Cointeracting bialgebras Loc Foissy October 2020 Wien Loc Foissy - PowerPoint PPT Presentation

Cointeracting bialgebras Theoretical consequences Applications Cointeracting bialgebras Loïc Foissy October 2020 – Wien Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs Let G a (proto)-algebraic monoid. The algebra C r G s of polynomial functions on G inherits a coproduct ∆ : C r G s Ý Ñ C r G s b C r G s « C r G ˆ G s such that: @ f P C r G s , @ x , y P G , ∆ p f qp x , y q “ f p xy q . This makes C r G s a bialgebra. It is a Hopf algebra if, and only if, G is a group. Moreover, G is isomorphic to the monoid char p C r G sq of characters of C r G s . Characters of a bialgebra B A character of a bialgebra B is an algebra morphism λ : B Ý Ñ C . The set of characters char p B q is given an associative convolution product: λ ˚ µ “ m C ˝ p λ b µ q ˝ ∆ . Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs Let G and G 1 be two (proto)-algebraic monoids, such that G 1 acts polynomialy on G by monoid endomorphisms (on the right). Then: Interacting bialgebras A “ p C r G s , m A , ∆ q is a bialgebra. 1 B “ p C r G 1 s , m B , δ q is a bialgebra. 2 B coacts on A by a coaction ρ : A Ý Ñ A b B . 3 A is a bialgebra in the category of B -comodules. 4 Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs In other words, for any f , g P A : p Id b ρ q ˝ ρ “ p ∆ b Id q ˝ ρ, ρ p fg q “ ρ p f q ρ p g q , ρ p 1 A q “ 1 A b 1 B , p ε A b Id B q ˝ ρ p f q “ ε A p f q 1 B , p ∆ b Id q ˝ ρ p f q “ m 1 , 3 , 24 ˝ p ρ b ρ q ˝ ∆ p f q . where A b 4 A b 3 " Ý Ñ m 1 , 3 , 24 : a 1 b a 2 b a 3 b a 4 Ý Ñ a 1 b a 3 b a 2 a 4 . Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs The algebra C r X s The group p C ˚ , ˆq acts on p C , `q by group automorphisms. A “ p C r X s , m , ∆ q with ∆ p X q “ X b 1 ` 1 b X , is a Hopf algebra. B “ p C r X , X ´ 1 s , m , δ q with δ p X q “ X b X , is a Hopf algebra. ρ p X q “ X b X defines a coaction of B on A , and A is a bialgebra in the category of B -comodules. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs The algebra C r X s The monoid p C , ˆq acts on p C , `q by group automorphisms. A “ p C r X s , m , ∆ q with ∆ p X q “ X b 1 ` 1 b X , is a Hopf algebra. B “ p C r X , X ´ 1 s , m , δ q with δ p X q “ X b X , is a bialgebra. ρ p X q “ X b X defines a coaction of B on A , and A is a bialgebra in the category of B -comodules. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs From now, we shall consider only examples where A “ B as algebras: we obtain objects p A , m , ∆ , δ q , with one product and two coproducts. The coaction ρ and the coproduct δ are equal. These objects will be called double bialgebras. The algebra C r X s p A , m q “ p B , m q “ p C r X s , m q , where m is the usual product of polynomials, and the coproducts are given by: ∆ p X q “ X b 1 ` 1 b X , δ p X q “ X b X . Then p C r X s , m , ∆ , δ q is a double bialgebra. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs From now, we shall consider only examples where A “ B as algebras: we obtain objects p A , m , ∆ , δ q , with one product and two coproducts. The coaction ρ and the coproduct δ are equal. These objects will be called double bialgebras. The algebra C r X s p A , m q “ p B , m q “ p C r X s , m q , where m is the usual product of polynomials, and the coproducts are given by: n ˆ n ˙ X k b X n ´ k , ∆ p X n q “ ÿ k k “ 0 δ p X n q “ X n b X n . Then p C r X s , m , ∆ , δ q is a double bialgebra. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs The Connes-Kreimer Hopf algebra of trees is based on rooted forests: 1 , , , , , , , , , , , , , , , , . . . As algebras, A “ B “ H CK with the disjoint union product. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs The first coproduct ∆ is given by admissible cuts (Connes-Kreimer coproduct). Example ∆ p q “ b 1 ` 1 b ` 2 b ` b , ∆ p q “ b 1 ` 1 b ` b ` b . Counit: # 1 if F “ 1 , ε p F q “ 0 otherwise. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs The second coproduct δ is given by contraction-extraction. Example δ p q “ b ` 2 b ` b , δ p q “ b ` 2 b ` b . Its counit is: # 1 if F “ . . . , ε 1 p F q “ 0 otherwise. (Calaque, Ebrahimi-Fard, Manchon, 2008). Then p H CK , m , ∆ , δ q is a double bialgebra. This construction can be extended to finite posets or to finite topologies. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs H G has for basis the set of graphs: q , q q ; _ , q qq _ , q q qq 1 ; q ; q q q , q q q ; q , q ä , q q ä , q q ä , q q q q q q q q q q , q q q q q q , q q q q q q å , q q ä _ q , _ q , q q q , q q q q . q q q q The product is the disjoint union. The unit is the empty graph 1. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs (Schmitt, 1994). The first coproduct ∆ is defined by ÿ ∆ p G q “ G | I b G | J . V p G q“ I \ J Examples ∆ p q q “ q b 1 ` 1 b q , q q “ q q b 1 ` 1 b q q ` 2 q b q , ∆ p q q b q ` 3 q b q q , ∆ p q q _ q “ q _ b 1 ` 1 b q qq _ ` 3 q q q q q b q ` q q b q ` 2 q b q q ` q b q q . ∆ p q q _ q “ q _ b 1 ` 1 b q q q _ ` 2 q q q q Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs (Schmitt, 1994– Manchon, 2011). The second coproduct δ is defined by ÿ δ p G q “ p G { „q b p G |„q , „ where: „ runs in the set of equivalences on V p G q which classes are connected. G | „ is the union of the equivalence classes of „ . G { „ is obtained by the contraction of the equivalence classes of „ . Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs Examples δ p q q “ q b q , q q “ q b q q ` q q b q q , δ p q q b q q q ` δ p qq _ q “ q b q _ ` 3 q q q q _ b q q q , q qq q b q q q ` δ p q q _ q “ q b q _ ` 2 q q q q _ b q q q . q qq Its counit is given by: # 1 if G has no edge , ε 1 p G q “ 0 otherwise . Then p H G , m , ∆ , δ q is a double bialgebra. Loïc Foissy Cointeracting bialgebras

