Composition Algebras Research Articles

On one-sided division infinite-dimensional normed real algebras

In this note we introduce the concept of Cayley homomorphism which is closely related with those of composition algebra and normalized orthogonal multiplication. The key result shows the existence of certain types of Cayley homomorphisms for infinite dimension. As an application we prove the existence of left division infinite-dimensional complete normed real algebras with left unity.

Composing Hierarchically Structured Images

This paper begins by examining the ‘classical’ raster-based composition model, establishing its weakness, and developing a new composition algebra based on line drawing. It then examines the role of composition in the Hierarchical Display Model, demonstrates that the forms of composition assumed in this model are inadequate to deal correctly with the interactions of two-dimensional representations of three-dimensional objects, and shows that our composition algebra resolves this difficulty. The composition steps required can be packaged with the use of a single attribute which constrains the apparent order of composition. This attribute is associated with the object which is to be taken ‘out of order’, obviating any needsfor layers.

Flexible composition algebras and okubo algebras

Flexible composition algebras and okubo algebras

Ternary composition algebras II. Automorphism groups and subgroups

We study an eight-dimensional ternary composition algebraε= (E, ⟨ ⟩, { }) of signature (8, 0) or (4, 4). Such an algebra is associated with an involutary outer automorphismMof the Lie algebra so (E)of ⟨, ⟩. The automorphisms and (in the (4, 4) case) counter-automorphisms ofεare obtained in terms of thed= 7 spin group determined byM. UsingMwe derive a ‘principle of duplicity’ for the ternary multiplication { } (related to the well-known principle of triality for the associated octonionic binary multiplication). However, rather than constructεout of the octonions as done (in effect) by previous authors, we give a four-dimensional ‘complex’ construction ofε. This non-octonionic view ofεhighlights certain (15-dimensional)d= 6 spin groups, rather than the customary (14-dimensional) octonionic automorphism groups. In the (4, 4) case we can choose to interpret ‘complex’ in terms of the split complex numbers, and are thereby led to consider the subgroup chain SO+(4, 4) ⊃ Spin+(3, 4) ⊃ SL (4; ℝ) (⋍ Spin+(3, 3)) instead of the chain SO+(4, 4) ⊃ Spin+(3, 4) ⊃ SU (2, 2) (⋍ Spin+(2, 4)).

Ternary composition algebras. I. Structure theorems: definite and neutral signatures

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.

Nonstandard signature of spacetime, superstrings, and the split composition algebras

It has been established that the covering group of the Lorentz group in the dimensions D=3, 4, 6 and 10, can be expressed in a unified way, based on the four composition-division algebras ℝ, ℂ, ℚ and $$\mathbb{O}$$ . If the division algebras in the construction of the covering groups of the Lorentz groups in D=3, 4, 6 and 10 are replaced by the split composition algebras, then the sequence of groups $$\widetilde{{\text{SO}}}{\text{(2,2)}}$$ , $$\widetilde{{\text{SO}}}{\text{(3,3)}}$$ and $$\widetilde{{\text{SO}}}{\text{(5,5)}}$$ results. Classical superstrings embedded in such spacetimes can be defined, and the split composition algebras provide a natural framework for their description.

A 2-dimensional quaternionic construction of an 8-dimensional ternary composition algebra

It is shown that a real 8-dimensional ternary composition algebra can be constructed out of a quaternionic 2-dimensional Hilbert space. The construction is seen to highlight a subgroup Spin(5)Pin(2) of the automorphism group Spin(7) of the algebra.

The metric of a space and quadratic algebras

A collection of algebras with product that preserves the metric of the underlying vector space is investigated. It is shown that this collection includes the generalized quaternions, the composition algebras, and the algebra of color. All the algebras in this collection that are either commutative or alternative are determined; the ramifications of nondegenerate metrics are detailed. If the algebra is alternative and not of dimension 1, 2, 4, or 8 over the base field, the metric must be degenerate.

A geometric construction of Moufang planes

An algebraic construction of degree 3 Jordan algebras (including the exceptional one) as trace 0 elements in a degree 4 Jordan algebra is translated to give a geometric construction of Barbilian planes coordinatized by composition algebras (including the Moufang plane) as skew polar line pairs and points on the quadratic surface determined by a polarity of projective 3-space over a smaller composition algebra.

Ternary composition algebras: 8 dimensions out of 4?

Just as real 8-dirnensional ternary composition algebras, of signatures (8, 0) and (4, 4), can be constructed out of 4-dimensional complex spacesC4, 0 andC2, 2, it is shown that complex 8-dimensional ternary composition algebras can be constructed out of aK-moduleK4, whereK is a certain commutative algebra of dimension 4 overR. The neutral signature real form of theK4-construction is associated with the symmetry-breaking chainSO(4, 4) ⊃ Spin+ (3, 4) ⊃SL(4;R , as compared to the chainSO(4, 4) ⊃ Spin+ (3, 4) ⊃SU(2, 2) associated with theC2, 2-construction. On breaking the symmetry further, by passing to the binary octonionic multiplication, the ternary viewpoint is seen to throw light on both the Zorn and the Gunaydin forms of octonionic multiplication and leads to several unorthodox forms as well.

Ternary composition algebras and Hurwitz’s theorem

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.

