foissy 2001] ‚ Puzzling situation: - Series with coefficients in a non-comm. algebra A do appear in physics, but their commutative representative Hopf algebra is not functorial in A cf. [Van Suijlekom 2007] for QED . - These series are related to some non-commutative Hopf algebras which are functorial in A .
Aim of the talk How are series related to non-commutative Hopf algebras?
Aim of the talk How are series related to non-commutative Hopf algebras? ‚ Toy model: the set of formal diffeomorphisms in one variable ! a n λ n ` 1 | a n P A ) ÿ Diff p A q “ a p λ q “ λ ` n ě 1 ` ˘ with composition law p a ˝ b qp λ q “ a b p λ q and unit e p λ q “ λ , when A is a unital associative algebra, but not commutative .
Aim of the talk How are series related to non-commutative Hopf algebras? ‚ Toy model: the set of formal diffeomorphisms in one variable ! a n λ n ` 1 | a n P A ) ÿ Diff p A q “ a p λ q “ λ ` n ě 1 ` ˘ with composition law p a ˝ b qp λ q “ a b p λ q and unit e p λ q “ λ , when A is a unital associative algebra, but not commutative . Examples of non-commutative coefficients A : M 4 p C q matrix algebra cf. QED renormalization [Brouder-AF-Krattenthaler 2001, 2006] T p E q tensor algebra cf. renormalization functor [Brouder-Schmitt 2002] and work in progress on bundles with Brouder and Dang L p H q linear operators on a Hilbert space
Aim of the talk How are series related to non-commutative Hopf algebras? ‚ Toy model: the set of formal diffeomorphisms in one variable ! a n λ n ` 1 | a n P A ) ÿ Diff p A q “ a p λ q “ λ ` n ě 1 ` ˘ with composition law p a ˝ b qp λ q “ a b p λ q and unit e p λ q “ λ , when A is a unital associative algebra, but not commutative . Examples of non-commutative coefficients A : M 4 p C q matrix algebra cf. QED renormalization [Brouder-AF-Krattenthaler 2001, 2006] T p E q tensor algebra cf. renormalization functor [Brouder-Schmitt 2002] and work in progress on bundles with Brouder and Dang L p H q linear operators on a Hilbert space ‚ Two problems: 1) define (pro)algebraic groups on non-commutative algebras 2) modify because Diff p A q is not a group !
Lie and (pro)algebraic groups on commutative algebras Group Function algebra G Lie group R r G s “ O p G q representations or (pro)algebraic Hopf com G p A q – Hom uCom p R r G s , A q dense in C 8 p G q algebraic group convolution group
Lie and (pro)algebraic groups on commutative algebras Group Function algebra G Lie group R r G s “ O p G q representations or (pro)algebraic Hopf com G p A q – Hom uCom p R r G s , A q dense in C 8 p G q algebraic group convolution group group-like infinitesimal elements Hopf algebra structure distributions duality supported at e Lie algebra Enveloping algebra algebra ext. U g – R r G s ˚ g “ T e G – Prim U g Hopf cocom (not com) g A “ g b A primitives U g A – U g b A uAs Ñ Lie : A ÞÑ A L Hom Lie p g , A L q – Hom uAs p U g , A q r a , b s “ ab ´ ba adjoint functors
Details on convolution groups and functorial Lie algebras Let A be a commutative algebra and H be a commutative Hopf algebra. Denote: multiplication m , unit u , coproduct ∆, counit ε and antipode S .
Details on convolution groups and functorial Lie algebras Let A be a commutative algebra and H be a commutative Hopf algebra. Denote: multiplication m , unit u , coproduct ∆, counit ε and antipode S . ‚ The set uCom p H , A q forms a group with Hom convolution α ˚ β “ m A p α b β q ∆ H unit e “ u A ε H α ´ 1 “ α S H inverse
Details on convolution groups and functorial Lie algebras Let A be a commutative algebra and H be a commutative Hopf algebra. Denote: multiplication m , unit u , coproduct ∆, counit ε and antipode S . ‚ The set uCom p H , A q forms a group with Hom convolution α ˚ β “ m A p α b β q ∆ H unit e “ u A ε H α ´ 1 “ α S H inverse ‚ Let G be a (pro)algebraic group represented by the Hopf algebra R r G s , and let x 1 , x 2 , ... be generators of R r G s (coordinate functions on G ). Then the isomorphism G p A q – Hom uCom p R r G s , A q is given by g ÞÝ Ñ α g : R r G s Ñ A x n ÞÑ α g p x n q “ x n p g q .
Details on convolution groups and functorial Lie algebras Let A be a commutative algebra and H be a commutative Hopf algebra. Denote: multiplication m , unit u , coproduct ∆, counit ε and antipode S . ‚ The set uCom p H , A q forms a group with Hom convolution α ˚ β “ m A p α b β q ∆ H unit e “ u A ε H α ´ 1 “ α S H inverse ‚ Let G be a (pro)algebraic group represented by the Hopf algebra R r G s , and let x 1 , x 2 , ... be generators of R r G s (coordinate functions on G ). Then the isomorphism G p A q – Hom uCom p R r G s , A q is given by g ÞÝ Ñ α g : R r G s Ñ A x n ÞÑ α g p x n q “ x n p g q . ‚ Let g be a Lie algebra with bracket r , s . Then g A “ g b A is also a Lie algebra with bracket r x b a , y b b s “ r x , y s b ab , and U g A – U g b A .
Convolution groups on non-commutative algebras Let A and H be unital associative algebras (not nec. commutative).
Convolution groups on non-commutative algebras Let A and H be unital associative algebras (not nec. commutative). ‚ Even if H is a Hopf algebra, the convolution α ˚ β “ m A p α b β q ∆ H is not well defined on Hom uAs p H , A q , because it is not an algebra morphism.
Convolution groups on non-commutative algebras Let A and H be unital associative algebras (not nec. commutative). ‚ Even if H is a Hopf algebra, the convolution α ˚ β “ m A p α b β q ∆ H is not well defined on Hom uAs p H , A q , because it is not an algebra morphism. ∆ f : H Ñ H f H , ‚ Solve requiring a modified coproduct where A f B is the free product algebra with concatenation a b b b a 1 b b 1 b ¨ ¨ ¨ instead of p aa 1 ¨ ¨ ¨ q b p bb 1 ¨ ¨ ¨ q as in A b B . Then m A : A b A Ñ A induces an algebra morphism m f A : A f A Ñ A , can define the convolution α ˚ β “ m f A p α f β q ∆ f H and get a group [Zhang 1991, Bergman-Hausknecht 1996] .
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