The Automorphism Group of an Octonion Algebra
In this chapter we study the group G = Aut(C) of automorphisms of an octonion algebra C over a field k. By “automorphism” we will in this chapter always understand a linear automorphism. Since automorphisms leave the norm invariant, Aut(C) is a subgroup o
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In this chapter we study the group G = Aut( C) of automorphisms of an octonion algebra C over a field k. By "automorphism" we will in this chapter always understand a linear automorphism. Since automorphisms leave the norm invariant, Aut(C) is a subgroup of the orthogonal group O(N) of the norm of C. Let K be an algebraic closure of k and put CK = K ®k C. The automorphism group G = Aut(CK) is a linear algebraic group in the sense of [Bor] , [Hu] and [Sp 81]; we refer to these books for the necessary background from the theory of algebraic groups. In the sequel we will, as a rule, denote algebraic groups by boldface letters. G is a closed subgroup of the algebraic group O(N) (the orthogonal group of the quadratic form on CK defined by N). We will prove in Th. 2.3.5 that G is a connected, simple algebraic group of type G2 . We will prove in Prop. 2.4.6 that the algebraic group G is defined over k; from this it follows that the automorphism group G is the group G(k) of k-rational points of the algebraic group G. The results in this chapter are valid in any characteristic, though the characteristic two case sometimes requires a separate treatment.
2.1 Automorphisms Leaving a Quaternion Subalgebra Invariant We begin by studying the automorphisms that map a given quaternion subalgebra D of C onto itself. Let t be such an automorphism. From t(D) = D it follows that t(Dl..) = Dl... As we did before, we write Dl.. = Da for a fixed a E Dl.. with N(a) #- 0; every element of C can then be written as x + ya with x, y E D. We have
t(x + ya) = t(x) + t(y)t(a) with t(x), t(y) E D and t(a) E Dl.. = Da. We define mappings u, v : D - D by t(x) = u(x) and t(y)t(a) = v(y)a (x,y ED). Hence T. A. Springer et al., Octonions, Jordan Algebras and Exceptional Groups © Springer-Verlag Berlin Heidelberg 2000
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2. The Automorphism Group of an Octonion Algebra
t(x + ya) = u(x) + v(y)a
(x,y ED).
Clearly, u = tiD is an automorphism of D, and v is an orthogonal transformation. Writing out
(x,W E D)
t((xa)w) = t(xa)t(w) with the aid of (1.22), we find
v(xw) = v(x)u(w)
(x,W
ED).
Since u is an automorphism, u(w) = u(w). Hence, writing y for w,
v(xy) = v(x)u(y)
(x,y ED).
Substituting x = e and defining p = v(e) we derive from this
v(y)
= pu(y)
(y ED).
Here p ED, and since v is an isometry, N (P) = 1. Thus we find for t:
t(x + ya)
= u(x) + (pu(y))a
(x,y ED).
(2.1)
Conversely, for any p E D with N (p) = 1 and any automorphism u of D, equation (2.1) defines an automorphism t of C which leaves D invariant; this is easily verified using (1.22) and the associativity of D. Since the associative algebra D is central simple, every automorphism of D is inner by the Skolem-Noether Theorem (see, e.g., [ArNT, Cor. 7.2Dj or [Ja 80, § 4.6, Cor. to Th. 4.9]). Hence by (2.1) every automorphism of C that leaves D invariant can be written as
t(x + ya) = cxc- 1 + (pcyc-1)a
(x,y ED)
(2.2)
with c E D, N(c) t- 0 and p E D, N(P) = 1. Here c is determined by t up to a nonzero scalar factor.
2.2 Connectedness and Dimension of the Automorphism Gro
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