Tangent Groups Research Articles

Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a C∞-hypersurface S without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure H12. As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in S, has negligible image with respect to the Hausdorff measure H12. In particular, we deduce that S cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset U of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz map f:U→S satisfies H12(f(U))=0. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all C∞-hypersurfaces in Hn with n≥2 are countably Hn−1×R-rectifiable according to Pauls’ definition, even with bi-Lipschitz maps.

Geometry of configurations in tangent groups

This article relates the Grassmannian complexes of geometric configurations to the tangent to the Bloch-Suslin complex and to the tangent to Goncharov's motivic complex. By means of morphisms, we bring the geometry of configurations in tangent groups, $T\mathcal{B}_2(F)$ and $T\mathcal{B}_3(F)$ that produce commutative diagrams. To show the commutativity of diagrams, we use combinatorial techniques that include permutations in symmetric group S6. We also create analogues of the Siegel's cross-ratio identity for the truncated polynomial ring F[e]ν.V

Infinitesimal and tangent to polylogarithmic complexes for higher weight

Motivic and polylogarithmic complexes have deep connections with $K$-theory. This article gives morphisms (different from Goncharov's generalized maps) between $\mathbb{k}$-vector spaces of Cathelineau's infinitesimal complex for weight $n$. Our morphisms guarantee that the sequence of infinitesimal polylogs is a complex. We are also introducing a variant of Cathelineau's complex with the derivation map for higher weight $n$ and suggesting the definition of tangent group $T\mathcal{B}_n(\mathbb{k})$. These tangent groups develop the tangent to Goncharov's complex for weight $n$.

Tangent maps and tangent groupoid for Carnot manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold (M,H) we get a smooth groupoid that encodes the smooth deformation of the pair M×M to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].

Tangents to Chow groups: on a question of Green–Griffiths

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch–Gersten–Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their “tangent sequences,” expressed in terms of absolute Kahler differentials. More generally, we place Green and Griffiths’ concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective K-theory, negative cyclic homology, and the relative algebraic Chern character.

Lagrangian dynamics on matched pairs

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler–Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler–Poincaré equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group S L ( 2 , C ) .

Iterative calculus on tangent floors

Abstract Tangent fibrations generate a “multi-floored tower”, while raising from one of its floors to the next one, one practically reiterates the previously performed actions. In this way, the "tower" admits a ladder-shaped structure. Raising to the first floors suffices for iteratively performing the subsequent steps. The paper mainly studies the tangent functor. We describe the structure of multiple vector bundle which naturally appears on the floors, tangent maps, sector-forms, the lift of vector fields to upper floors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant differentiation. Finally, we will point out within the floors special subbundles - the osculating bundles, which play an essential role in classical theories.

Who ordered the anti-de Sitter tangent group?

Abstract General relativity can be unambiguously formulated with Lorentz, de Sitter and anti-de Sitter tangent groups, which determine the fermionic representations. We show that besides of the Lorentz group only anti-de Sitter tangent group is consistent with all physical requirements.

Gravity with de Sitter and unitary tangent groups

Einstein Gravity can be formulated as a gauge theory with the tangent space respecting the Lorentz symmetry. In this paper we show that the dimension of the tangent space can be larger than the dimension of the manifold and by requiring the invariance of the theory with respect to 5d Lorentz group (de Sitter group) Einstein theory is reproduced unambiguously. The other possibility is to have unitary symmetry on a complex tangent space of the same dimension as the manifold. In this case the resultant theory is Einstein-Strauss Hermitian gravity. The tangent group is important for matter couplings. We show that in the de Sitter case the 4 dimensional space time vector and scalar are naturally unified by a hidden symmetry being components of a 5d vector in the tangent space. With a de Sitter tangent group spinors can exist only when they are made complex or taken in doublets in a way similar to N=2 supersymmetry.

Homology of tangent groups considered as discrete groups and scissors congruence

Extensions of results of Sah (1986, 1989) and Bökstedt et al., joined with previous results of the author, give a new approach to a homological result 1 1 Another direct proof of this homological result is due to Dupont and Sah. known, by Dupont, to be equivalent to the Sydler's theorem on scissors congruences of polyhedras in Euclidean three-space.

Quasi-Riemannian gravity and spontaneous breaking of the Lorentz gauge symmetry in more than four dimensions.

It is shown that a pure gravity theory in d dimensions, with an action quadratic in torsion and curvature, may lead to a spontaneous breaking of the SO(1,d-1) gauge symmetry. The physical vacuum, corresponding to a minimum of the self-interaction torsion potential, is characterized by a constant nonvanishing torsion background, and in the lowest-order expansion of the gravitational field around this configuration, an effective quasi-Riemannian theory is obtained. In particular, all nine parameters of the corresponding quasi-Riemannian action are determined, as a function of the torsion self-interaction coupling strength, for the tangent space groups SO(1,d-4) x SO(3) and SO(1,d-2). Finally, the possibility that a breaking of the local Lorentz symmetry may be associated to a change of sign of the effective four-dimensional gravitational coupling constant is discussed.

A graded generalization of Lie triples

A graded generalization of Lie triples is defined such that the well-known relations between Lie triples and Lie algebras remain valid. For instance, the graded generalizations of Lie algebras, considered recently in physics and mathematics, become such triples with respect to the graded double commutator, and every such triple can be constructed by means of an involutive automorphism of degree zero on such an algebra as eigenspace of eigenvalue −1. In case of a Z 2 -graduation there is an elementary example, considered first in the second quantization in quantum field theory, which is constructed on a graded vector space V with a graded symmetric bilinear form <, >. This triple has a realization in the Clifford algebra constructed over ( V, <, >). An elementary construction of representations which in the (Lie) group case leads to inhomogenizations and tangent groups can be generalized to these triples as well.

Generalized tangent representations for groups and symmetric spaces of matrices

Any two representations of dimensions n resp. r of a given group G allow the construction of a third representation φ in the space of rectangular n × r matrices Kn,r over the same ground field K. The φ-semidirect product of Kn,r and G then has (n + r) dimensional representation. The inhomogenizations of G and in case of matrix Lie groups G the tangent groups are special cases of this construction. The contragredient as well as the Lie algebraical versions of these results are included. In the final section the construction is generalized to symmetric spaces and their local algebraical structures, the Lie triples, by defining semidirect products resp. semidirect sums with respect to a representation