From linear algebra via affine algebra to projective algebra

Bertram, Wolfgang

doi:10.1016/j.laa.2003.06.021

Cited by 16 publications

(34 citation statements)

References 7 publications

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“…2.2.2 for the multiplication maps and then prove (PG2) in a suitable affinization, see [8]. In the course of that proof, one sees that (PG2) is indeed the geometric analog of the Fundamental Formula.…”

Section: Examplementioning

confidence: 88%

Is There a Jordan Geometry Underlying Quantum Physics?

Bertram

2008

Int J Theor Phys

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There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics, see, e.g.. From a purely mathematical side, the point of view of Jordan algebra theory might give new strength to such approaches: there is a "Jordan geometry" belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.

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Section: Examplementioning

confidence: 88%

Is There a Jordan Geometry Underlying Quantum Physics?

Bertram

2008

Int J Theor Phys

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“…In case (X + , X − ) = (KP n , (KP n ) * ) is an ordinary projective geometry, the action of the translation group on the hyperplane at infinity is the trivial action. However, already in the case of more general Grassmannian geometries this is no longer true, as can be seen from the explicit formulas for this case given in [2]. Corollary 3.8.…”

Section: Theorem 37 the Actionsmentioning

confidence: 99%

“…Then, by elementary linear algebra, the pair (C × C, P) has the main features of a generalized projective geometry (Proposition 8.2, cf. also [2]), and in fact there is a homomorphism into the geometry (F × F , G) with g = gl R (V ) which induces isomorphisms on subgeometries that are homogeneous under the elementary projective groups (Theorem 8.4). Such geometries, called Grassmannian geometries, correspond to special Jordan pairs, i.e., to subpairs of associative pairs.…”

Section: Introductionmentioning

confidence: 99%

Projective completions of Jordan pairs

Bertram

Neeb

2004

Journal of Algebra

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We prove that the projective completion (X + , X − ) of the Jordan pair (g 1 , g −1 ) corresponding to a 3-graded Lie algebra g = g 1 ⊕ g 0 ⊕ g −1 can be realized inside the space F of inner 3-filtrations of g in such a way that the remoteness relation on X + × X − corresponds to transversality of flags. This realization is used to give geometric proofs of structure results which will be used in Part II of this work in order to define on X + and X − the structure of a smooth manifold (in arbitrary dimension and over general base fields or -rings).

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“…Remark: the structure of the structure maps. In the case of the Grassmann geometry one can give a fairly explicit formula for the structure maps Π r , and one can find algebraic identities that, in some sense, characterize these maps (see [Be02], [Be04]). For k > 2 , no results of this kind seem to be known.…”

Section: Proofmentioning

confidence: 99%

Inner ideals and intrinsic subspaces of linear pair geometries

Bertram¹,

Löwe²

2008

Advg

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Abstract. We introduce the notion of intrinsic subspaces of linear and affine pair geometries, which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [BeNe04], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic subspaces defined by the Peirce-decomposition which is related to 5 -gradings of the projective Lie algebra. These examples, as well as the examples of general and Lagrangian flag geometries, lead to the conjecture that geometries of intrinsic subspaces tend to be themselves linear pair geometries.Contents:

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From linear algebra via affine algebra to projective algebra

Cited by 16 publications

References 7 publications

Is There a Jordan Geometry Underlying Quantum Physics?

Is There a Jordan Geometry Underlying Quantum Physics?

Projective completions of Jordan pairs

Inner ideals and intrinsic subspaces of linear pair geometries

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