Crossed modules of Hopf algebras: an approach via monoids Gabriella - PowerPoint PPT Presentation

Crossed modules of Hopf algebras: an approach via monoids Gabriella B¨ ohm Wigner Research Centre for Physics, Budapest Quantum groups and their analysis Summer school and workshop at University of Oslo 6 th of August 2019

Crossed module of groups [Whitehead 1941] a • a generalization of a normal subgroup N ⊳ G to non-injective N → G a • a diverse applications a • a equivalent to: ◮ strict 2-group (= category object in the category of groups) a • equivalent to:a ◮ simplicial group whose Moore complex has length 1 a • a concise categorical proof [G Janelidze 2003] of groupoids [Brown, ˙ I¸ cen 2003] of Hopf algebras [Fern´ andez Vilaboa et al. 2006] [Aguiar 1997] [Majid 2012] [Faria Martins 2016] [Gran, Sterck, Vercruysse 2019] [Emir 2019] a • a working definitions a • a bits of the equivalent forms a • a no abstract categorical treatment Unified treatment ? a simplified review of [GB arXiv:1803.03418 1803.04124 1803.04622]

Idea view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects

Idea view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects

� � � � � Monoids in monoidal categories Definition. A monoidal category consists of a • a a category C juxtaposition � C I a • a functors C × C ✶ � − (= ≡ ) a • a coherent natural isomorphisms ( − =) ≡ � − a • a coherent natural isomorphisms I − − I (omitted throughout). Examples: (set , × ), (span , � ), ( vec , ⊗ ), (clg , ⊗ ). Definition. A monoid in a monoidal category consists of m 1 � u 1 � a • a an object A AAA AA A AA m � A u a • a morphisms AA I s.t. commute. 1 m � m 1 u � m � A � A A A m m Examples: ordinary monoids, small categories, algebras, bialgebras. f � A ′ in C s.t. f . m = m ′ . ff and f . u = u ′ . Definition. A monoid morphism is A

� � � � � � � � � � � � � � � � Factorization of monoids g fg � CC f � C m � C is invertible For monoid morphisms A B s.t. q := AB monoid morphism ⇔ monoid morphisms s.t. g g f f ba � DD A C B A C B BA gf � m c c CC D a a b b D D m � ( ∗ ) C m q − 1 � ab � DD AB . commutes. g c � D f � C c � D c �→ C A C B q − 1 � AB ab � DD m � D a � D b C A B �→ multiplicative iff ( ∗ ) commutes

Idea view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects

� � � � � Category objects For a category with pullbacks and a given object B , the category of spans s B A is monoidal via the pullback � B . t Definition. A category object is a monoid in the category of spans: s � A c B A . B A � i t

� � � � � � � � � �� Relative category objects � A In a categorical Hopf algebra B A � B A in [Fern´ andez Vilaboa et al. 2006], the cotensor product � B is not a pullback: s � B t for coalgebra maps A C B C := { � i a i ⊗ c i ∈ A ⊗ C | � 2 ) ⊗ c i = � i a i ⊗ t ( c i i a i 1 ⊗ s ( a i 1 ) ⊗ c i a • A � 2 } a • is a subcoalgebra iff a �→ a 1 ⊗ s ( a 2 ) and c �→ t ( c 1 ) ⊗ c 2 are coalgebra maps a • (then the counits ε induce coalgebra maps in the bottom row) D a c a • a factorization exists j � A ⊗ C ε ⊗ 1 � C A A ⊗ C A � B C 1 ⊗ ε j a • iff d �→ a ( d 1 ) ⊗ c ( d 2 ) is a coalgebra map Idea: only a relative pullback wrt a suitable admissible class of spans.

� � � � � � � � Admissible class of spans Definition. A class S of spans in any category is admissible if g ′ � Y ′ ∈ S ∀ f ′ , g ′ and g � Y ∈ S ⇒ X ′ f ′ g � Y f f X A X A g � Y ∈ S ⇒ X g � Y ∈ S ∀ h . h � A f f h X A A B Examples: a • a The class of all spans in any category is admissible. g � Y | a �→ f ( a 1 ) ⊗ g ( a 2 ) is a coalgebra map } f a • a In coalg the class C := { X A a • a is admissible. g � Y | a �→ f ( a 1 ) ⊗ g ( a 2 ) is a coalgebra map } f a • a In bialg the class B := { X A a • a is admissible.

