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Wolfgang Bertram

Publications and source records attributed to Wolfgang Bertram.

At least 37 records · Page 2Linked to original sources
arXiv2012

Homotopes of Symmetric Spaces II. Structure Variety and Classification

We classify homotopes of classical symmetric spaces (studied in Part I of this work). Our classification uses the fibered structure of homotopes: they are fibered as symmetric spaces, with flat fibers, over a non-degenerate base; the base spaces correspond to inner ideals in Jordan pairs. Using that inner ideals in classical Jordan pairs are always complemented (in the sense defined by O. Loos and E. Neher), the classification of homotopes is obtained by combining the classification of inner ideals with the one of isotopes of a given inner ideal.

math.DG↗
arXiv2012

The projective geometry of a group

We show that the pair given by the power set and by the "Grassmannian"(set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize several results from preceding joint work with M. Kinyon (arXiv:0903.5441), which concerned abelian groups, to the case of general non-abelian groups. Most notably, pairs of subgroups parametrize torsor and semitorsor structures on the power set. The rôle of associative algebras and -pairs from loc. cit. is now taken by analogs of near-rings.

math.GR↗
arXiv2012

A general construction of Weil functors

We construct the Weil functor $T^A$ corresponding to a general Weil algebra $A = K \oplus N$: this is a functor from the category of manifolds over a general topological base field or ring $K$ (of arbitrary characteristic) to the category of manifolds over $A$. This result simultaneously generalizes results known for ordinary, real manifolds, and previous results by the first author for the case of the higher order tangent functors ($A = T^k K$) and for the case of jet rings ($A = K[X]/(X^{k+1})$). We investigate some algebraic aspects of these general Weil functors ("K-theory of Weil functors", action of the "Galois group" $\Aut_K(A)$), which will be of importance for subsequent applications to general differential geometry.

math.DG↗
arXiv2011

Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors

We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, "cubic" approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).

math.DG↗
arXiv2010

Associative Geometries. I: Torsors, linear relations and Grassmannians

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

math.RA↗
arXiv2010

Associative Geometries. II: Involutions, the classical torsors, and their homotopes

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.

math.RA↗
arXiv2010

Associative Geometries. I: Torsors, linear relations and Grassmannians

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

math.RA↗
arXiv2010

Associative Geometries. II: Involutions, the classical torsors, and their homotopes

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.

math.RA↗
arXiv2009

Symmetric bundles and representations of Lie triple systems

We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. We show that such a bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing "unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.

math.DG↗
arXiv2008

On the Hermitian projective line as a home for the geometry of Quantum Theory

In the paper "Is there a Jordan geometry underlying quantum physics?" (Int. J. Theor. Phys., to appear; arXiv:0801.3069 [math-ph]), generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum Theory. In the present note, we refine this proposition by discussing further structural features of Quantum Theory: the link with associative involutive algebras, and with Jordan-Lie and Lie-Jordan algebas. The associated geometries are (Hermitian) projective lines over an associative algebra; their axiomatic definition and theory will be given in subsequent work with M. Kinyon.

math-ph↗
arXiv2008

Is there a Jordan geometry underlying quantum physics?

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, 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.

math-ph↗
arXiv2007

Difference Problems and Differential Problems

We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a significant contribution to the understanding of the foundations of differential and integral calculus.

math.GM↗
arXiv2007

Jordan structures and non-associative geometry

In this survey paper we give an overview over constructions of geometries associated to Jordan structures (algebras, triple systems and pairs), featuring analogs of these constructions with the Lie functor on the one hand and with the approach of non-commutative geometry on the other hand.

math.RA↗
arXiv2007

Homotopes and Conformal Deformations of Symmetric Spaces

Homotopy is an important feature of associative and Jordan algebraic structures: such structures always come in families whose members need not be isomorphic among other, but still share many important properties. One may regard homotopy as a special kind of deformation of a given algebraic structure. In this work, we investigate the global counterpart of this phenomenon on the geometric level of the associated symmetric spaces -- on this level, homotopy gives rise to conformal deformations of symmetric spaces. These results are valid in arbitrary dimension and over general base fields and -rings.

math.RA↗
arXiv2006

Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the characteristic. Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

math.DG↗
arXiv2006

Inner Ideals and Intrinsic Subspaces of Linear Pair Geometries

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 [Bertram-Neeb, J. Alg. 277], 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.

math.RA↗
arXiv2004

Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ``compact-like'' duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as "standard models" -- they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.

math.GR↗
arXiv2003

Projective completions of Jordan pairs. Part I: The generalized projective geometry of a Lie algebra

A geometric realization of the projective completion of the Jordan pair corresponding to a three-graded Lie algebra is given which permits to develop a geometric structure theory of the projective completion. This will be used in Part II of this work to define a manifold structure on the projective completion (in arbitrary dimension and over quite general base fields and -rings).

math.RA↗