2-Crossed Modules of Hopf Algebras 2-Crossed Modules of Hopf Algebras Kadir EMİR Joint work with: João Faria Martins 11.08.2016 Category Theory ’2016 Dalhousie and St Mary’s, Halifax Nova Scotia, Canada
2-Crossed Modules of Hopf Algebras Abstract Abstract In this talk, we ❖ define the Moore complex of any simplicial (cocommutative) Hopf algebra by using Hopf kernels which are defined quite different from the kernels of groups or various well-known algebraic structures, ❖ see that these Hopf kernels only make sense in the case of cocommutativity , ❖ give some applications of the Moore complex such as iterated Peiffer pairings and simplicial decomposition of simplicial cocommutative Hopf algebras, ❖ introduce the notion of 2-crossed modules of (cocommutative) Hopf algebras and continue to talk about its categorical properties such as its relations with simplicial objects, Lie algebras and groups.
2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions Hopf Algebraic Conventions Let H be a Hopf algebra. We will use Sweedler’s notation to denote the coproduct in H : x ′ ⊗ x ′′ where x ∈ H ∆ ( x ) = � ( x ) And therefore a Hopf algebra H is cocommutative if, for each x ∈ H we have: x ′ ⊗ x ′′ = � x ′′ ⊗ x ′ � ( x ) ( x ) The identity of a Hopf algebra is 1 H and the counit map is ǫ : H → � . The identity map is µ : λ ∈ � �→ λ 1 H ∈ H.
2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions Hopf Algebraic Conventions Let H be a Hopf algebra. We will use Sweedler’s notation to denote the coproduct in H : x ′ ⊗ x ′′ where x ∈ H ∆ ( x ) = � ( x ) And therefore a Hopf algebra H is cocommutative if, for each x ∈ H we have: x ′ ⊗ x ′′ = � x ′′ ⊗ x ′ � ( x ) ( x ) The identity of a Hopf algebra is 1 H and the counit map is ǫ : H → � . The identity map is µ : λ ∈ � �→ λ 1 H ∈ H. Definition 1 If I is a bialgebra, we say that I is an H -module algebra if there exist a left action ρ : H ⊗ I → I , denoted by x ⊗ v ∈ H ⊗ I �→ x ◮ ρ v ∈ I , such that, for each x ∈ H and u, v ∈ I . 1 H ◮ ρ v = v and x ◮ ρ 1 I = ǫ ( x )1 I x ◮ ρ ( uv ) = � ( x ′ ◮ ρ u )( x ′′ ◮ ρ v ) ( x )
2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions Hopf Algebraic Conventions Let H be a Hopf algebra. We will use Sweedler’s notation to denote the coproduct in H : x ′ ⊗ x ′′ where x ∈ H ∆ ( x ) = � ( x ) And therefore a Hopf algebra H is cocommutative if, for each x ∈ H we have: x ′ ⊗ x ′′ = � x ′′ ⊗ x ′ � ( x ) ( x ) The identity of a Hopf algebra is 1 H and the counit map is ǫ : H → � . The identity map is µ : λ ∈ � �→ λ 1 H ∈ H. Definition 1 If I is a bialgebra, we say that I is an H -module algebra if there exist a left action ρ : H ⊗ I → I , denoted by x ⊗ v ∈ H ⊗ I �→ x ◮ ρ v ∈ I , such that, for each x ∈ H and u, v ∈ I . 1 H ◮ ρ v = v and x ◮ ρ 1 I = ǫ ( x )1 I x ◮ ρ ( uv ) = � ( x ′ ◮ ρ u )( x ′′ ◮ ρ v ) ( x ) A left action of H on I is said to make I an H -module coalgebra if, for all x ∈ H and v ∈ I , we have: ∆ ( x ◮ ρ v ) = � � ( x ′ ◮ ρ v ′ ) ⊗ ( x ′′ ◮ ρ v ′′ ) ( x ) ( v ) ǫ ( x ◮ ρ v ) = ǫ ( x ) ǫ ( v )
2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions Hopf Algebraic Conventions Definition 2 (Majid, [5]) Let I, H be cocommutative Hopf algebras, where I is an H module algebra and coalgebra, under the the action ρ : H ⊗ I → I . We have (!) a bialgebra , moreover a Hopf algebra I ⊗ ρ H , called the “smash product of H and I , with underlying vector space I ⊗ H , identity 1 I ⊗ 1 H and (for all u, v ∈ I and x, y ∈ H ): � u ( x ′ ◮ ρ v ) � ( u ⊗ ρ x )( v ⊗ ρ y ) = � ⊗ ρ x ′′ y ( x ) ∆ ( u ⊗ ρ x ) = � � ( u ′ ⊗ ρ x ′ ) ⊗ ρ ( u ′′ ⊗ ρ x ′′ ) ( u ) ( x ) S ( u ⊗ ρ x ) = � 1 I ⊗ ρ S ( x ) �� � S ( u ) ⊗ ρ 1 H
2-Crossed Modules of Hopf Algebras Preliminaries Hopf Algebraic Conventions Hopf Algebraic Conventions Definition 2 (Majid, [5]) Let I, H be cocommutative Hopf algebras, where I is an H module algebra and coalgebra, under the the action ρ : H ⊗ I → I . We have (!) a bialgebra , moreover a Hopf algebra I ⊗ ρ H , called the “smash product of H and I , with underlying vector space I ⊗ H , identity 1 I ⊗ 1 H and (for all u, v ∈ I and x, y ∈ H ): � u ( x ′ ◮ ρ v ) � ( u ⊗ ρ x )( v ⊗ ρ y ) = � ⊗ ρ x ′′ y ( x ) ∆ ( u ⊗ ρ x ) = � � ( u ′ ⊗ ρ x ′ ) ⊗ ρ ( u ′′ ⊗ ρ x ′′ ) ( u ) ( x ) S ( u ⊗ ρ x ) = � 1 I ⊗ ρ S ( x ) �� � S ( u ) ⊗ ρ 1 H Example 3 If H is a cocommutative Hopf algebra (normally only in that case (!)) , then H itself has a natural H -module algebra and coalgebra structure, which is given by the “adjoint action" (where x, y ∈ H ): � x ′ yS ( x ′′ ) . ρ ( x ⊗ y ) = = x � ad y ( x )
2-Crossed Modules of Hopf Algebras Preliminaries Crossed Module of Hopf Algebras Crossed Modules of Cocommutative Hopf Algebras Definition 4 (Majid) A crossed module of (cocommutative) Hopf algebras [4, 5] is given by a Hopf algebra map: ∂ : I → H where I is an H -module bialgebra such that satisfying (for all x ∈ H and u, v ∈ I ): ∂ ( x ◮ ρ v ) = x � ad ∂ ( v ) ∂ ( u ) ◮ ρ v = u � ad v .
