Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs On Skew-Homomorphisms B. Kuzma 1 G. Dolinar G. Nagy P . Szokol 1 UP FAMNIT May 28, 2015 Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Skew-morphism Given a function Φ: A → A , let Φ 0 = Id . Φ k ( x ) = Φ(Φ k − 1 ( x )) . Φ − 1 the inverse of Φ . Definition Φ: A → A is skew-morphism if for some function κ : A → N Φ( ab ) = Φ( a )Φ κ ( a ) ( b ) Remark For Φ bijective can also take κ : A → Z . Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Skew-morphism Given a function Φ: A → A , let Φ 0 = Id . Φ k ( x ) = Φ(Φ k − 1 ( x )) . Φ − 1 the inverse of Φ . Definition Φ: A → A is skew-morphism if for some function κ : A → N Φ( ab ) = Φ( a )Φ κ ( a ) ( b ) Remark For Φ bijective can also take κ : A → Z . Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background A map M is a 2-cell embedding of a simple connected graph Γ into oriented surface. M is Cayley if exists subgroup A ≤ Aut + ( M ) acting regularly on V (Γ) . M is regular if ∀ ( a , b ) , ( c , d ) ∈ E (Γ) there is Φ ∈ Aut + ( M ) such that = ( c , d ) . � � Φ( a ) , Φ( b ) n (2002) Cayley map M is regular iff there exists certain unital Jajcay, Širᡠskew-morphism. Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background A map M is a 2-cell embedding of a simple connected graph Γ into oriented surface. M is Cayley if exists subgroup A ≤ Aut + ( M ) acting regularly on V (Γ) . M is regular if ∀ ( a , b ) , ( c , d ) ∈ E (Γ) there is Φ ∈ Aut + ( M ) such that = ( c , d ) . � � Φ( a ) , Φ( b ) n (2002) Cayley map M is regular iff there exists certain unital Jajcay, Širᡠskew-morphism. Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background A map M is a 2-cell embedding of a simple connected graph Γ into oriented surface. M is Cayley if exists subgroup A ≤ Aut + ( M ) acting regularly on V (Γ) . M is regular if ∀ ( a , b ) , ( c , d ) ∈ E (Γ) there is Φ ∈ Aut + ( M ) such that = ( c , d ) . � � Φ( a ) , Φ( b ) n (2002) Cayley map M is regular iff there exists certain unital Jajcay, Širᡠskew-morphism. Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background A map M is a 2-cell embedding of a simple connected graph Γ into oriented surface. M is Cayley if exists subgroup A ≤ Aut + ( M ) acting regularly on V (Γ) . M is regular if ∀ ( a , b ) , ( c , d ) ∈ E (Γ) there is Φ ∈ Aut + ( M ) such that = ( c , d ) . � � Φ( a ) , Φ( b ) n (2002) Cayley map M is regular iff there exists certain unital Jajcay, Širᡠskew-morphism. Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background Cyclic extensions of groups! A , C ≤ G subgroups with A ∩ C = 1 and G = AC . Assume C = � c � . Given g = ac ∈ G , we have ca = Φ( a ) c i for some i ∈ Z Then, c ( ab ) = Φ( ab ) c k = ( ca ) b = Φ( a ) c i b = Φ( a )Φ i ( b ) c t By uniqueness, c k = c t and for some i = κ ( a ) ∈ Z Φ( ab ) = Φ( a )Φ i ( b ) Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background Cyclic extensions of groups! A , C ≤ G subgroups with A ∩ C = 1 and G = AC . Assume C = � c � . Given g = ac ∈ G , we have ca = Φ( a ) c i for some i ∈ Z Then, c ( ab ) = Φ( ab ) c k = ( ca ) b = Φ( a ) c i b = Φ( a )Φ i ( b ) c t By uniqueness, c k = c t and for some i = κ ( a ) ∈ Z Φ( ab ) = Φ( a )Φ i ( b ) Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background Cyclic extensions of groups! A , C ≤ G subgroups with A ∩ C = 1 and G = AC . Assume C = � c � . Given g = ac ∈ G , we have ca = Φ( a ) c i for some i ∈ Z Then, c ( ab ) = Φ( ab ) c k = ( ca ) b = Φ( a ) c i b = Φ( a )Φ i ( b ) c t By uniqueness, c k = c t and for some i = κ ( a ) ∈ Z Φ( ab ) = Φ( a )Φ i ( b ) Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background Cyclic extensions of groups! A , C ≤ G subgroups with A ∩ C = 1 and