Addendum to "Existence of Hermitian n-Symmetric Spaces and of Non-commutative Naturally Reductive Spaces".
There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.
The Levi-Civita functional equation (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for...
Let be a del Pezzo surface of degree , and let be the simple Lie group of type . We construct a locally closed embedding of a universal torsor over into the -orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Néron-Severi torus of identified with a maximal torus of extended by the group of scalars. Moreover, the -invariant hyperplane sections of the torsor defined by the roots of are the inverse images...
Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra . Furthermore, assume that the linear adjoint action of on is diagonalizable with non-purely imaginary eigenvalues. Let . We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a submanifold of (+)*, and a...
We define partial spectral integrals on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by , where is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that converges a.e. to f(x) as R → ∞.
Let be a right-invariant sub-Laplacian on a connected Lie group and let denote the associated “spherical partial sums,” where is the spectral resolution of We prove that converges a.e. to as under the assumption
Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.