Applications of non-associative Hopf algebras to Loop theory Jos e - PowerPoint PPT Presentation

Applications of non-associative Hopf algebras to Loop theory Jos´ e M. P´ erez-Izquierdo Dpto. Matem´ aticas y Computaci´ on Universidad de La Rioja, Spain Loops 2019 Conference Budapest University of Technology and Economics, Hungary

Linearizing ◮ O. Loos: Symmetric spaces I: General theory (1969). Idea of non-associative Hopf algebras. ◮ J.-P. Serre: Lie algebras and Lie groups (1964). He discuss local Lie groups by using distributions with support at e .

Distributions with support at a point Let ( Q , e ) be a smooth pointed manifold and ( U , ( x 1 , . . . , x n )) a coordinate neighborhood at e and the vector space � R � ∂ e 1 1 · · · ∂ e n D e ( Q ) = n | e � e 1 ,..., e n ≥ 0 with ∂ 0 1 · · · ∂ 0 n | e ( f ) := δ e := f ( e ). Given ϕ : Q 1 → Q 2 , e 2 = ϕ ( e 1 ) a smooth map, consider the linear map ϕ ′ : D e 1 ( Q 1 ) → D e 2 ( Q 2 ) ϕ ′ ( µ ): g �→ µ ( g ◦ ϕ ) µ �→ We obtain a functor from the category of smooth pointed manifolds to the category of vector spaces ( linearization )

Manifolds Distributions T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Distributions Diagonal Comultiplication D ( e , e ) ( Q × Q ) ∼ Q → Q × Q ∆: D e ( Q ) → = D e ( Q ) ⊗ D e ( Q ) � x �→ ( x , x ) µ �→ µ (1) ⊗ µ (2) T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Distributions Diagonal Comultiplication D ( e , e ) ( Q × Q ) ∼ Q → Q × Q ∆: D e ( Q ) → = D e ( Q ) ⊗ D e ( Q ) � x �→ ( x , x ) µ �→ µ (1) ⊗ µ (2) Constant Counit Q → e D e ( e ) ∼ ǫ : D e ( Q ) → = R x �→ e µ �→ ǫ ( µ ) T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Distributions Diagonal Comultiplication D ( e , e ) ( Q × Q ) ∼ Q → Q × Q ∆: D e ( Q ) → = D e ( Q ) ⊗ D e ( Q ) � x �→ ( x , x ) µ �→ µ (1) ⊗ µ (2) Constant Counit Q → e D e ( e ) ∼ ǫ : D e ( Q ) → = R x �→ e µ �→ ǫ ( µ ) Product Product D e ( Q ) ⊗ D e ( Q ) → D e ( Q ) Q × Q → Q ( x , y ) �→ xy µ ⊗ ν �→ µν T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Distributions Diagonal Comultiplication D ( e , e ) ( Q × Q ) ∼ Q → Q × Q ∆: D e ( Q ) → = D e ( Q ) ⊗ D e ( Q ) � x �→ ( x , x ) µ �→ µ (1) ⊗ µ (2) Constant Counit Q → e D e ( e ) ∼ ǫ : D e ( Q ) → = R x �→ e µ �→ ǫ ( µ ) Product Product D e ( Q ) ⊗ D e ( Q ) → D e ( Q ) Q × Q → Q ( x , y ) �→ xy µ ⊗ ν �→ µν Inverse Antipode Q → Q D e ( Q ) → D e ( Q ) x − 1 x �→ µ �→ S ( µ ) T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Distributions Diagonal Comultiplication D ( e , e ) ( Q × Q ) ∼ Q → Q × Q ∆: D e ( Q ) → = D e ( Q ) ⊗ D e ( Q ) � x �→ ( x , x ) µ �→ µ (1) ⊗ µ (2) Constant Counit Q → e D e ( e ) ∼ ǫ : D e ( Q ) → = R x �→ e µ �→ ǫ ( µ ) Product Product D e ( Q ) ⊗ D e ( Q ) → D e ( Q ) Q × Q → Q ( x , y ) �→ xy µ ⊗ ν �→ µν Division Division D e ( Q ) ⊗ D e ( Q ) → D e ( Q ) Q × Q → Q µ ⊗ ν �→ µ \ ν ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Definition (Non-associative Hopf alg.?) Diagonal Q → Q × Q x �→ ( x , x ) Constant Q → e x �→ e Product Q × Q → Q ( x , y ) �→ xy Division Q × Q → Q ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Definition (Non-associative Hopf alg.?) Diagonal 1. (Bialgebra) Unital algebra + ∆ and ǫ homomorphisms of unital algebras Q → Q × Q x �→ ( x , x ) Constant Q → e x �→ e Product To prove it for D e ( Q ) Q × Q → Q linearize. . . ( x , y ) �→ xy ( x , y ) → xy Division ↓ ↓ ( x , x , y , y ) → ( xy , xy ) Q × Q → Q ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Definition (Non-associative Hopf alg.?) Diagonal 1. (Bialgebra) Unital algebra + ∆ and ǫ homomorphisms of unital algebras 2. � µ (1) \ ( µ (2) ν ) = ǫ ( µ ) ν = � µ (1) ( µ (2) \ ν ) Q → Q × Q x �→ ( x , x ) Constant Q → e x �→ e Product To prove it for D e ( Q ) Q × Q → Q linearize. . . ( x , y ) �→ xy ( x , y ) → ( x , x , y ) Division ↓ ↓ y ∼ x \ ( xy ) Q × Q → Q ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Definition (Non-associative Hopf alg.?) Diagonal 1. (Bialgebra) Unital algebra + ∆ and ǫ homomorphisms of unital algebras 2. � µ (1) \ ( µ (2) ν ) = ǫ ( µ ) ν = � µ (1) ( µ (2) \ ν ) Q → Q × Q x �→ ( x , x ) 3. � ( µν (1) ) /ν (2) = ǫ ( ν ) µ = � ( µ/ν (1) ) ν (2) Constant Q → e x �→ e Product To prove it for D e ( Q ) Q × Q → Q linearize. . . ( x , y ) �→ xy ( x , y ) → ( x , y , y ) Division ↓ ↓ x ∼ ( xy ) / y Q × Q → Q ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Definition (Non-associative Hopf alg.?) Diagonal 1. (Bialgebra) Unital algebra + ∆ and ǫ homomorphisms of unital algebras 2. � µ (1) \ ( µ (2) ν ) = ǫ ( µ ) ν = � µ (1) ( µ (2) \ ν ) Q → Q × Q x �→ ( x , x ) 3. � ( µν (1) ) /ν (2) = ǫ ( ν ) µ = � ( µ/ν (1) ) ν (2) Constant ◮ (Coasociativity) (∆ ⊗ I ) ◦ ∆ = ( I ⊗ ∆) ◦ ∆ Q → e x �→ e Product To prove it for D e ( Q ) Q × Q → Q linearize. . . ( x , y ) �→ xy x → ( x , x ) Division ↓ ↓ ( x , x ) → ( x , x , x ) Q × Q → Q ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