G = AC . Assume C = � c � . Given g = ac ∈ G , we have ca = Φ( a ) c i for some i ∈ Z Then, c ( ab ) = Φ( ab ) c k = ( ca ) b = Φ( a ) c i b = Φ( a )Φ i ( b ) c t By uniqueness, c k = c t and for some i = κ ( a ) ∈ Z Φ( ab ) = Φ( a )Φ i ( b ) Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Background Cyclic extensions of groups! A , C ≤ G subgroups with A ∩ C = 1 and G = AC . Assume C = � c � . Given g = ac ∈ G , we have ca = Φ( a ) c i for some i ∈ Z Then, c ( ab ) = Φ( ab ) c k = ( ca ) b = Φ( a ) c i b = Φ( a )Φ i ( b ) c t By uniqueness, c k = c t and for some i = κ ( a ) ∈ Z Φ( ab ) = Φ( a )Φ i ( b ) Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Case κ ( G ) = 0 Lemma Φ: M n ( F ) → M n ( F ) skew-morphism. If κ ( G ) = 0 for some G ∈ GL n ( F ) then Φ( X ) = MX . Proof. Φ( X ) = Φ( G · G − 1 X ) = Φ( G )Φ κ ( G ) ( G − 1 X ) = Φ( G ) · G − 1 X . Define M := Φ( G ) G − 1 . Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Case Φ linear. Theorem Φ: M n ( F ) → M n ( F ) linear skew-morphism. Then exists s ∈ N and λ ∈ F such that where N ∈ GL n ( F ) and N 1 − s = NM s = λ I . Φ( X ) = MXN . Corollary Φ: M n ( F ) → M n ( F ) linear, unital skew-morphism. Then Φ( X ) = N − 1 XN . Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Case Φ linear. Theorem Φ: M n ( F ) → M n ( F ) linear skew-morphism. Then exists s ∈ N and λ ∈ F such that where N ∈ GL n ( F ) and N 1 − s = NM s = λ I . Φ( X ) = MXN . Corollary Φ: M n ( F ) → M n ( F ) linear, unital skew-morphism. Then Φ( X ) = N − 1 XN . Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs Case Φ linear. Theorem Φ: M n ( F ) → M n ( F ) linear skew-morphism. Then exists s ∈ N and λ ∈ F such that where N ∈ GL n ( F ) and N 1 − s = NM s = λ I . Φ( X ) = MXN . Corollary Φ: M n ( F ) → M n ( F ) linear, unital skew-morphism. Then Φ( X ) = N − 1 XN . Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs General surjective Φ Example There exists a nonlinear unital, bijective skew-morphism Φ: M 2 ( Z 2 ) → M 2 ( Z 2 ) . �� 0 1 � 0 1 �� 0 1 �� 0 1 � 1 1 �� 0 1 �� � �� �� � �� = 2 , = 3 , φ = , κ φ = , κ 1 0 1 1 1 0 1 1 1 0 1 1 �� 1 1 � 1 1 �� 1 1 �� 1 1 � 0 1 �� 1 1 �� � �� �� � �� = 2 , = 1 , φ = , κ φ = , κ 1 0 0 1 1 0 0 1 1 0 0 1 �� 1 0 �� 1 0 � 1 0 �� 1 0 �� 1 0 � 1 0 �� � �� �� � �� = 1 , = 3 , = = φ , κ φ , κ 0 1 1 1 1 1 1 1 0 1 0 1 �� 0 0 � 0 0 �� 0 0 �� 0 0 � 0 0 �� 0 0 �� � �� �� � �� = 2 , = 1 , φ = , κ φ = , κ 1 0 1 0 1 0 0 1 1 1 0 1 �� 0 0 � 0 0 �� 0 0 �� 0 1 � 0 1 �� 0 1 �� � �� �� � �� = 3 , = 0 , φ = , κ φ = , κ 1 1 0 1 1 1 0 0 0 1 0 0 �� 0 1 � 1 1 �� 0 1 �� 1 1 � 1 1 �� 1 1 �� � �� �� � �� = 1 , = 0 , = = φ , κ φ , κ 0 1 0 0 0 1 0 0 1 1 0 0 �� 1 1 � 0 1 �� 1 1 �� 1 0 � 1 0 �� 1 0 �� � �� �� � �� = 3 , = 2 , φ = , κ φ = , κ 1 1 0 0 1 1 1 0 0 0 1 0 �� 1 0 � 1 0 �� 1 0 �� 0 0 � 0 0 �� 0 0 �� � �� �� � �� = 0 , = 1 . φ = , κ φ = , κ 0 0 1 0 0 0 0 0 0 0 0 0 Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs General surjective Φ Theorem Φ: M n ( F ) → M n ( F ) surjective skew-morphism. Then rk Φ( X ) = rk X. If κ ( GL n ) ≥ 1 and κ ( G ) > 1 for some G ∈ GL n ( F ) THEN Φ s = id for some s ≥ 1 . Assume κ ( GL n ) = { 1 } . Then, Cof ( a , b ) := ( − b , a ) � S − 1 X σ S , X ∈ GL n ( n ≥ 3 ) Φ( X ) = γ S − 1 X σ G , X ∈ M n \ GL n , Restricted Skew-Morphisms
Definitions and Preliminaries Skew-morphisms on M n ( F ) Ideas of proofs General surjective Φ Theorem Φ: M n ( F ) → M n ( F ) surjective skew-morphism. Then rk Φ( X ) = rk X. If κ ( GL n ) ≥ 1 and κ ( G ) > 1 for some G ∈ GL n ( F ) THEN Φ s = id for some s ≥ 1 . Assume κ ( GL n ) = { 1 } . Then, Cof ( a , b ) := ( − b , a ) � S − 1 X σ S , X ∈ GL n ( n ≥ 3 ) Φ( X ) = γ S − 1 X σ G , X ∈ M n \ GL n , Restricted Skew-Morphisms
Recommend
More recommend
