Bireflectionality of the weak orthogonal and the weak symplectic groups

Ellers, Erich W.; Frank, Rolfdieter; Nolte, Wolfgang

doi:10.1016/0021-8693(84)90090-5

Cited by 11 publications

(9 citation statements)

References 7 publications

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“…We refer to [15,16,39] for proofs and to [4,[9][10][11][12]14,29,37], [43, p. 18], [46], [50, 1.6.3], [53, pp. 156-159] for further details, generalisations and additional references.…”

Section: Metric Vector Spacesmentioning

confidence: 99%

Projective Metric Geometry and Clifford Algebras

Havlicek

2021

Results Math

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Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

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“…We refer to [15,16,39] for proofs and to [4,[9][10][11][12]14,29,37], [43, p. 18], [46], [50, 1.6.3], [53, pp. 156-159] for further details, generalisations and additional references.…”

Section: Metric Vector Spacesmentioning

confidence: 99%

Projective Metric Geometry and Clifford Algebras

Havlicek

2021

Results Math

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“…We observe that v, w are linearly independent (otherwise F (π 1 ) = F (π 2 ), a contradiction). Using [6, Theorem W] or [2] or [3], we may assume that B(π) = B(π 1 ) + B(π 2 ). Then and B(ρπ) = B(ι 1 ) + B(ι 2 ).…”

Section: Let π Be An Element Inmentioning

confidence: 99%

The special orthogonal group is trireflectional

Ellers

Villa

2004

Archiv der Mathematik

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Let K be a field of even characteristic, V a finite-dimensional vector space over K, and SO(V ) the special orthogonal group. Then SO(V ) is trireflectional, provided dim V > 2 and SO(V ) = O + (4, 2). Problem statement.Let V be a finite-dimensional vector space over a field K. A mapping q : V −→ K is called a quadratic form if q(αx) = α 2 q(x) for all α ∈ K and all x ∈ V and if there is a symmetric bilinear form b q : V × V −→ K such that b q (x, y) := q(x+y)−q(x)−q(y) for x, y ∈ V . We call the bilinear form b q nondegenerate or nonsingular if b q (x, y) = 0 for all y ∈ V implies x = 0. The orthogonal group of q is defined by

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“…Let G be a group. An element g ∈ G is called reversible if g −1 = xgx −1 for some x ∈ G. The terminology 'real' has also been used extensively in the literature to refer the reversible elements, for example, see [Ell77], [Ell83], [EFN84], [FZ82], [KN87b], and [KN87a], [ST05], [ST08], [TZ05], [Won66]. A non-trivial element g ∈ G is called an involution if g 2 = 1.…”

Section: Introductionmentioning

confidence: 99%

Reversible quaternionic hyperbolic isometries

Bhunia

Gongopadhyay

2020

Linear Algebra and its Applications

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Let G be a group. An element g ∈ G is called reversible if it is conjugate to g −1 within G, and called strongly reversible if it is conjugate to g −1 by an order two element of G. Let H n H be the n-dimensional quaternionic hyperbolic space. Let PSp(n, 1) be the isometry group of H n H . In this paper, we classify reversible and strongly reversible elements in Sp(n) and Sp(n, 1). Also, we prove that all the elements of PSp(n, 1) are strongly reversible.

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Bireflectionality of the weak orthogonal and the weak symplectic groups

Cited by 11 publications

References 7 publications

Projective Metric Geometry and Clifford Algebras

Projective Metric Geometry and Clifford Algebras

The special orthogonal group is trireflectional

Reversible quaternionic hyperbolic isometries

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