scholarly journals Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

2016 ◽  
Vol 2016 (715) ◽  
Author(s):  
Bart De Bruyn ◽  
Hendrik Van Maldeghem
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Polar Spaces ◽  
Dual Polar Spaces

AbstractSuppose 𝕆 is an alternative division algebra that is quadratic over some subfield 𝕂 of its center

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

scholarly journals Minimal scattered sets and polarized embeddings of dual polar spaces

2007 ◽  
Vol 28 (7) ◽  
pp. 1890-1909 ◽  
Author(s):  
Bart De Bruyn ◽  
Antonio Pasini
Keyword(s):  
Polar Spaces ◽  
Dual Polar Spaces

scholarly journals On the Automorphisms of the Four-dimensional Real Division Algebras

2017 ◽  
Vol 9 (2) ◽  
pp. 95
Author(s):  
Andre S. Diabang ◽  
Alassane Diouf ◽  
Mankagna A. Diompy ◽  
Alhousseynou Ba
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Automorphisms Groups

In this paper, we study partially the automorphisms groups of four-dimensional division algebra. We have proved that there is an equivalence between Der(A)=su(2) and Aut(A)=SO(3). For an unitary four-dimensional real division algebra, there is an equivalence between dim(Der(A))=1 and Aut(A)=SO(2).

scholarly journals Opposite Relation on Dual Polar Spaces and Half-Spin Grassmann Spaces

2009 ◽  
Vol 54 (3-4) ◽  
pp. 301-308 ◽  
Author(s):  
Mariusz Kwiatkowski ◽  
Mark Pankov
Keyword(s):  
Polar Spaces ◽  
Dual Polar Spaces

scholarly journals Non-Classical Hyperplanes of Finite Thick Dual Polar Spaces

10.37236/7348 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Complete Classification ◽  
Polar Spaces ◽  
Dual Polar Spaces

We obtain a classification of the nonclassical hyperplanes of all finite thick dual polar spaces of rank at least 3 under the assumption that there are no ovoidal and semi-singular hex intersections. In view of the absence of known examples of ovoids and semi-singular hyperplanes in finite thick dual polar spaces of rank 3, the condition on the nonexistence of these hex intersections can be regarded as not very restrictive. As a corollary, we also obtain a classification of the nonclassical hyperplanes of $DW(2n-1,q)$, $q$ even. In particular, we obtain a complete classification of all nonclassical hyperplanes of $DW(2n-1,q)$ if $q \in \{ 8,32 \}$.

scholarly journals Dual polar spaces of arbitrary rank

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Simon Huggenberger
Keyword(s):  
Polar Spaces ◽  
Arbitrary Rank ◽  
Dual Polar Spaces

A Note on Subnormal Subgroups of Division Algebras

1978 ◽  
Vol 30 (01) ◽  
pp. 161-163 ◽  
Author(s):  
Gary R. Greenfield
Keyword(s):  
Division Algebra ◽  
Multiplicative Group ◽  
Subnormal Subgroup ◽  
Local Fields ◽  
Division Algebras ◽  
Subnormal Subgroups

Let D be a division algebra and let D* denote the multiplicative group of nonzero elements of D. In [3] Herstein and Scott asked whether any subnormal subgroup of D* must be normal in D*. Our purpose here is to show that division algebras over certain p-local fields do not satisfy such a “subnormal property”.

scholarly journals MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250052 ◽  
Author(s):  
CÉCILE COYETTE
Keyword(s):  
Division Algebra ◽  
Laurent Series ◽  
Elementary Proof ◽  
Formal Series ◽  
Direct Approach ◽  
Division Algebras ◽  
Sufficient Condition ◽  
Series Ring

The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].

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