Half- automorphisms of Cayley -Dickson loops Peter Plaumann - PowerPoint PPT Presentation

Half- automorphisms of Cayley -Dickson loops Peter Plaumann Erlangen University Trento, Italy (GTG Meeting) June 17, 2017 Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Scott’s Theorem In the year 1957 W.R.Scott introduced in the concept of a half–homomorphism between semi-groups. He calls a mapping ϕ : S 1 → S 2 between semi-groups S 1 and S 2 a half–homomorphism if the alternative ∀ x , y ∈ S 1 : ϕ ( xy ) = ϕ ( x ) ϕ ( y ) or ϕ ( xy ) = ϕ ( y ) ϕ ( x ) (1) holds and proves the following Theorem Every half-isomorphism of a cancellation semi-group S 1 into a cancellation semi-group S 2 is either an isomorphism or an anti-isomorphism. We call a half-isomorphism proper if it is neither an isomorphism or an anti-isomorphism. Then one finds in Scott’s paper the following result which in this article we call Scott’s Theorem . There is no proper half-homomorphism from a group G 1 into a group G 2 . Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Scott’s Theorem for Moufang loops S.Gagola III, M.L.Merlini Giuliani - Half–isomorphisms of Moufang loops of odd order J. of Alg. and Appl. (2012) - On half-automorphisms of certain Moufang loops with even order J. of Algebra, (2013) M. Kinyon, I. Stuhl, P. Vojtechovsky - Half-automorphisms of Moufang loops, Journal of Algebra, (2016), Peter Plaumann Half- automorphisms of Cayley -Dickson loops

table of octonion loop Let us study the example of octonion loop: ◦ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 2 4 8 6 3 1 5 7 14 9 16 10 11 12 13 15 3 3 5 4 7 6 8 1 2 15 13 9 11 14 16 12 10 4 4 6 7 1 8 2 3 5 12 14 15 9 16 10 11 13 5 5 7 2 8 4 3 6 1 13 11 14 16 12 15 10 9 6 6 1 5 2 7 4 8 3 10 12 13 14 15 9 16 11 7 7 8 1 3 2 5 4 6 11 16 12 15 10 13 9 14 8 8 3 6 5 1 7 2 4 16 15 10 13 9 11 14 12 9 9 10 11 12 16 14 15 13 4 6 7 1 5 2 3 8 10 10 12 16 14 15 9 13 11 2 4 5 6 3 1 8 7 11 11 13 12 15 10 16 9 14 3 8 4 7 6 5 1 2 12 12 14 15 9 13 10 11 16 1 2 3 4 8 6 7 5 13 13 15 10 16 9 11 14 12 8 7 2 5 4 3 6 1 14 14 9 13 10 11 12 16 15 6 1 8 2 7 4 5 3 15 15 16 9 11 14 13 12 10 7 5 1 3 2 8 4 6 16 16 11 14 13 12 15 10 9 5 3 6 8 1 7 2 4 Peter Plaumann Half- automorphisms of Cayley -Dickson loops

some properties of octonion loop In the octonion loop O the following statements are true: (1) O is a loop of order 16 (2) O is a Moufang loop. (3) In O there are one central element z = − 1 of order 2 and 14 elements of order 4. (4) If H is a subloop of O , then z ∈ H . (5) Z is a normal subloop of O which coincides with the nucleus, the commutant, the center, the associator subloop, the commutator subloop and the Frattini subloop of O . (6) O / Z is an elementary abelian group of order 8. Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Elementary mapping Let Q be a diassociative loop and let c be an element of Q . Then the mapping τ c : Q → Q defined by � x − 1 if x ∈ { c , c − 1 } τ c ( x ) = (2) ∈ { c , c − 1 } if x / x is called an elementary mapping . Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Elementary mapping -I Obviously the following proposition holds. Proposition. For a diassociative loop Q of exponent 4 put X = { x ∈ Q | x 2 � = 1 } . Then (1) If c ∈ X , then the mapping τ c is an involution. (2) For a � = b ∈ X one has τ a = τ b if and only if ab = 1. (3) The group Γ( Q ) of all elementary mappings of the loop Q is 2 - elementary abelian group of orden 2 k , where k = | X | 2 Peter Plaumann Half- automorphisms of Cayley -Dickson loops

