On the category of cocommutative Hopf algebras Florence Sterck - PowerPoint PPT Presentation

On the category of cocommutative Hopf algebras Florence Sterck Joint work with Marino Gran and Joost Vercruysse Université catholique de Louvain and Université Libre de Bruxelles 8 July 2019 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 1 / 26

Overview Hopf algebras 1 Semi-abelian 2 Crossed modules 3 Commutator 4 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 2 / 26

Hopf algebras Hopf algebra A Hopf algebra H over a field K is given by An algebra ( H , m : H ⊗ H → H , u : K → H ) 1 id H ⊗ m H ⊗ H H ⊗ H ⊗ H H ⊗ H u ⊗ id H m id H ⊗ u m ⊗ id H m K ⊗ H H H ⊗ K H ⊗ H H ∼ ∼ m = = A coalgebra ( H , ∆ : H → H ⊗ H , ǫ : H → K ) 2 id H ⊗ ∆ H ⊗ H ⊗ H H ⊗ H H ⊗ H ǫ ⊗ id H id H ⊗ ǫ ∆ ⊗ id H ∆ ∆ H ⊗ H K ⊗ H H ⊗ K H H ∼ ∼ ∆ = = We use Sweedler’s notation, ∆( x ) = x 1 ⊗ x 2 . Some conditions of compatibility 3 An antipode S : H → H 4 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 3 / 26

Hopf algebras Hopf algebra A Hopf algebra H over a field K is given by An algebra m : H ⊗ H → H , u : K → H 1 A coalgebra ∆ : H → H ⊗ H , ǫ : H → K 2 Some conditions of compatibility, 3 ∆ ⊗ ∆ 1 ⊗ σ ⊗ 1 H ⊗ H H ⊗ H ⊗ H ⊗ H H ⊗ H ⊗ H ⊗ H m m ⊗ m ∆ H ⊗ H H u ⊗ u m K K H ⊗ H H K H ⊗ H u u ǫ ⊗ ǫ ǫ ǫ ∆ H K H where σ ( x ⊗ y ) = y ⊗ x . ( H , m , u , ∆ , ǫ ) is a bialgebra . An antipode S : H → H 4 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 4 / 26

Hopf algebras Hopf algebra A Hopf algebra H over a field K is given by A algebra m : H ⊗ H → H , u : K → H 1 A coalgebra ∆ : H → H ⊗ H , ǫ : H → K 2 Some conditions of compatibility (bialgebra), 3 An antipode S : H → H 4 ∆ S ⊗ id H m H ⊗ H H ⊗ H H H id H ⊗ S u ǫ K ∆ A Hopf algebra H is called cocommutative if H H ⊗ H where σ ( x ⊗ y ) = y ⊗ x . ∆ σ H ⊗ H In Sweedler’s notation : x 1 ⊗ x 2 = x 2 ⊗ x 1 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 5 / 26

Hopf algebras Examples : Let G be a group, kG = { � g α g g | g ∈ G } the group algebra is a Hopf 1 algebra, ∆( g ) = g ⊗ g , ǫ ( g ) = 1 , S ( g ) = g − 1 Let g be a Lie algebra, U ( g ) is a Hopf algebra with 2 ∆( x ) = 1 ⊗ x + x ⊗ 1 , ǫ ( x ) = 0 , S ( x ) = − x . Hopf K , coc objects : cocommutative Hopf algbras arrows : morphisms of Hopf algebras i.e. morphisms of algebras and coalgebras u f f ⊗ f ∆ H ′ ⊗ H ′ K H H H ′ H ⊗ H H H ⊗ H m m f ⊗ f f u f ǫ ǫ H ′ ⊗ H ′ H H H ′ f H ′ ∆ K Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 6 / 26

