556 research outputs found
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}).Comment: V2: minor changes, and chapter 3: new results included; to appear in
Forum Mathematicu
Weil Spaces and Weil-Lie Groups
We define Weil spaces, Weil manifolds, Weil varieties and Weil Lie groups
over an arbitrary commutative base ring K (in particular, over discrete rings
such as the integers), and we develop the basic theory of such spaces, leading
up the definition of a Lie algebra attached to a Weil Lie group. By definition,
the category of Weil spaces is the category of functors from K-Weil algebras to
sets; thus our notion of Weil space is similar to, but weaker than the one of
Weil topos defined by E. Dubuc (1979). In view of recent result on Weil
functors for manifolds over general topological base fields or rings by A.
Souvay, this generality is the suitable context to formulate and to prove
general results of infinitesimal differential geometry, as started by the
approach developed in Bertram, Mem. AMS 900
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 r\^ole of associative algebras and -pairs from loc. cit. is now
taken by analogs of near-rings
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.Comment: 28 pages, 2nd corrected versio
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