� � � � �� � � � � � 2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Cocommutative Hopf Algebra Simplicial Cocommutative Hopf Algebra A simplicial (cocommutative) Hopf algebra � is a collection of (cocommutative) Hopf algebras H n ( n ∈ � ) together with Hopf algebra maps called faces and degeneracies: d n i : H n → H n − 1 , 0 ≤ i ≤ n s n +1 H n → H n +1 0 ≤ j ≤ n : , j which are to satisfy the following simplicial identities: (i) d i d j = d j − 1 d i if i < j (ii) s i s j = s j +1 s i if i ≤ j (iii) d i s j = s j − 1 d i if i < j d j s j = d j +1 s j = id d i s j = s j d i − 1 if i > j + 1 Any simplicial cocommutative Hopf algebra can be pictured as: d 2 d 1 d 1 ���� H 2 = H 3 H 1 H 0 � d 0 d 0 s 0 s 0 s 1 We will denote the category of simplicial cocommutative Hopf algebras by Simp .
� � � � �� � � � � � 2-Crossed Modules of Hopf Algebras The Moore Complex and Applications Simplicial Cocommutative Hopf Algebra Simplicial Cocommutative Hopf Algebra A simplicial (cocommutative) Hopf algebra � is a collection of (cocommutative) Hopf algebras H n ( n ∈ � ) together with Hopf algebra maps called faces and degeneracies: d n i : H n → H n − 1 , 0 ≤ i ≤ n s n +1 H n → H n +1 0 ≤ j ≤ n : , j which are to satisfy the following simplicial identities: (i) d i d j = d j − 1 d i if i < j (ii) s i s j = s j +1 s i if i ≤ j (iii) d i s j = s j − 1 d i if i < j d j s j = d j +1 s j = id d i s j = s j d i − 1 if i > j + 1 Any simplicial cocommutative Hopf algebra can be pictured as: d 2 d 1 d 1 ���� H 2 = H 3 H 1 H 0 � d 0 d 0 s 0 s 0 s 1 We will denote the category of simplicial cocommutative Hopf algebras by Simp . An n-truncated simplicial (cocommutative) Hopf algebra T r n � = { H n , . . . , H 1 , H 0 } is a simplicial cocommutative Hopf algebra obtained by forgetting any dimensions of greater than n in � . We denote category of n-truncated simplicial Hopf algebras by Tr n Simp .
2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex Hopf Kernels Hopf kernels appear in [1]. Let f, g : A → B be two Hopf algebra maps. Define subalgebras: LEqual ( f, g ) = { x ∈ A | ( f ⊗ id ) ∆ ( x ) = ( g ⊗ id ) ∆ ( x ) } REqual ( f, g ) = { x ∈ A | ( id ⊗ f ) ∆ ( x ) = ( id ⊗ g ) ∆ ( x ) } which implies f ( x ) = g ( x ) , the property of equalizer of group maps.
2-Crossed Modules of Hopf Algebras The Moore Complex and Applications The Moore Complex Hopf Kernels Hopf kernels appear in [1]. Let f, g : A → B be two Hopf algebra maps. Define subalgebras: LEqual ( f, g ) = { x ∈ A | ( f ⊗ id ) ∆ ( x ) = ( g ⊗ id ) ∆ ( x ) } REqual ( f, g ) = { x ∈ A | ( id ⊗ f ) ∆ ( x ) = ( id ⊗ g ) ∆ ( x ) } which implies f ( x ) = g ( x ) , the property of equalizer of group maps. △ The problem is: ! ∆ � LEqual ( f, g ) � ∆ � REqual ( f, g ) � ⊆ LEqual ( f, g ) ⊗ A ⊆ A ⊗ REqual ( f, g ) and and S � LEqual ( f, g ) � S � REqual ( f, g ) � = REqual ( f, g ) and = LEqual ( f, g ) So they don’t form a (sub) Hopf algebra structure.
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