Introduction to lie algebras and representation theory

representations, parameterized by t > 0: J− en = √ n(t + n− 1) en−1, J+ en = √ (n + 1)(t + n) en+1, J0 en = (t + 2n)en Want J− + J+ + cJ0 = X

https://www.math.tamu.edu/~manshel/papers/Orthogonal-poly.pdf

Introduction to orthogonal polynomials

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

This article is devoted to the perturbative renormalization of the abelian Higgs-Kibble model, within the class of renormalizable gauges which are odd under charge conjugation. The Bogoliubov Parasiuk Hepp-Zimmermann renormalization scheme is used throughout, including the renormalized action principle proved by Lowenstein and Lam. The whole study is based on the fulfillment to all orders of perturbation theory of the Slavnov identities which express the invariance of the Lagrangian under a supergauge type family of non-linear transformations involving the Faddeev-Popov ghosts. Direct combinatorial proofs are given of the gauge independence and unitarity of the physicalS operator. Their simplicity relies both on a systematic use of the Slavnov identities as well as suitable normalization conditions which allow to perform all mass renormalizations, including those pertaining to the ghosts, so that the theory can be given a setting within a fixed Fock space. Some simple gauge independent local operators are constructed.

/pdf/renormalization-of-the-abelian-higgs-kibble-model-14fr0vp7oj.pdf

Renormalization of the Abelian Higgs-Kibble Model

We describe H ULLAC , an integrated code for calculating atomic structure and cross sections for collisional and radiative atomic processes. This code evolved and has been used over the years, but so far, there was no coherent, comprehensive, and in-depth presentation of it. It is based on relativistic quantum mechanical calculations including configuration interaction. The collisional cross sections are calculated in the distorted wave approximation. The theory and code are presented, emphasizing the various novel methods that has been developed to obtain accurate results very efficiently. In particular we describe the parametric potential method used for both bound and free orbitals, the factorization–interpolation method applied in the derivation of collisional rates, the phase amplitude approach for calculating the continuum orbitals and the N JGRAF graphical method used in the calculation of the angular momentum part of the matrix elements. Special effort has been made to insure the simplicity of use, which is demonstrated in an example.

HULLAC, an integrated computer package for atomic processes in plasmas

Theinterpretation of the Meixner--Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra ${\frak{su}}(1,1)$ and the Clebsch--Gordan decomposition lead to generalizations of the convolution identities for these polynomials. Using the Racah coefficients, convolution identities for continuous Hahn, Hahn, and Jacobi polynomials are obtained. From the quantized universal enveloping algebra for ${\frak{su}}(1,1)$, convolution identities for the Al-Salam and Chihara polynomials and the Askey--Wilson polynomials are derived by using the Clebsch--Gordan and Racah coefficients. For the quantized universal enveloping algebra for ${\frak{su}}(2)$, q-Racah polynomials are interpreted as Clebsch--Gordan coefficients, and the linearization coefficients for a two-parameter family of Askey--Wilson polynomials are derived.

Convolutions for orthogonal polynomials from Lie and quantum algebra representations.

Kac distinguished between typical and atypical finite‐dimensional irreducible representations of the Lie superalgebras sl(m/n) and provided an explicit character formula appropriate to all the typical representations. Here, the range of validity of some character formulas for atypical representations that have been proposed are discussed. Several of them are of the Kac–Weyl type, but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl(3/4). Having ruled out, therefore, all such formulas, a completely new extension of the Kac–Weyl character formula is proposed. The validity of this formula in the case of all covariant tensor irreducible representations is proved, and some evidence in support of the conjecture that it covers all irreducible representations of sl(m/n) is presented.

Character formulas for irreducible modules of the Lie superalgebras sl(m/n)

(1990). A character formula for singly atypical modules of the lie superalgebra sl(m/n) Communications in Algebra: Vol. 18, No. 10, pp. 3453-3480.

A character formula for singly atypical modules of the lie superalgebra sl(m/n)

The shift operator technique is used to give a complete analysis of all finite‐ and infinite‐dimensional irreducible representations of the exceptional Lie superalgebrasD(2,1;α). For all cases, the star or grade star conditions for the algebra are investigated. Among the finite‐dimensional representations there are no star and only a few grade star representations, but an infinite class of infinite‐dimensional star representations is found. Explicit expressions are given for the ‘‘doublet’’ representation of D(2,1;α). The one missing label problem D(2,1;α)→su(2)+su(2)+su(2) is discussed in detail and solved explicitly.

J. Van der Jeugt

Papers

Convolutions for orthogonal polynomials from Lie and quantum algebra representations.

Character formulas for irreducible modules of the Lie superalgebras sl(m/n)

A character formula for singly atypical modules of the lie superalgebra sl(m/n)

Irreducible representations of the exceptional Lie superalgebras D(2,1;α)

Convolutions for orthogonal polynomials from Lie and quantum algebra representations