Tensor Products of Composition Algebras, Albert Forms and Some Exceptional Simple Lie Algebras

In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra $(\mathcal {A}, - )$, we associate a quadratic form $Q$ called the Albert form of $(\mathcal {A}, - )$. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index $F_{4,1}^{21}$, ${}^2E_{6,1}^{29}$, $E_{7,1}^{48}$ and $E_{8,1}^{91}$. That description is used to obtain a classification of the indicated Lie algebras over ${\mathbf {R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3$.

Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras

In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra ( A , − ) (\mathcal {A},\, - ) , we associate a quadratic form Q Q called the Albert form of ( A , − ) (\mathcal {A},\, - ) . The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index F 4 , 1 21 F_{4,1}^{21} , 2 E 6 , 1 29 {}^2E_{6,1}^{29} , E 7 , 1 48 E_{7,1}^{48} and E 8 , 1 91 E_{8,1}^{91} . That description is used to obtain a classification of the indicated Lie algebras over R ( ( T 1 , … , T n ) ) , n ⩽ 3 {\mathbf {R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3 .

Generalized Minkowski metric and composition algebras

New variables («new» as regards the space-time variables) are obtained starting from the requirement that their allowable values should leave the value of the Minkowski metric unchanged. To this purpose, the space-time variables are embedded into real and then complex nonassociative algebras (with an identity element) endowed with nondegenerate quadratic forms generalizing the Minkowski metric. The algebras are proved to fulfil the above-mentioned requirement if and only if they are composition algebras. Using the Hurwitz theorem, it is shown that in the real case there is only one allowable algebra, which is isomorphic to the eight-dimensional split real Cayley algebra, while, in the complex case, there are two allowable algebras: one is isomorphic to the four-dimensional complex quaternion algebra, the other is isomorphic to the eight-dimensional complex Cayley algebra. In the case of these three algebras, new variables and their allowable values are obtained. The essential role played in this context by the (multiplicative) noncommutativity and nonassociativity is pointed out in some final remarks.

Automating the design of graphical presentations of relational information

The goal of the research described in this paper is to develop an application-independent presentation tool that automatically designs effective graphical presentations (such as bar charts, scatter plots, and connected graphs) of relational information. Two problems are raised by this goal: The codification of graphic design criteria in a form that can be used by the presentation tool, and the generation of a wide variety of designs so that the presentation tool can accommodate a wide variety of information. The approach described in this paper is based on the view that graphical presentations are sentences of graphical languages. The graphic design issues are codified as expressiveness and effectiveness criteria for graphical languages. Expressiveness criteria determine whether a graphical language can express the desired information. Effectiveness criteria determine whether a graphical language exploits the capabilities of the output medium and the human visual system. A wide variety of designs can be systematically generated by using a composition algebra that composes a small set of primitive graphical languages. Artificial intelligence techniques are used to implement a prototype presentation tool called APT (A Presentation Tool), which is based on the composition algebra and the graphic design criteria.

Composition algebras of polynomials

Composition algebras of polynomials

Zur Struktur der vierdimensionalen quadratischen Algebren

Assuming properties, which are essential for division algebras, but mostly invariant to extensions of the ground field, we investigate the structure of quadratic division algebras of dimension four over an arbitrary field of characteristic not two. We relate the size of the group of automorphisms of such an algebra A to algebraic laws valid in A, characterize Lie-admissibility by means of the skew-commutative vector algebra of A and outline the possibilities of describing A by irreducible identities of degree 3. Some results of the last chapter apply to arbitrary dimensions. We show, that a simple quadratic algebra with the right (left) inverse property for invertible elements is a composition algebra. Finally we conclude, that a quadratic division algebra of dimension four with a right (left) nucleus different from the center is associative.

Quadratic Forms Permitting Triple Composition

In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, $Q\left (\{{xyz} \} \right ) = Q(x)Q(y)Q(z)$. In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples $\{xyz \} = (xy)z$ 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.

Quadratic forms permitting triple composition

In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, Q ( { x y z } ) = Q ( x ) Q ( y ) Q ( z ) Q\left (\{{xyz} \} \right ) = Q(x)Q(y)Q(z) . In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples { x y z } = ( x y ) z \{xyz \} = (xy)z or x ( y z ) x(yz) determined by a composition algebra, with the quadratic form Q Q the usual norm form. For any fixed Q Q this leads to 1 1 isotopy class in dimensions 1 1 and 2 2 , 3 3 classes in the dimension 4 4 quaternion case, and 6 6 classes in the dimension 8 8 octonion case.

Generalized quaternions and spacetime symmetries

The construction of a class of associative composition algebras qn on R4 generalizing the well-known quaternions Q provides an explicit representation of the universal enveloping algebra of the real three-dimensional Lie algebras having tracefree adjoint representations (class A Bianchi type Lie algebras). The identity components of the four-dimensional Lie groups GL(qn,1) ⊆qn (general linear group in one generalized quaternion dimension) which are generated by the Lie algebra of this class of quaternion algebras are diffeomorphic to the manifolds of spacetime homogeneous and spatially homogeneous spacetimes having simply transitive homogeneity isometry groups with tracefree Lie algebra adjoint representations. In almost all cases the complete group of isometries of such a spacetime is isomorphic to a subgroup of the group of left and right translations and automorphisms of the appropriate generalized quaternion algebra. Similar results hold for the single class B Lie algebra of Bianchi type V, characterized by its ‘‘pure trace’’ adjoint representation.