� � � � � � � � � � � � � � � � � Relative pullback p A p C � C ∈ S s � B t Definition. The S -relative pullback of any A is A C A � B C D c h p C � . s.t. in A � B C C a t p A � • the blue square commutes � B A a c � C ∈ S and the exterior commutes then ∃ ! h • if A D s p A g � E p C g � E ∈ S ⇒ A � g � E ∈ S f f f • A & C A � B C D A � B C D B C D g p A � A g p C � C ∈ S ⇒ E g f � A � f � A � f � A � • E D B C & E D B C D B C ∈ S Examples. • If S = { all spans } then S -relative pullback=pullback. • If A s � B t are bialgebra maps s.t. a �→ a 1 ⊗ s ( a 2 ), c �→ t ( c 1 ) ⊗ c 2 are C ( ε ⊗ 1) . j � A � B C C coalgebra maps then is a B -relative pullback. t (1 ⊗ ε ) . j � � B A s

� � � � � � � Relative category A s � B , Theorem. Let S be an admissible class of spans in a category s.t. if A t B C C ∈ S , then there exists the S -relative pullback A � B C . Then for any B for which B B B ∈ S , there is a monoidal category: t s � B s.t. A s � B , B t a • a objects are the spans B C ∈ S A A C a • a morphisms are the span morphisms a • a monoidal product is the S -relative pullback with the unit B B . B Example. For a cocommutative bialgebra B there is a monoidal category: t s � B of bialgebras s.t. a �→ a 1 ⊗ s ( a 2 ) and a • a objects are the spans B A a • a a �→ t ( a 1 ) ⊗ a 2 are coalgebra maps a • a morphisms are the maps of bialgebra spans a • a monoidal product is the cotensor product over B with the unit B B B . Definition. For S and B as in the theorem, an S -relative category — with object of objects B — is a monoid in the above monoidal category: s � A c B A � B A . i t

Idea view groups groupoids Hopf algebras as distinguished monoids categories bialgebras i.e. distinguished monoids in the category of sets spans coalgebras and apply the factorization theory of monoids to relate relative category objects ← → crossed modules ← → simplicial objects

� � � � Relative category in the category of bialgebras In bialg, a B -relative category is given by bialgebra maps B A = { � i a i ⊗ c i ∈ A ⊗ C | � 2 ) ⊗ c i = � s i a i ⊗ t ( c i c i � A i a i 1 ⊗ s ( a i 1 ) ⊗ c i B A � 2 } t s.t. a �→ a 1 ⊗ s ( a 2 ) & a �→ t ( a 1 ) ⊗ a 2 are coalgebra maps ( ⇒ B is cocommutative) s.t. i and c are maps of spans s.t. c is associative with the unit i . � j � A i For A � B I = { y ∈ A | y 1 ⊗ s ( y 2 ) = y ⊗ 1 } B , j ⊗ i � A ⊗ A m � A , q = ( A � B I ) ⊗ B y ⊗ b �→ yi ( b ) has the inverse a �→ a 1 i ( z ( s ( a 2 ))) ⊗ s ( a 3 ) whenever B is a Hopf algebra with antipode z .

� � � � � Split epimorphisms versus actions [Radford 1985] split epimorphism monoid and comonoid in mod( B ) s c ⊲ � Y ) s.t. � A ⇔ ( Y , B ⊗ Y B A � B A i t ( bb ′ ) ⊲ y = b ⊲ ( b ′ ⊲ y ) 1 ⊲ y = 1 ← b ⊲ ( yy ′ ) = ( b 1 ⊲ y )( b 2 ⊲ y ′ ) b ⊲ 1 = ε ( b )1 Hopf algebra − ( b ⊲ y ) 1 ⊗ ( b ⊲ y ) 2 = b 1 ⊲ y 1 ⊗ b 2 ⊲ y 2 ε ( b ⊲ y ) = ε ( b ) ε ( y ) q − 1 � ( A � s i ⊗ j � A ⊗ A m � A 1 ⊗ ε � A � A � B �→ ( A � B I , B ⊗ ( A � B I ) B I ) ⊗ B B I ) i b ⊗ y �→ i ( b 1 ) yi ( z ( b 2 )) ε ⊗ 1 ⊲ � Y ) B 1 ⊗− � Y ⊗ B =: A ( Y , B ⊗ Y �→ a �→ a 1 ⊗ s ( a 2 ) is a coalgebra map ⇔ B is cocommutative.

� � � � � � � � � � � � � � Reflexive graphs versus pre-crossed modules s c � A . Hopf algebra → B A � A i B t bialgebra map ⇔ bialgebra maps s.t. j j 1 ⊗ k � B ⊗ B i i A � B I A B A � B I A B B ⊗ ( A � B I ) m i ⊗ j � t t A ⊗ A B k k B B m � A m q − 1 B I ) ⊗ B k ⊗ 1 � B ⊗ B ( A � . commutes; i.e. . k ( b 1 ⊲ y ) b 2 = bk ( y ) 1 st Peiffer condition . a �→ t ( a 1 ) ⊗ a 2 is a coalgebra map ⇔ y �→ k ( y 1 ) ⊗ y 2 is a coalgebra map