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.
We study the problem of -boundedness () of operators of the form for a commuting system of self-adjoint left-invariant differential operators on a Lie group of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when is a homogeneous group and are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.
In this paper we study Markov semigroups generated by Hörmander-Dunkl type operators on Heisenberg group.