Semi-abelian Semi-abelian Definition (Janelidze, Marki, Tholen (2002, JPAA)) A category C is semi-abelian if and only if pointed 1 regular 2 finitely complete 1 regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 protomodular 3 exact 4 binary coproducts 5 Examples : Grp, Lie K , CompGrp, ... Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 7 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed i.e. ∃ a zero object, 0 , such that ∀ X ∈ C , ∃ ! X → 0 and 0 → X 1 In Hopf K , coc , the base field K is the zero object, with ǫ and u . regular 2 finitely complete 1 regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 protomodular 3 exact 4 binary coproducts 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 8 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed 1 regular 2 finitely complete finite products and equalizers 1 A ⊗ B π A π B ψ ( x ) = f ( x 1 ) ⊗ g ( x 2 ). ψ A B ∆( ψ ( x )) = f ( x 1 ) 1 ⊗ g ( x 2 ) 1 ⊗ f ( x 1 ) 1 ⊗ g ( x 2 ) 2 g = f ( x 1 ) ⊗ g ( x 3 ) ⊗ f ( x 2 ) ⊗ g ( x 4 ) f X ( ψ ⊗ ψ ) · ∆ = f ( x 1 ) ⊗ g ( x 2 ) ⊗ f ( x 3 ) ⊗ g ( x 4 ) π A = Id ⊗ ǫ π B = ǫ ⊗ Id The equalizer of f , g : A → B is given by Eq ( f , g ) = { a ∈ A | a 1 ⊗ f ( a 2 ) ⊗ a 3 = a 1 ⊗ g ( a 2 ) ⊗ a 3 } . regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 protomodular 3 exact 4 binary coproducts 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 9 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed 1 regular 2 finitely complete 1 regular epi/mono factorization 2 f In Hopf K , coc , A B regular epimorphisms = surjective f inc morphisms f ( A ) pullback stability of regular epimorphisms 3 protomodular 3 exact 4 binary coproducts 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 10 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed 1 regular 2 finitely complete 1 regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 π C A × B C C To prove it we use a result of Newman, π A there is a bijection between Hopf subalge- bras and left ideals and two-sided coideals A B f protomodular 3 exact 4 binary coproducts 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 11 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed 1 regular 2 finitely complete 1 regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 protomodular 3 s k 0 K A A f v u w Hopf K , coc = Grp(CoAlg K , coc ) s ′ 0 K ′ A ′ A ′ k ′ f ′ exact 4 binary coproducts 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 12 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed 1 regular 2 finitely complete 1 regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 protomodular 3 exact : 4 Since we have pointed, N → H is normal iff h 1 nS ( h 2 ) ∈ N regular and protomodular ∀ h ∈ H , n ∈ N. f surjective f f ( N ) N g 1 f ( n ) S ( g 2 ) = f ( h 1 ) f ( n ) f ( S ( h 2 )) = f ( h 1 nS ( h 2 )) ∈ f ( N ) H G where f ( h ) = g . f binary coproducts 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 13 / 26

Semi-abelian Hopf K , coc is semi-abelian pointed 1 regular 2 finitely complete 1 regular epi/mono factorization 2 pullback stability of regular epimorphisms 3 protomodular 3 exact 4 binary coproducts as in the category of algebras. 5 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 14 / 26

Semi-abelian Theorem (Gran, Sterck, Vercruysse (2019, JPAA)) Hopf K , coc is semi-abelian. Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 15 / 26

Semi-abelian Theorem (Gran, Sterck, Vercruysse (2019, JPAA)) Hopf K , coc is semi-abelian. Consequences : Noether’s isomorphism theorems 1 classical homological lemmas 2 commutator theory 3 categorical notion of action, semi-direct product and crossed modules 4 Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 15 / 26

Semi-abelian Theorem (Gran, Sterck, Vercruysse (2019, JPAA)) Hopf K , coc is semi-abelian. Consequences : Noether’s isomorphism theorems 1 classical homological lemmas 2 commutator theory 3 categorical notion of action, semi-direct product and crossed modules 4 Theorem (Janelidze (2003, GMJ)) If C is a semi-abelian category, then XMod( C ) ∼ = Grpd( C ) Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 15 / 26

Crossed modules Crossed modules of groups Internal groupoids in Grp Crossed modules i s µ : A → B a group morphism, m G 1 × G 0 G 1 G 1 G 0 e A a B -group, B × A → A , such that t µ ( b a ) = b µ ( a ) b − 1 , where s , t , e , i are the "source", µ ( a ) a ′ = aa ′ a − 1 . "target", "identity", "inverse" mor- phisms, and m is the multiplica- tion/composition of "composable" morphisms. Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 16 / 26

Crossed modules Crossed modules of groups Crossed modules Internal groupoids in Grp µ : A → B a group morphism, A a B -group, B × A → A , such that i s m G 1 × G 0 G 1 G 1 G 0 µ ( b a ) = b µ ( a ) b − 1 , e t µ ( a ) a ′ = aa ′ a − 1 . s m ( A ⋊ B ) × B ( A ⋊ B ) A ⋊ B B e t m (( a , b ) , ( a ′ , b ′ )) = ( aa ′ , b ′ ); where s ( a , b ) = b ; t ( a , b ) = µ ( a ) b ; e ( b ) = (1 A , b ) . Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 17 / 26

Crossed modules Crossed modules of groups Crossed modules Internal groupoids in Grp µ : A → B a group morphism, A a B -group, B × A → A , such that i s m G 1 × G 0 G 1 G 1 G 0 µ ( b a ) = b µ ( a ) b − 1 , e t µ ( a ) a ′ = aa ′ a − 1 . µ := t | Ker ( s ) : Ker ( s ) → G 0 ; G 0 × Ker ( s ) → Ker ( s ) : ( g , k ) → e ( g ) ke ( g ) − 1 . Florence Sterck (UCLouvain-ULB) On the category of cocommutative Hopf algebras 8 July 2019 18 / 26