Manifolds Definition (Non-associative Hopf alg.?) Diagonal 1. (Bialgebra) Unital algebra + ∆ and ǫ homomorphisms of unital algebras 2. � µ (1) \ ( µ (2) ν ) = ǫ ( µ ) ν = � µ (1) ( µ (2) \ ν ) Q → Q × Q x �→ ( x , x ) 3. � ( µν (1) ) /ν (2) = ǫ ( ν ) µ = � ( µ/ν (1) ) ν (2) Constant ◮ (Coasociativity) (∆ ⊗ I ) ◦ ∆ = ( I ⊗ ∆) ◦ ∆ Q → e ◮ (Cocommutativity) ∆ = ∆ op x �→ e Product To prove it for D e ( Q ) Q × Q → Q linearize. . . ( x , y ) �→ xy x → ( x , x ) Division ↓ ↓ ( x , x ) = ( x , x ) Q × Q → Q ( x , y ) �→ x \ y T e Q = R � ∂ 1 | e , . . . , ∂ n | e � = Prim( D e ( Q )) = { µ | ∆( µ ) = µ ⊗ δ e + δ e ⊗ µ } .

linearizing. . . If Q is a group ( x , y , z ) → ( x , yz ) µ ⊗ ν ⊗ η → µ ⊗ νη ↓ ↓ ↓ ↓ ( xy , z ) → ( xy ) z = x ( yz ) µν ⊗ η → ( µν ) η = µ ( νη ) D e ( Q ) is associative

linearizing. . . If Q is a Moufang loop µ ν η µ ν η a x y a x y = = µ (1) ( ν ( µ (2) η )) (( µ (1) ν ) µ (2) ) η a ( x ( ay )) (( ax ) a ) y Hopf-Moufang

linearizing. . . If Q is. . . More interactions between Hopf like objects and non-associative algebra

Quantum quasigroups and loops The natural objects to study Definition (J.D.H. Smith, 2016) A quantum quasigroup (resp. quantum loop) in a symmetric monoidal category ( V , ⊗ , 1 ) is a bimagma (resp. biunital bimagma) ( A , ∇ , ∆) in V for which the left composite morphism ∆ ⊗ 1 A 1 A ⊗∇ A ⊗ A − − − → A ⊗ A ⊗ A − − − − → A ⊗ A and its dual composite 1 A ⊗ ∆ ∇⊗ 1 A A ⊗ A − − − → A ⊗ A ⊗ A − − − − → A ⊗ A are both invertible. This definition is selfdual

Tangent space = Primitive elements

Tangent algebras of certain local smooth loops Lie’s theorems The categories of local Lie groups and f.d. real Lie algebras are equivalent Malcev, 1955; Kuzmin 1970 The categories of local smooth Moufang loops and f.d. real Malcev algebras are equivalent It was made global by Kerdman (1979) and P. Nagy (1993) Mikheev and Sabinin, 1982 The categories of local smooth Bol loops and f.d. real Bol algebras are equivalent Moufang loop: a ( x ( ay )) = (( ax ) a ) y Bol loop: a ( x ( ay )) = ( a ( xa )) y Bol algebra: a vector space with a skew-commutative product [ x , y ] and a trilinear product [ x , y , z ] that satisfy [ a , a , b ] = 0 [ a , b , c ] + [ b , c , a ] + [ c , a , b ] = 0 [ x , y , [ a , b , c ]] = [[ x , y , a ] , b , c ] + [ a , [ x , y , b ] , c ] + [ a , b , [ x , y , c ]] [ a , b , [ c , d ]] = [[ a , b , c ] , d ] + [ c , [ a , b , d ]] + [ c , d , [ a , b ]] + [[ a , b ] , [ c , d ]]

Tangent algebras of general local smooth loops ◮ Kikkawa: On local loops in affine manifolds (1964). ◮ Sabinin: The geometry of loops (1972). Geodesic loops.