Atypical Representations of Uq(sl(N))U_{q}(sl(N)) at Roots of Unity

We show how to adapt the Gelfand-Zetlin basis for describing the atypical representation of ${\cal U}_{\displaystyle{q}}(sl(N))$ when $q$ is root of unity. The explicit construction of atypical representation is presented in details for $N=3$.Comment: 18 pages, Tex-file and 2 figures. Uuencoded, compressed and tared archive of plain tex file and postscript figure file. Upon uudecoding, uncompressing and taring, tex the file atypique.te

Duality and Representations for New Exotic Bialgebras

We find the exotic matrix bialgebras which correspond to the two non-triangular nonsingular 4x4 R-matrices in the classification of Hietarinta, namely, R_{S0,3} and R_{S1,4}. We find two new exotic bialgebras S03 and S14 which are not deformations of the of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of R_{S1,4} in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GL_{p,q}(2).Comment: 33 pages, LaTeX2e, using packages: cite,amsfonts,amsmath,subeqn; reference updated; v3: corrections in subsection 3.

A Careers Perspective on Entrepreneurship

[Excerpt] What if being an entrepreneur were treated like any other occupation—teacher, nurse, manager? What if the decision to found a new venture were thought of as one of many options that individuals consider as they try to structure a meaningful and rewarding career? How would the field of entrepreneurship research be different? In our view, there is much to be learned by conceiving of entrepreneurship not solely as a final destination, but as a step along a career trajectory. Doing so opens the study of entrepreneurship to a wider range of scholarly insights, and promises important insights for entrepreneurial practice, training, and policy. This special issue takes an important step in this direction

L\'evy Processes on Uq(g)U_q(g) as Infinitely Divisible Representations

L\'evy processes on bialgebras are families of infinitely divisible representations. We classify the generators of L\'evy processes on the compact forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then we show how the processes themselves can be reconstructed from their generators and study several classical stochastic processes that can be associated to these processes.Comment: 13 pages, LATEX file, ASI-TPA/13/99 (TU Clausthal); 6/99 (Preprint-Reihe Mathmatik, Univ. Greifswald)

Exotic Bialgebra S03: Representations, Baxterisation and Applications

The exotic bialgebra S03, defined by a solution of the Yang-Baxter equation, which is not a deformation of the trivial, is considered. Its FRT dual algebra $s03_F$ is studied. The Baxterisation of the dual algebra is given in two different parametrisations. The finite-dimensional representations of $s03_F$ are considered. Diagonalisations of the braid matrices are used to yield remarkable insights concerning representations of the L-algebra and to formulate the fusion of finite-dimensional representations. Possible applications are considered, in particular, an exotic eight-vertex model and an integrable spin-chain model.Comment: 24 pages, Latex; V2: revised subsection 4.1, added 9 references, to appear in Annales Henri Poincare in the volume dedicated to D. Arnaudo

The Spectrum of Yang Mills on a Sphere

In this note, we determine the representation content of the free, large N, SU(N) Yang Mills theory on a sphere by decomposing its thermal partition function into characters of the irreducible representations of the conformal group SO(4,2). We also discuss the generalization of this procedure to finding the representation content of N=4 Super Yang Mills.Comment: 18 pages v2. references added. typos fixe

Explicit Character Formulae for Positive Energy UIRs of D=4 Conformal Supersymmetry

This paper continues the project of constructing the character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). In the first paper we gave the bare characters which represent the defining odd entries of the characters. Now we give the full explicit character formulae for N=1 and for several important examples for N=2 and N=4.Comment: 48 pages, TeX with Harvmac, overlap in preliminaries with arXiv:hep-th/0406154; some comments and references adde

Invariant Differential Operators and Characters of the AdS_4 Algebra

The aim of this paper is to apply systematically to AdS_4 some modern tools in the representation theory of Lie algebras which are easily generalised to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models. Here we introduce the necessary representations of the AdS_4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS_4 Verma modules. These are used to obtain the AdS_4 invariant differential operators. Using this we display a new structure - a diagram involving four partially equivalent reducible representations one of which contains all finite-dimensional irreps of the AdS_4 algebra. We study in more detail the cases involving UIRs, in particular, the Di and the Rac singletons, and the massless UIRs. In the massless case we discover the structure of sets of 2s_0-1 conserved currents for each spin s_0 UIR, s_0=1,3/2,... All massless cases are contained in a one-parameter subfamily of the quartet diagrams mentioned above, the parameter being the spin s_0. Further we give the classification of the so(5,C) irreps presented in a diagramatic way which makes easy the derivation of all character formulae. The paper concludes with a speculation on the possible applications of the character formulae to integrable models.Comment: 30 pages, 4 figures, TEX-harvmac with input files: amssym.def, amssym.tex, epsf.tex; version 2 1 reference added; v3: minor corrections; v.4: minor corrections, v.5: minor corrections to conform with version in J. Phys. A: Math. Gen; v.6.: small correction and addition in subsections 4.1 & 4.