Collineations of Polar Spaces

1985 ◽  
Vol 37 (2) ◽  
pp. 296-309 ◽  
Author(s):  
Donald G. James
Keyword(s):  
Projective Space ◽  
Quadratic Forms ◽  
Fundamental Theorem ◽  
Valuation Ring ◽  
Finite Dimension ◽  
Vector Spaces ◽  
Polar Spaces ◽  
Hermitian Forms ◽  
Bijective Mapping ◽  
Integral Structure

The fundamental theorem of projective geometry describes the bijective collineations between two projective spaces PV and PV′ of finite dimension (greater than one) over division rings k and k′ in terms of an isomorphism φ:k → k′ and a φ-semilinear bijective mapping between the underlying vector spaces V and V′. Tits [9, Theorem 8.611] has given an extensive generalization of this theorem to embeddable polar spaces induced by polarities coming from either (σ, )-hermitian forms or from (σ, )-quadratic forms with Witt indices at least two. In another direction, Klingenberg [7] and later André [1] and Rado [8], have generalized the fundamental theorem by considering non-injective collineations. Now the isomorphism φ must be replaced by a place φ:k → k′ ∪ ∞ and an integral structure over the valuation ring A = φminus1(k′) is induced into the projective space PV.

Periodicity theorems and conjectures in hermitian K-theory

2009 ◽  
Vol 4 (1) ◽  
pp. 67-75 ◽  
Author(s):  
Max Karoubi
Keyword(s):  
Quadratic Forms ◽  
Fundamental Theorem ◽  
Hermitian Forms ◽  
K Theory

AbstractIn this appendix to [4], we state a periodicity conjecture using form parameters. This conjecture contains as particular case the fundamental theorem in hermitian K-theory proved in [6] but also some results of Bak [1], Barge and Lannes [2], Sharpe [8], for lower hermitian K-groups. The interest of this conjecture lies also in the parallel use of hermitian forms and quadratic forms with form parameters introduced by Bak.

scholarly journals The generating rank of a polar Grassmannian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ilaria Cardinali ◽  
Luca Giuzzi ◽  
Antonio Pasini
Keyword(s):  
Quadratic Forms ◽  
Vector Spaces ◽  
Hermitian Forms ◽  
Division Rings ◽  
Generating Sets ◽  
Witt Index

Abstract In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

On the axiomatic foundations of the theory of Hermitian forms

1970 ◽  
Vol 67 (2) ◽  
pp. 243-250 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Recent Work ◽  
Quadratic Forms ◽  
Hermitian Form ◽  
Major Objective ◽  
Hermitian Forms ◽  
Division Rings ◽  
General Terms ◽  
Definition Of ◽  
Simple Definition

In recent work on some topological problems (7), I was forced to adopt a complicated definition of ‘Hermitian form’ which differed from any in the literature. A recent paper by Tits(5) on quadratic forms over division rings contains a new and simple definition of these. A major objective of this paper is to formulate both these definitions in somewhat more general terms, and to show that they are equivalent.

Sign in / Sign up

Export Citation Format

Share Document