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Journal ArticleDOI

Clifford-like parallelisms

01 Apr 2019-Journal of Geometry (Springer International Publishing)-Vol. 110, Iss: 1, pp 1-18
TL;DR: This work characterises the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms and establishes necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.
Abstract: Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space $$({\mathbb P},{\mathrel {\parallel _{\ell }}},{\mathrel {\parallel _{r}}})$$ over a quaternion skew field we characterise the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms $$\mathrel {\parallel _{\ell }}$$ and $$\mathrel {\parallel _{r}}$$ , in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,479 citations

Journal ArticleDOI
01 Mar 2021
TL;DR: In this article, the authors studied the action of the full automorphism group of a Clifford-like parallelism on parallel classes in a 3D projective double space and proved that the linear part of the group of the Clifford parallelism is the same for all Cliffordlike parallelisms which can be associated to it.
Abstract: We recall the notions of Clifford and Clifford-like parallelisms in a 3-dimensional projective double space In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones

4 citations

Posted Content
TL;DR: In this article, the automorphism group of a Clifford-like parallelism over a quaternion skew field was studied, and it was shown that over certain skew fields, it is possible to choose a group of automorphisms in such a way that the group is either properly contained in the left parallelism or coincides with the right parallelism.
Abstract: In this paper we focus on the description of the automorphism group $\Gamma_{\parallel}$ of a Clifford-like parallelism $\parallel$ on a $3$-dimensional projective double space $\bigl(\mathbb{P}(H_F),{\mathrel{\parallel_{\ell}}},{\mathrel{\parallel_{r}}}\bigr)$ over a quaternion skew field $H$ (with centre a field $F$ of any characteristic). We compare $\Gamma_{\parallel}$ with the automorphism group $\Gamma_{\ell}$ of the left parallelism $\mathrel{\parallel_{\ell}}$, which is strictly related to $\mathrm{Aut}(H)$. We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose $\parallel$ in such a way that $\Gamma_{\parallel}$ is either properly contained in $\Gamma_{\ell}$ or coincides with $\Gamma_{\ell}$ even though ${\parallel} eq{\mathrel{\parallel_{\ell}}}$.

3 citations


Cites background from "Clifford-like parallelisms"

  • ...The exposition of this topic in [12] serves as major basis for this article....

    [...]

  • ...of [12] where the construction of Cliffordlike parallelisms appears frequently in the more general framework of “blending”; this point of view will be disregarded here)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the automorphism group of a Clifford-like parallelism over a quaternion skew field was studied, and it was shown that over certain skew fields, it is possible to choose a group of automorphisms in such a way that the group is either properly contained in the left parallelism or coincides with the right parallelism.
Abstract: In this paper we focus on the description of the automorphism group $\Gamma_{\parallel}$ of a Clifford-like parallelism $\parallel$ on a $3$-dimensional projective double space $\bigl(\mathbb{P}(H_F),{\mathrel{\parallel_{\ell}}},{\mathrel{\parallel_{r}}}\bigr)$ over a quaternion skew field $H$ (with centre a field $F$ of any characteristic). We compare $\Gamma_{\parallel}$ with the automorphism group $\Gamma_{\ell}$ of the left parallelism $\mathrel{\parallel_{\ell}}$, which is strictly related to $\mathrm{Aut}(H)$. We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose $\parallel$ in such a way that $\Gamma_{\parallel}$ is either properly contained in $\Gamma_{\ell}$ or coincides with $\Gamma_{\ell}$ even though ${\parallel} eq{\mathrel{\parallel_{\ell}}}$.

1 citations

References
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Journal ArticleDOI
TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,479 citations

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TL;DR: In this article, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,090 citations

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