elementary mappings - II In O there are 14 elements of order 4 which give us precisely 7 different sets { x , x − 1 } with 2 elements. The set I of these 2-element sets consists of { 2 , 6 } , { 3 , 7 } , { 8 , 5 } , { 9 , 12 } , { 14 , 10 } , { 15 , 11 } , { 16 , 13 } . (3) Theorem Let O be the octonion loop and Γ( O ) be the group generated by all elementary mappings. Then every half-automorphism of O is the product of an automorphism of O and an element γ ∈ Γ( O ). Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Sketch of Proof Suppose that ϕ : O → O is a proper half-automorphism. Since O is a Moufang loop by Gagola -Giuliani results there exists a triple ( x , y , z ) such that ϕ restricted to � x , y � is an automorphism, ϕ restricted to � x , z � is an anti-automorphism, and [ x , y ] � = 1 , [ x , z ] � = 1. Since ϕ is proper, the loop � x , y , z � cannot be a group by the theorem of Scott. Hence � x , y , z � = O . In what follows we always will consider these generators x , y , z of the loop O In particular yz � = zy . In O there are 16 elements: {± 1 , ± x , ± y , ± z , ± xy , ± xz , ± yz , ± x ( yz ) } . Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Proof -I Note, that for any half-automorphism ψ of O and for any t ∈ O one has: ψ (1) = 1 , ψ ( − t ) = − ψ ( t ) since − 1 is the unique element of order 2 in O and central. Note also that the following relations hold: xy = − yx , xz = − zx , yz = − zy , x ( yz ) = z ( xy ) = y ( zx ) = − ( yz ) x = − ( xy ) z = − ( zx ) y . In order to understand the behavior of proper half-automorphism ϕ on O it is enough to consider only the set S = { x , y , z , xy , xz , yz , x ( yz ) } of 7 elements and to analyse the behavior of ϕ on all possible subgroups generated by 2 elements of S . There are 21 different pairs, some of them generate the same subgroups. Indeed: Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Proof - II (i) � x , y � = � x , xy � = � xy , y � (ii) � x , z � = � x , xz � = � xz , z � (iii) � y , z � = � y , yz � = � yz , z � (iv) � x , yz � = � x , x ( yz ) � = � yz , x ( yz ) � (v) � y , xz � = � y , x ( yz ) � = � xz , x ( yz ) � (vi) � z , yx � = � z , x ( yz ) � = � yx , x ( yz ) � (vii) � xy , yz � = � xy , xz � = � xz , yz � One obtains ( vii ) using the fact that u 2 = [ u , v ] = ( u , v , t ) = − 1 and the Moufang identity in O in the following form. For any elements u , v , t ∈ S we have: ut · vt = − tu · vt = − ( t ( uv ) t ) = t 2 · ( uv ) = − ( uv ) (4) Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Proof - III Thus we get 7 different 2-generated subgroup of O , every one of them is isomorphic to Q 8 . For example, � x , y � = {± 1 , ± x , ± y , ± xy } . There are two possibilities: ϕ restricted to � y , z � is an automorphism or ϕ restricted to � y , z � is an antiautomorphism. Analysing both cases step by step finally we have HAut ( O ) = Γ( O ) Aut ( O ) . Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Remark For a diassociative loop Q the inversion mapping ( J : x �→ x − 1 ) : Q → Q is an anti-automorphism of Q . Above we had defined 7 elementary half-automorphisms { τ 1 , . . . τ 7 } which correspond to the set S of the elements of O . Note, that in the octonion loop O one has 7 � τ i = J . i =1 Obviosly, if ϕ is an anti-automorphism then ϕ ◦ J is an automorphism. Note that the elementary maps of the octonion loop O considered as permutations of the set of elements of O generate the elementary abelian group Γ( O ) of order 2 7 . But not all elements of Γ( O ) are proper half-automorphisms, for example J is not proper. Here we do not discuss the intersection HAut ( O ) ∩ Γ( O ). Peter Plaumann Half- automorphisms of Cayley -Dickson loops

normality of Γ Proposition If Q is a diassociative loop of exponent 4 and Γ( Q ) is a subgroup of HAut ( Q ) then Γ( Q ) is normal in HAut ( Q ) Proof. For an element c ∈ Q of order 4 and for α ∈ HAut ( Q ) one has α ◦ ( c , c − 1 ) ◦ α − 1 = ( α ( c ) , α ( c − 1 )). Hence α ◦ τ c ◦ α − 1 = τ α ( c ) . Peter Plaumann Half- automorphisms of Cayley -Dickson loops

Cayley -Dickson loops Theorem. In the Cayley-Dickson loops CDL n , n ≥ 4 the following statements are true: CDL n is a loop of order 2 n . (1) (2) CDL n is a diassociative loop. (3) In CDL n there are one central element z = − 1 of order 2 and 2 n − 2 elements of order 4. (4) If H is a subloop of CDL n , then z ∈ H . (5) Z = � z � is a normal subloop of CDL n which coincides with the nucleus, the commutant, the center, the associator subloop, the commutator subloop and the Frattini subloop of CDL n . CDL n / Z is an elementary abelian group of order 2 n − 1. (7) Peter Plaumann Half- automorphisms of Cayley -Dickson loops