The Derivation Algebra of a Real Division Algebra

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Introduction. Let A be an algebra (not necessarily associative) over a field F. For each a E A let La(x) = ax and Ra(x) = xa for x E A.Then A is said to be a division algebra if for every a ? 0 the transformations La and Ra are invertible. A finite dimensional associative division algebra over the reals is either the reals, the complex numbers, or the quaternions. If the hypothesis of associativity is replaced by the weaker assumption that the algebra is alternative, then as shown by Bruck and Kleinfeld [2], the algebra is the octonions or is associative. Milnor and Bott [8] and Kervaire [7] independently proved that every finite dimensional division algebra over the reals must have dimension 1, 2, 4, or 8, but the problem remains of trying to describe the real division algebras which fail to satisfy the associative or alternative laws.Recently real division algebras have appeared in physics papers [3] and [9]. The division algebras which arise in physical situations are likely to exhibit many symmetries or equivalently many automorphisms. Since the automorphism group is a real Lie group whose Lie algebra is the algebra of derivations, the "largeness" of the derivation algebra thus reflects the symmetries of the division algebra. In this paper we investigate the derivation algebra DerA of a real division algebra A and obtain the following classification result: THEOREM.Assume A is a real division algebra. (i) If dimA =1 or 2, then DerA = 0. (ii) If dim A = 4, then Der A is isomorphic to su(2) or dim DerA = Oor 1. DERIVATION ALGEBRA 1137 COROLLARY 2. With A as above, let L denote a subalgebra of DerA and let N be the solvable radical of L. Then N is abelian. Proof. Let K be the algebraic closure of the field F and let NK denote N ?FK, and AK = A ?FK. Then, NK is a solvable Lie algebra of linear transformations over K, and thus by Lie's Theorem the elements of NK can be simultaneously upper triangularized. But as a consequence, the elements of [NKNK] are strictly upper triangular, and hence nilpotent on AK. Since [NN] C [NKNK], it follows that the derivations in [NN] are nilpotent on A. Therefore [NN] = 0 by Lemma 1. O COROLLARY 3. DerA contains no split semisimple subalgebras.Proof. If DerA contains a split semisimple subalgebra, then DerA contains a copy of sl(2, F). If e, h, f denote the standard basis elements of sl(2, F), then e and f act nilpotently on each finite-dimensional sl(2, F)module. Thus, they are nilpotent on A which gives a contradiction. O COROLLARY 4. If S is a semisimple subalgebra of DerA, then S contains no ad-nilpotent ele...

An investigation of real division algebras using derivations

“…The easiest example is obtained by taking the values of the constants prescribed in (4.5) with the sole exception that 7]± is some negative number besides -1. This algebra was shown to be a division algebra in ( [1], Theorem 20), and it has su(Z) as its derivation algebra according to Theorem 4.4. (2).…”

“…For example, if we choose e = l = ^ = 7, β <0, β -£ δ <0, ζ = p = l = θ = σ, then A is isomorphic to the division algebra of ( [1], Theorem 20) using the map u^u 9 x 1^ e lf x 2 <-> e 2 , x 3 <-e 4 , y 1 -e 3 , y 2 -e β , y z -> e Q , y, «-> e 7 .…”

“…Specifically, products in A are given by [1,Theorem 20] that this algebra is a real division algebra, and we want to calculate its derivations for appropriate conditions on the /3's. In particular, we shall establish THEOREM 6.9.…”

“…By [1,Corollary 16 diag{0, 0, ai, -ai, βi, -βί, (a + β)i y -{a + β)i) diag {0, 0, 0, 0, yί, -yi, yi, -yi} for nonzero real numbers α, β, y. 2 and d 2 = (1/7)32 are also a basis for Der A and…”

Real Commutative Division Algebras

“…First attempts towards a systematic classification of all real division algebras, up to isomorphism, were made by Benkart and Osborn [5]. To date this intriguing problem is still far from being solved in full generality.…”