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Dissident algebras

100%
Dieterich E.
Colloquium Mathematicum
|
1999
|
tom 82
|
nr 1
13-23
EN
Given a euclidean vector space V = (V,〈〉) and a linear map η: V ∧ V → V, the anti-commutative algebra (V,η) is called dissident in case η(v ∧ w) ∉ ℝv ⊕ ℝw for each pair of non-proportional vectors (v,w) ∈ $V^2$. For any dissident algebra (V,η) and any linear form ξ: V ∧ V → ℝ, the vector space ℝ × V, endowed with the multiplication (α,v)(β,w) = (αβ -〈v,w〉+ ξ(v ∧ w), αw + βv + η(v ∧ w)), is a quadratic division algebra. Up to isomorphism, each real quadratic division algebra arises in this way. Vector product algebras are classical special cases of dissident algebras. Via composition with definite endomorphisms they produce new dissident algebras, thus initiating a construction of dissident algebras in all possible dimensions m ∈ {0,1,3,7} and of real quadratic division algebras in all possible dimensions n ∈ {1,2,4,8}. For m ≤ 3 and n ≤ 4, this construction yields complete classifications. For m=7 it produces a 28-parameter family of pairwise non-isomorphic dissident algebras. For n=8 it produces a 49-parameter family of pairwise non-isomorphic real quadratic division algebras.
2
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Existence and construction of two-dimensional invariant subspaces for pairs of rotations

100%
Dieterich E.
Colloquium Mathematicum
|
2009
|
tom 114
|
nr 2
203-211
EN
By a rotation in a Euclidean space V of even dimension we mean an orthogonal linear operator on V which is an orthogonal direct sum of rotations in 2-dimensional linear subspaces of V by a common angle α ∈ [0,π]. We present a criterion for the existence of a 2-dimensional subspace of V which is invariant under a given pair of rotations, in terms of the vanishing of a determinant associated with that pair. This criterion is constructive, whenever it is satisfied. It is also used to prove that every pair of rotations in V has a 2-dimensional invariant subspace if and only if the dimension of V is congruent to 2 modulo 4.
3
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A general approach to finite-dimensional division algebras

100%
Dieterich E.
Colloquium Mathematicum
|
2012
|
tom 126
|
nr 1
73-86
EN
We present a short and rather self-contained introduction to the theory of finite-dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In Sections 2-3 we develop the general theory over an arbitrary ground field k, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories 𝒟ₙ(k), and the role played by the irreducible morphisms. Sections 4-5 deal with the classical case of real division algebras, emphasizing the double sign decomposition of the level subcategories 𝒟ₙ(ℝ) for n ∈ {2,4,8} and the problem of describing their blocks, along with an account of known partial solutions to this problem.
4
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Dissident maps on the seven-dimensional Euclidean space

63%
Dieterich E., Lindberg L.
Colloquium Mathematicum
|
2003
|
tom 97
|
nr 2
251-276
EN
Our article contributes to the classification of dissident maps on ℝ ⁷, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on ℝ ⁷. The first class is formed by all composed dissident maps, obtained from a vector product on ℝ ⁷ by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on ℝ ⁷.
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