For a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ, C) of the solvmanifold Γ. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ, C).
We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.The proof uses a criterion for absolute continuity involving spaces with : A measure on a compact metrisable group is absolutely continuous with respect to Haar measure on if for some a certain subspace of which is related to has sufficiently many continuous linear...
Given a representation of a local unitary group and another local unitary group , either the Theta correspondence provides a representation of or we set . If is fixed and varies in a Witt tower, a natural question is: for which is ? For given dimension there are exactly two isometry classes of unitary spaces that we denote . For let us denote the minimal of the same parity of such that , then we prove that where is the dimension of .
We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.
The aim of this paper is the study of abelian Lie algebras as subalgebras of the nilpotent Lie algebra gn associated with Lie groups of upper-triangular square matrices whose main diagonal is formed by 1. We also give an obstruction to obtain the abelian Lie algebra of dimension one unit less than the corresponding to gn as a Lie subalgebra of gn. Moreover, we give a procedure to obtain abelian Lie subalgebras of gn up to the dimension which we think it is the maximum.