Quadratic Division Algebras Research Articles

Rota\u2013Baxter Operators on Quadratic Algebras

We prove that all Rota–Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota–Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota–Baxter operators and the solutions to the alternative Yang–Baxter equation on the Cayley–Dickson algebra. We also investigate the Rota–Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.

How to Obtain Non-Flexible Quadratic Algebras as Mutation Algebras of Hurwitz Algebras

We present an elementary way to obtain new families of division algebras of degree n out of division algebras with a multiplicative norm. Families of four- and eight-dimensional non-flexible quadratic division algebras over a field F of characteristic not 2 are easily constructed as a-mutation algebras of Hurwitz division algebras A over F, choosing \({a \in A \setminus F}\) suitable. In particular, we thus obtain real eight-dimensional division algebras with large derivation algebras.

Classification of the four-dimensional power-commutative real division algebras

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterized.

The degree of an eight-dimensional real quadratic division algebra is 1, 3, or 5

A celebrated theorem of Hopf (1940) [11], Bott and Milnor (1958) [1], and Kervaire (1958) [12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified (Dieterich (2005) [6], Dieterich (1998) [3], Dieterich and Öhman (2002) [9]), the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp. It was conjectured in Dieterich et al. (2006) [7].

Liftings of dissident maps

We study dissident maps η on R m for m ∈ { 3 , 7 } by investigating liftings Φ : R m → R m of the selfbijection η P : P ( R m ) → P ( R m ) , η P [ v ] = ( η ( v ∧ R m ) ) ⊥ induced by η . Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting Φ whose component functions are homogeneous polynomials of degree d , relatively prime and without non-trivial common zero. We prove that 1 ⩽ d ⩽ m - 2 . We achieve a complete description of all dissident maps of degree one and we solve their isomorphism problem (Theorems 4.8 and 4.13). As a consequence, we achieve a complete description of all real quadratic division algebras of degree one and we solve their isomorphism problem (Theorems 5.1 and 5.3). Moreover we present examples of eight-dimensional real quadratic division algebras of degree 3 and 5 (Proposition 6.3). This extends earlier results of Osborn [Trans. Amer. Math. Soc. 105 (1962) 202–221], Hefendehl [Geometriae Dedicata 9 (1980) 129–152], Hefendehl-Hebeker [Arch. Math. 40 (1983) 50–60], Cuenca Mira et al. [Lin. Alg. Appl. 290 (1999) 1–22], Dieterich [Proc. Amer. Math. Soc. 128 (2000) 3159–3166] and Dieterich and Lindberg [Colloq. Math. 97 (2003) 251–276] on the classification of real quadratic division algebras.

On the doubling of quadratic algebras

The concept of doubling, which was introduced around 1840 by Graves and Hamilton, associates with any quadratic algebraA over a eld k of characteristic not 2 its doubleV(A) =A A with multiplication (w;x)(y;z) = (wy zx;xy +zw). This yields an endofunctor on the category of all quadratic k-algebras which is faithful but not full. We study in which respect the division property of a quadratic k-algebra is preserved under doubling and, provided this is the case, whether the doubles of two non-isomorphic quadratic division algebras are again non-isomorphic. Generalizing a theorem of Dieterich (9) from R to arbitrary square-ordered ground elds k we prove that the division property of a quadratic k-algebra of dimension smaller than or equal to 4 is preserved under doubling. Generalizing an aspect of the celebrated (1,2,4,8)-theorem of Bott, Milnor (4) and Kervaire (21) from R to arbitrary ground elds k of characteristic not 2 we prove that the division property of an 8-dimensional doubled quadratic k-algebra is never preserved under doubling. Finally, we contribute to a solution of the still open problem of classifying all 8-dimensional real quadratic division algebras by extending an approach of Dieterich and Lindberg (12) and proving that, under a mild additional assumption, the doubles of two non-isomorphic 4-dimensional real quadratic division algebras are again non-isomorphic.

ON THE CLASSIFICATION OF $4$ -DIMENSIONAL QUADRATIC DIVISION ALGEBRAS OVER SQUARE-ORDERED FIELDS

A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all 4-dimensional quadratic division algebras over a square-ordered field k is shown to be equivalent to the problem of finding normal forms for all pairs (X, Y) of 3 × 3 matrices over k, X being antisymmetric and Y being positive definite, under simultaneous conjugation by SO3(k). A solution is derived for the subproblem of this matrix pair problem defined by requiring Y+Yt to be orthogonally diagonalizable. The classifying list is given in terms of a 9-parameter family of configurations in k3,\sformed by a pair of points and an ellipsoid in normal position. Each 4-dimensional quadratic division algebra A over a square-ordered field k is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism α of its purely imaginary hyperplane. Calling A diagonalizable in case α is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable 4-dimensional quadratic division k-algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those 4-dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real 4-dimensional quadratic division algebras. Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a 4-parameter family of pairs of definite 3 × 3 matrices over k, embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.

Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)

In his remarkable article “Quadratic division algebras” (Trans. Amer. Math. Soc. 105 (1962), 202–221), J. M. Osborn claims to solve ‘the problem of determining all quadratic division algebras of order 4 over an arbitrary field F F of characteristic not two … \ldots modulo the theory of quadratic forms over F F ’ (cf. p. 206). While we shall explain in which respect he has not achieved this goal, we shall on the other hand complete Osborn’s basic results (by a reasoning which is finer than his) to derive in the real ground field case a classification of all 4-dimensional quadratic division algebras and the construction of a 49-parameter family of pairwise nonisomorphic 8-dimensional quadratic division algebras. To make these points clear, we begin by reformulating Osborn’s fundamental observations on quadratic algebras in categorical terms.

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.

Bonded quadratic division algebras

Bonded quadratic division algebras

Quadratic Division Algebras

Quadratic Division Algebras