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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 : S1 → S2 between semi-groups S1 and S2 a half–homomorphism if the alternative ∀x, y ∈ S1 : 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 S1 into a cancellation semi-group S2 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 G1 into a group G2.

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 τc : Q → Q defined by τc(x) = x−1 if x ∈ {c, c−1} x if x / ∈ {c, c−1} (2) 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 | x2 = 1}. Then (1) If c ∈ X, then the mapping τc is an involution. (2) For a = b ∈ X one has τa = τb 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 2k, 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

• f

{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 u2 = [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) = t2 · (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 Q8. 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 {τ1, . . . τ7} which correspond to the set S of the elements of O. Note, that in the octonion loop O one has

7

• i=1

τi = J. 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 27. 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 ◦ τc ◦ α−1 = τα(c).

Peter Plaumann

Half- automorphisms of Cayley -Dickson loops

Cayley -Dickson loops

• Theorem. In the Cayley-Dickson loops CDLn, n ≥ 4 the following

statements are true: (1) CDLn is a loop of order 2n. (2) CDLn is a diassociative loop. (3) In CDLn there are one central element z = −1 of order 2 and 2n − 2 elements of order 4. (4) If H is a subloop of CDLn, then z ∈ H. (5) Z = z is a normal subloop of CDLn which coincides with the nucleus, the commutant, the center, the associator subloop, the commutator subloop and the Frattini subloop of CDLn. (7) CDLn/Z is an elementary abelian group of order 2n − 1.

Peter Plaumann

Half- automorphisms of Cayley -Dickson loops

Class of diassociative loops

Let us define a class L of diassociative loops, which obey the properties: (i) Z(L) = {1, z}, x2 = z, x ∈ Z(L), (ii) [x, y] = z, x ∈ Z(L), y ∈ {1, z, x, xz}. for every L ∈ L and for x, y, z ∈ L Obviosly, L includes the class of Cayley -Dickson loops. Note that the class L is not a variety. The minimal variety generated by L is the variety V (L) given by the identities: x4 = [x2, y] = (x2, y, z) = (y, x2, z) = (y, z, x2) = [[x, y], z] = ([x, y], z, t) = (z, [x, y], t) = (z, t, [x, y]) = [x, y]2 = 1. It is easy to see that if L ∈ V (L) then any two elements x, y ∈ L generate a cyclic group of order 2n, n = 0, 1, 2; or 2−group of order 8 : D4 or Q8. In the case L ∈ L we have that a group, generated by x, y ∈ L is a cyclic

• r Q8.

Peter Plaumann

Half- automorphisms of Cayley -Dickson loops

Half-automorphisms on L

Theorem 1. If L ∈ L, then τx is a half-automorphism for all x ∈ L, x ∈ Z(L). Definition A diassociative commutative loop of exponent 2 is called a Steiner loop. Theorem 2. If L ∈ L, then L/Z(L) is a Steiner loop and for any Steiner loop S there exists a loop L ∈ L such that L/Z(L) ≃ S.

Peter Plaumann

Half- automorphisms of Cayley -Dickson loops

last page

In this talk the Chapter ”Half-automorphisms of Cayley -Dickson loop”, by M.L.Merlini Giuliani, Peter Plaumann and Liudmila Sabinina, in the book ”Lie groups, Differential equations, and Geometry”, Springer, 2018 was used THANK YOU

Peter Plaumann

Half- automorphisms of Cayley -Dickson loops