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Quadratic Forms Permitting Triple Composition
Abstract
In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, . In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples or x(yz) determined by a composition algebra, with the quadratic form Q the usual norm form. For any fixed Q this leads to 1 isotopy class in dimensions 1 and 2, 3 classes in the dimension 4 quaternion case, and 6 classes in the dimension 8 octonion case.
... The key reference for a.v. triple systems will be [25]. In the finite-dimensional case the absolute value comes from an inner product, so the results in [25] can be applied and are fundamental for our study. ...
... triple systems will be [25]. In the finite-dimensional case the absolute value comes from an inner product, so the results in [25] can be applied and are fundamental for our study. Following the philosophy of [25], if A = A 0 ⊕ A 1 is a two-graded a.v. ...
... In the finite-dimensional case the absolute value comes from an inner product, so the results in [25] can be applied and are fundamental for our study. Following the philosophy of [25], if A = A 0 ⊕ A 1 is a two-graded a.v. algebra, then its odd part A 1 is an a.v. ...
We study two-graded absolute valued algebras. These are two-graded algebras satisfying the absolute value multiplicative property only on homogeneous elements. Thus, hexagonions (also called sedenions) and other sixteen-dimensional algebras arise as examples of these algebras. The even parts of two-graded absolute valued algebras are the absolute valued algebras, while the odd parts are exactly the absolute valued triple systems. So, in a way these two-graded algebras give a unifying viewpoint of both structures. We also study the simplicity and give several ways to construct two-graded absolute valued algebras. We also provide a description of isomorphism classes for two-graded absolute valued algebras of dimensions 1, 2 and 4.
... triple system has dimension 1 in the complex case and 1, 2, 4 or 8 in the real one, and that the absolute values are the usual euclidean norms. So, the Ref. [14] gives the first approach to the classification of finite-dimensional a.v. triple systems. ...
... triple systems will be real and of dim(T ) ∈ {2, 4, 8}. Taking into account [14] and the triality property, (see also [13, Section 4] and [17, Section 2.2]), we can prove that the triple product in T = R n as euclidean space, n ∈ {2, 4, 8}, can adopt one of the following twelve forms ...
We introduce homotopical techniques in the frameworks of two-graded absolute valued algebras and absolute valued triple systems, which will simplify the study of these structures. To this end, we previously refine and concrete the known descriptions of two-graded absolute valued algebras and absolute valued triple systems, as well as characterize the fact that an absolute valued triple system is the odd part of an absolute valued two-graded algebra.
... In this section we classify isoparametric triple systems of algebra type with m 1 =m 2 +l. We will see that such triples are built up from composition triples where in this paper (in contrast to [8]!) a composition triple is a triple system (@BULLET@BULLET@BULLET): XχXχX-*X on a finite-dimensional euclidean vector space (X y < , @BULLET» which permits composition, i.e., <(ΛJ, y, z) y (x y y y z)y=ζx y #Xy, JvX^, z} holds for every x y y y z^X. Let L(x y y)^End X be defined by L(x y y)z=(x y y y z) and let L(x y j/)* denote the adjoint of L(x y y). ...
... The classification of isotopy classes of composition triples (over arbitrary fields) was carried out by K. McCrimmon. As a special case of [8] Theorem 7.6 we get). This implies the corollary. ...
... Instead of the alternative low, the weakly alternative low exists: (aa)b − a(ab) = b(aa) − (ba)a. This identity is equivalent to both the flexible low a(ba) = (ab)a and the power-associative low a(aa)=(aa)a [1], [19], [6] ...
In the article, the main ideas of the induction construction arXiv:1204.0194,
arXiv:1110.4737, arXiv:1202.0941, arXiv:1208.4466 are considered in the form of
sketches. The article establishes a link between Clifford algebras and
alternative-elastic algebras at the level of connectors.
... Dorfmeister and Neher's approach was via the isoparametric triple systems [3], which are algebraic in nature. The proof also relies on the fairly involved algebraic classification result [8] about composition triples. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs {3, 4} and {7, 8}, based on more geometric considerations. ...
Ozeki and Takeuchi \cite[I]{OT} introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and M\"unzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and M\"unzner collectively as OT-FKM type throughout the paper. Dorfmeister and Neher \cite{DN} then employed isoparametric triple systems \cite{DN1}, which are algebraic in nature, to prove that Condition A alone implies the isoparametric hypersurface is of OT-FKM type. Their proof for the case of multiplicity pairs and rests on a fairly involved algebraic classification result \cite{Mc} about composition triples. In light of the classification \cite{CCJ} that leaves only the four exceptional multiplicity pairs and unsettled, it appears that Condition A may hold the key to the classification when the multiplicity pairs are and . Thus Condition A deserves to be scrutinized and understood more thoroughly from different angles. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs and , based on more geometric considerations. We make it explicit and apparent that the octonian algebra governs the underlying isoparametric structure.
