Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invariants of $\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.Comment: 20 pages, 2 Appendice

Electronic marking and identification techniques to discourage document copying

Modern computer networks make it possible to distribute documents quickly and economically by electronic means rather than by conventional paper means. However, the widespread adoption of electronic distribution of copyrighted material is currently impeded by the ease of illicit copying and dissemination. In this paper we propose techniques that discourage illicit distribution by embedding each document with a unique codeword. Our encoding techniques are indiscernible by readers, yet enable us to identify the sanctioned recipient of a document by examination of a recovered document. We propose three coding methods, describe one in detail, and present experimental results showing that our identification techniques are highly reliable, even after documents have been photocopied

Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup

Integrating heterogeneous distributed COTS discrete-event simulation packages: An emerging standards-based approach

This paper reports on the progress made toward the emergence of standards to support the integration of heterogeneous discrete-event simulations (DESs) created in specialist support tools called commercial-off-the-shelf (COTS) discrete-event simulation packages (CSPs). The general standard for heterogeneous integration in this area has been developed from research in distributed simulation and is the IEEE 1516 standard The High Level Architecture (HLA). However, the specific needs of heterogeneous CSP integration require that the HLA is augmented by additional complementary standards. These are the suite of CSP interoperability (CSPI) standards being developed under the Simulation Interoperability Standards Organization (SISO-http://www.sisostds.org) by the CSPI Product Development Group (CSPI-PDG). The suite consists of several interoperability reference models (IRMs) that outline different integration needs of CSPI, interoperability frameworks (IFs) that define the HLA-based solution to each IRM, appropriate data exchange representations to specify the data exchanged in an IF, and benchmarks termed CSP emulators (CSPEs). This paper contributes to the development of the Type I IF that is intended to represent the HLA-based solution to the problem outlined by the Type I IRM (asynchronous entity passing) by developing the entity transfer specification (ETS) data exchange representation. The use of the ETS in an illustrative case study implemented using a prototype CSPE is shown. This case study also allows us to highlight the importance of event granularity and lookahead in the performance and development of the Type I IF, and to discuss possible methods to automate the capture of appropriate values of lookahead