H. F. Jones, Groups, Representations and Physics, 2nd ed., IOP Publishing (1998). A fairly easy going introduction. H. Georgi, Lie Algebras in Particle Physics, Perseus Books (1999). Describes the basics of Lie algebras for classical groups. J. Fuchs and C. Schweigert, Symmetries, Lie Algebras and Representations, 2nd ed., CUP (2003). This is more comprehensive and more mathematically sophistic...

The infinitesimal generators of Lévy processes in Euclidean space are pseudo-differential operators with symbols given by the Lévy-Khintchine formula. This classical analysis relies heavily on Fourier analysis which in the case when the state space is a Lie group becomes much more subtle. Still the notion of pseudo-differential operators can be extended to connected, simply connected nilpotent ...

We consider the Fer and the Magnus expansions for the numerical solution of the nonlinear matrix Lie-group ODE y 0 = (t; y)y; y(0) = y0; where y evolves in a matrix Lie group G and (t; y) is in the Lie algebra g. Departing from a geometrical approach, that distinguishes between those operations performed in the group and those performed in the tangent space, we construct Lie-group invariant met...

We classify realizations of the Lie algebras of the rotation O(3) and Euclid E(3) groups within the class of first-order differential operators in arbitrary finite dimensions. It is established that there are only two distinct realizations of the Lie algebra of the group O(3) which are inequivalent within the action of a diffeomorphism group. Using this result we describe a special subclass of ...

The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin’s pure braid group, a Lie algebra analyzed in work of Kohno [12, 13, 14], and Falk-Randell [9]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criteri...

A set of coordinates in the non parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic p...