Split extension classifiers in the category of cocommutative Hopf - PowerPoint PPT Presentation

Split extension classifiers in the category of cocommutative Hopf algebras Marino Gran Universit´ e catholique de Louvain joint work with G. Kadjo, F . Sterck and J. Vercruysse Category Theory 2019 University of Edinburgh 13 July 2019

Outline “Abelian” versus “semi-abelian” Cocommutative Hopf algebras Split extension classifiers A description in the case of Hopf algebras

Outline “Abelian” versus “semi-abelian” Cocommutative Hopf algebras Split extension classifiers A description in the case of Hopf algebras

� � � “Abelian” versus “semi-abelian” Definition A category C is abelian if ◮ C has a 0-object ◮ C has finite products ◮ any arrow f in C has a factorisation f = i ◦ p f X Y p i � � I where p is a normal epi and i is a normal mono .

� � � Ab is the typical example of abelian category : ◮ Ab has a 0-object : the trivial group { 0 } ◮ Ab has finite products ◮ any homomorphism f in Ab has a factorisation f = i ◦ p f X Y p i � � f ( X ) where p is a surjective homomorphism (= normal epi) and i is an inclusion as a normal subgroup (= normal mono).

� � � Grp is not abelian : ◮ Grp has a 0-object : the trivial group ◮ Grp has finite products ◮ Problem : an arrow f in Grp does not have a factorisation f = i ◦ p f X Y p i � � f ( X ) with p a surjective homomorphism and i an inclusion as a normal subgroup.

Question : is there a list of simple axioms to develop a unified treatment of the categories Grp, Rng, Lie K ,...?

Question : is there a list of simple axioms to develop a unified treatment of the categories Grp, Rng, Lie K ,...? S. Mac Lane, Duality for groups, Bull. Amer. Math. Soc. (1950)

Several proposals of “non-abelian contexts” for radical theory : S. A. Amitsur (1954), A.G. Kurosh (1959) non-abelian homological algebra : A. Fr¨ olich (1961), M. Gerstenhaber (1970), G. Orzech (1972) commutator theory : P . Higgins (1956), S.A. Huq (1968), etc.

� � � � � Definition (G. Janelidze, L. M´ arki, W. Tholen, JPAA, 2002) A finitely complete category C is semi-abelian if ◮ C has a 0-object ◮ C has A + B ◮ C is (Barr)-exact ◮ C is (Bourn)-protomodular : k � K � A � B 0 f u v w � K ′ � A ′ � B ′ 0 k ′ f ′ u , w isomorphisms ⇒ v isomorphism.

Examples Grp, Rng, Lie K , XMod (more generally, any variety of Ω -groups)

Examples Grp, Rng, Lie K , XMod (more generally, any variety of Ω -groups) Loop, Grp ( Comp ) , Set op ∗ , Heyt, etc.

Examples Grp, Rng, Lie K , XMod (more generally, any variety of Ω -groups) Loop, Grp ( Comp ) , Set op ∗ , Heyt, etc. [ C is abelian ] ⇔ [ C and C op are semi-abelian]!

Examples Grp, Rng, Lie K , XMod (more generally, any variety of Ω -groups) Loop, Grp ( Comp ) , Set op ∗ , Heyt, etc. [ C is abelian ] ⇔ [ C and C op are semi-abelian]! Many new connections have been discovered between semi-abelian (co)homology and commutator theory in universal algebra.

Outline “Abelian” versus “semi-abelian” Cocommutative Hopf algebras Split extension classifiers A description in the case of Hopf algebras

� � � � � � � � � � � � � Let K be a field. Bialgebras A K -bialgebra ( A , m , u , ∆ , ǫ ) is both a K -algebra ( A , m , u ) and a K -coalgebra ( A , ∆ , ǫ ) , where m , u , ∆ , ǫ are linear maps such that 1 A ⊗ m � 1 A ⊗ u � u ⊗ 1 A A ⊗ A ⊗ A A ⊗ A A ⊗ K A ⊗ A K ⊗ A m ⊗ 1 A m m r A l A � A A ⊗ A A m and 1 A ⊗ ǫ ǫ ⊗ 1 A � ∆ A A ⊗ A A ⊗ K A ⊗ A K ⊗ A ∆ ∆ ⊗ 1 A ∆ r − 1 l − 1 A A A ⊗ A 1 A ⊗ ∆ � A ⊗ A ⊗ A A commute, and m and u are K -coalgebra morphisms.

� � � A Hopf algebra ( A , m , u , ∆ , ǫ, S ) is a K -bialgebra with an antipode, a linear map S : A → A making the following diagram commute : 1 A ⊗ S � A ⊗ A A ⊗ A S ⊗ 1 A ∆ m � K � A A u ǫ

� � � � � A Hopf algebra ( A , m , u , ∆ , ǫ, S ) is a K -bialgebra with an antipode, a linear map S : A → A making the following diagram commute : 1 A ⊗ S � A ⊗ A A ⊗ A S ⊗ 1 A ∆ m � K � A A u ǫ ( A , m , u , ∆ , ǫ, S ) is cocommutative if the following triangle commutes : A ∆ ∆ tw � A ⊗ A A ⊗ A ∼ = In Sweedler’s notations : ∆( a ) = a 1 ⊗ a 2 = a 2 ⊗ a 1 , for any a ∈ A .