Axioms are proposed for a certain ‘‘alternative’’ kind of ternary composition algebra, termed a 3Cn algebra. The axioms are shown to be (for n>2) in a simple correspondence with the axioms for a ternary vector cross product algebra. The axioms imply that n=1, 2, 4, or 8 (from which the usual Hurwitz theorem is deduced). The existence of 3C8 algebras is demonstrated by an explicit construction in four‐dimensional Hilbert space, without appeal to the properties of the algebra of octonions.
We rephrase the classical theory of composition alge- bras over fields, particularly the Cayley-Dickson Doubling Process and Zorn's Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.
Previous work on ternary composition algebras of definite signature is generalized to the only other case allowed by the axioms, namely that of neutral signature. One new feature is the presence, in the neutral case, of `counter-automorphisms'. The structure of the `exceptional' eight-dimensional algebras is probed, and, in various guises, a fundamental dichotomy is unearthed. Canonical forms are obtained, and various further properties are derived.
Using some theorems on composition algebras over rings of genus zero and elementary results from algebraic geometry, all composition algebras over a ring of fractions related to the projective line over a field are enumerated and partly classified.
Absolute valued triple system are the natural ternary extension of absolute valued algebras. In this article we classify strongly power-associative absolute valued triple systems.
We work to find a basis of graded identities for the octonion algebra. We do
so for the and gradings, both of them derived
of the Cayley-Dickson process, the later grading being possible only when the
characteristic of the scalars is not two.
Non-split nonassociative quaternion algebras over fields were first discovered over the real numbers independently by Dickson and Albert. They were later classified over arbitrary fields by Waterhouse. These algebras naturally appeared as the most interesting case in the classification of the four-dimensional nonassociative algebras which contain a separable field extension of the base field in their nucleus. We investigate algebras of constant rank 4 over an arbitrary ring R which contain a quadratic etale subalgebra S over R in their nucleus. A generalized Cayley-Dickson doubling process is introduced to construct a special class of these algebras.
Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely
by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.
The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 n for all n 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.
A conic bundle or quadric bundle in characteristic 2 can have generic fiber which is nowhere smooth over the function field of the base variety. In that case, the generic fiber is called a quasilinear quadric. We solve some of the main problems of birational geometry for quasilinear quadrics. First, if there is a rational map from one quasilinear quadric to another of the same dimension, and also a map back, then the two quadrics are birational. Next, we determine exactly which quasilinear quadrics are ruled, that is, birational over the base field to the product of some variety with the projective line. Both statements are conjectured for quadrics in any characteristic. The proofs begin by extending Karpenko and Merkurjev's theorem on the essential dimension of quadrics to arbitrary quadrics (smooth or not) in characteristic 2.
Nondegenerate forms N of degree d on a unital nonassociative algebra A over a ring R which permit composition, i.e., satisfy N(1)=1 and N(xy)=N(x)N(y) for all x,y in A, are studied. These forms were first classified by Schafer over fields of characteristic 0 or >d. We investigate cubic and quartic nondegenerate forms which permit composition over certain rings and curves. Classes of highly degenerate cubic forms N over fields which permit composition are constructed.
The simple Lie algebras of types A, B, C, D, except D
4, have been determined by W. Landherr [1] and [2] and by the present author [1], [2], [3] and [4]. The main result of these investigations was the establishment of a 1 — 1 correspondence between the Lie algebras and the simple associative algebras with involution (, J). Here is associative, J is an involution (anti-automorphism of period two) and simplicity means that there are no ideals invariant under J except and 0. Isomorphism, homomorphism, etc. for such pairs are defined as usual for operator groups. The Lie algebras of type D
4 were first studied by the author’s student, C. L. Caroll, Jr., in his University of North Carolina dissertation (1943, unpublished). In this paper Caroll introduced the notion of type D
4I, D
4II, D
4III, and D
4VI for these Lie algebras, and constructed examples of all of these types. His work was based on the half-spin representation of the split D
4.
The principal objective of the present paper is the study of the automorphisms and groups of automorphisms of composition algebras, that is, the algebras arising from quadratic forms which permit composition. These algebras are mainly quaternion algebras and Cayley algebras. The problem of determining the quadratic forms which permit composition (Huryitz’s problem) has been treated by many authors (2). In spite of this, there does not appear in any one place a complete solution of this problem in its most general form — which amounts to the determination of the algebras for an arbitrary field and not just to the determination of the possible dimensionalities. We give such a solution here for the case of characteristic not two. Aside from its intrinsic interest and applications to other fields (for example Jordan algebras, absolute valued algebras) we have still another reason for treating the Hurvitz problem again, namely: The analysis of the composition algebras is essential for our study of their automorphisms.
Composition algebras and their automorphisms 55-80. 2._Triality and Lie algebras of type D4
- N Jacobson
N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo (2) 7 (1958),
55-80.
2._Triality
and Lie algebras of type D4, Rend. Circ. Mat. Palermo (2) 13(1964), 1-25.