Definition Cointeracting bialgebras A first example: the polynomial algebra C r X s Theoretical consequences Rooted trees Applications Graphs Questions Theoretical consequences? Examples and applications? Loïc Foissy Cointeracting bialgebras

Cointeracting bialgebras Actions and morphisms Theoretical consequences Antipode Applications We consider a double bialgebra p A , m , ∆ , δ q . Proposition Let B be a bialgebra and let E A Ñ B be the set of bialgebra morphisms from A to B . The monoid of characters M A of p A , m , δ q acts on E A Ñ B : φ Ð λ “ p φ b λ q ˝ δ. Loïc Foissy Cointeracting bialgebras

Cointeracting bialgebras Actions and morphisms Theoretical consequences Antipode Applications If p A , ∆ q is a connected coalgebra: Theorem There exists a unique φ 1 : A Ý Ñ C r X s , compatible with 1 both bialgebraic structure. The following maps are bijections, inverse one from the 2 other: " M A $ E A Ñ C r X s Ý Ñ M A Ý Ñ E A Ñ C r X s & ε 1 ˝ φ Ý Ñ φ Ý Ñ φ 1 Ð λ, λ “ φ p¨qp 1 q . % Loïc Foissy Cointeracting bialgebras

Cointeracting bialgebras Actions and morphisms Theoretical consequences Antipode Applications Let us apply this result on the double bialgebra of forests. is primitive, φ 1 p q is primitive, so φ 1 p q “ λ X . As As φ 1 p qp 1 q “ ε 1 p q “ 1, φ 1 p q “ X . ∆ p q “ b 1 ` 1 b ` b , ∆ p φ 1 p qq “ φ 1 p q b 1 ` 1 b φ 1 p q ` X b X , so φ 1 p q “ X 2 2 ` λ X . As φ 1 p qp 1 q “ ε 1 p q “ 0, we obtain λ “ ´ 1 2. φ 1 p q “ X p X ´ 1 q . 2 Loïc Foissy Cointeracting bialgebras