Example Any group G gives the group-algebra � K [ G ] = { α g g | g ∈ G , } , g which becomes a cocommutative Hopf algebra with S ( g ) = g − 1 . ∆( g ) = g ⊗ g , ǫ ( g ) = 1 ,

Example Any group G gives the group-algebra � K [ G ] = { α g g | g ∈ G , } , g which becomes a cocommutative Hopf algebra with S ( g ) = g − 1 . ∆( g ) = g ⊗ g , ǫ ( g ) = 1 , In the category Hopf K , coc of cocommutative Hopf algebras there is the full subcategory GrpHopf K ⊂ Hopf K , coc of group Hopf algebras (= generated by grouplike elements).

Theorem (M. Gran, F. Sterck and J. Vercruysse, JPAA, 2019) The category Hopf K , coc is semi-abelian.

Theorem (M. Gran, F. Sterck and J. Vercruysse, JPAA, 2019) The category Hopf K , coc is semi-abelian. Remark The fact that Hopf K , coc is protomodular follows from Hopf K , coc ∼ = Grp ( Coalg K , coc )

Theorem (M. Gran, F. Sterck and J. Vercruysse, JPAA, 2019) The category Hopf K , coc is semi-abelian. Remark The fact that Hopf K , coc is protomodular follows from Hopf K , coc ∼ = Grp ( Coalg K , coc ) The most difficult part is to prove that Hopf K , coc is a regular category (this was explained by F . Sterck in her talk).

In particular, this result implies Theorem (M. Takeuchi, Manuscr. Math., 1972) The category Hopf comm K , coc is abelian.

In particular, this result implies Theorem (M. Takeuchi, Manuscr. Math., 1972) The category Hopf comm K , coc is abelian. Indeed : Hopf comm K , coc = Ab ( Hopf K , coc ) .

In particular, this result implies Theorem (M. Takeuchi, Manuscr. Math., 1972) The category Hopf comm K , coc is abelian. Indeed : Hopf comm K , coc = Ab ( Hopf K , coc ) . A ∈ Hopf K , coc is abelian ⇔ ∆: A → A ⊗ A is a normal mono

In particular, this result implies Theorem (M. Takeuchi, Manuscr. Math., 1972) The category Hopf comm K , coc is abelian. Indeed : Hopf comm K , coc = Ab ( Hopf K , coc ) . A ∈ Hopf K , coc is abelian ⇔ ∆: A → A ⊗ A is a normal mono ⇔ A is commutative : ab = ba ⇔ A ∈ Hopf comm K , coc

� There is an adjunction ab Hopf comm K , coc = Ab ( Hopf K , coc ) � Hopf K , coc ⊥ U

� � There is an adjunction ab Hopf comm K , coc = Ab ( Hopf K , coc ) � Hopf K , coc ⊥ U In general, if C is semi-abelian, Ab ( C ) is abelian ab Ab ( C ) � C ⊥ U with unit of the adjunction η A � � A A [ A , A ]

� � � Commutators For general normal Hopf subalgebras M , N of A ∈ Hopf K , coc � A M N one can compute the categorical commutator : [ M , N ] Huq = �{ m 1 n 1 S ( m 2 ) S ( n 2 ) | m ∈ M , n ∈ N }� A (where ∆( m ) = m 1 ⊗ m 2 and ∆( n ) = n 1 ⊗ n 2 ).

� � � � � � � In Hopf K , coc the condition [ M , N ] Huq = 0 is equivalent to the existence of a (unique) morphism p : M ⊗ N → A making the diagram M ⊗ N ( 1 M , 0 ) ( 0 , 1 N ) M N p A commute, where p ( m ⊗ n ) = mn , for any m ⊗ n ∈ M ⊗ N .

� � � � � � � In Hopf K , coc the condition [ M , N ] Huq = 0 is equivalent to the existence of a (unique) morphism p : M ⊗ N → A making the diagram M ⊗ N ( 1 M , 0 ) ( 0 , 1 N ) M N p A commute, where p ( m ⊗ n ) = mn , for any m ⊗ n ∈ M ⊗ N . This allows one to apply methods of commutator theory to Hopf K , coc .

Outline “Abelian” versus “semi-abelian” Cocommutative Hopf algebras Split extension classifiers A description in the case of Hopf algebras

� Split extensions In a semi-abelian category C a split extension is a diagram s � X � A � 0 κ 0 � B (1) p where κ = Ker ( p ) and p ◦ s = 1 B .

� Split extensions In a semi-abelian category C a split extension is a diagram s � X � A � 0 κ 0 � B (1) p where κ = Ker ( p ) and p ◦ s = 1 B . Example In the category Grp of groups each split extension (1) is determined by a morphism χ : B → Aut ( X ) where the action of B on X is given by χ ( b )( x ) = s ( b ) xs ( b ) − 1 for any b ∈ B and x ∈ X .

� Given any X ∈ Grp there is a universal split extension i 2 i 1 � X � X ⋊ Aut ( X ) � 0 0 � Aut ( X ) p 2 (with kernel X ) with the following universal property :

� � � � � � � Given any X ∈ Grp there is a universal split extension i 2 i 1 � X � X ⋊ Aut ( X ) � 0 0 � Aut ( X ) p 2 (with kernel X ) with the following universal property : for any other split extension, there is a unique morphism s � X � A κ 0 B 0 p ∃ ! χ ∃ ! χ i 2 � X � X ⋊ Aut ( X ) � 0 . 0 � Aut ( X ) i 1 p 2

� � � � � � Given X ∈ Grp, the group Aut ( X ) is the split extension classifier : s � X � A κ 0 B 0 p ∃ ! χ ∃ ! χ i 2 � X � X ⋊ Aut ( X ) � 0 . 0 � Aut ( X ) i 1 p 2 The category Grp has representable actions in the sense of F . Borceux, G. Janelidze, G.M. Kelly, Comment. Math. Univ. Carolin. 2005.