quantum groups

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1: 17.17 Physical Applications

… ►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys. ►Quantum groups also apply $q$-series extensively. Quantum groups are really not groups at all but certain Hopf algebras. They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. …

2: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

3: Bibliography K

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Quantum Groups. Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94370-6, MathReview (Yu. N. Bespalov), ZentralBlatt

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§17.17.

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Askey-Wilson polynomials as zonal spherical functions on the $\mathrm{SU}(2)$ quantum group. SIAM J. Math. Anal. 24 (3), pp. 795–813.

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ISSN 0036-1410, Document, Link, MathReview Entry

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(18.28.27).

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Compact quantum groups and $q$-special functions. In Representations of Lie Groups and Quantum Groups, Pitman Res. Notes Math. Ser., Vol. 311, pp. 46–128.

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Notes:

Sections 3 and 4 also available as arXiv:math/9403216

External Links:

MathReview (Eric M. Opdam)

Referenced by:

(18.27.14_3).

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4: Bibliography V

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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii), Ch.35.

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5: Bibliography W

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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Notes:

Expanded and improved ed. Translated from the German by J. J. Griffin.

External Links:

ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt

Referenced by:

§34.12.

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6: 12.17 Physical Applications

… ► … ►Miller (1974) treats separation of variables by group theoretic methods. Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. …

7: Bibliography H

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Quantum theory for mathematicians. Graduate Texts in Mathematics, Vol. 267, Springer, New York.

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External Links:

ISBN 978-1-4614-7115-8; 978-1-4614-7116-5, Document, Link, MathReview Entry

Referenced by:

§1.18(x).

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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

External Links:

ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

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§1.2(vi).

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Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. 2nd edition, Noordhoff International Publishing, Leyden.

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Notes:

The first edition was published in 1965 by Prentice-Hall. The 2nd edition was reissued in 1983 by Springer, and in 1986 by Martinus Nijhoff Publishers (Kluwer Academic Publishers Group).

External Links:

MathReview (Chia-Shun Yih), ZentralBlatt

Referenced by:

§10.73(i).

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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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Document

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§8.24(iii).

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8: Bibliography P

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A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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Document, ZentralBlatt

Referenced by:

§29.18(iv).

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Introduction to quantum mechanics. Dover Publications, Inc., New York.

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Notes:

With applications to chemistry, Reprint of the 1935 original

External Links:

ISBN 0-486-64871-0, MathReview Entry

Referenced by:

§1.18(v), §18.39(i), §18.39(ii), §18.39(ii).

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Ideas of Quantum Chemistry. second edition, Elsevier, Amsterdam-New York.

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ISBN 978-0-444-59436-5

Referenced by:

§18.39(ii).

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Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer-Verlag, New York.

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Notes:

Pólya’s contribution translated from the German by Dorothee Aeppli

External Links:

ISBN 0-387-96413-4, MathReview (Daniel Turzík)

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§26.20.

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9: Bibliography D

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The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.

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External Links:

Document, MathReview (T. Erber)

Referenced by:

§12.11(i), §12.11(i), §12.17.

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On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.

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Notes:

Variable-precision Algol program, maximum precision 15D

External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§28.36(ii), §28.36(iii).

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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.

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Notes:

Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997

External Links:

ISBN 0-8218-1198-3, MathReview, ZentralBlatt

Referenced by:

§21.9.

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Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141 (1), pp. 55–147.

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ISSN 0020-9910, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.

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ISSN 0002-9947, Link, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.38(iii).

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10: Bibliography G

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The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

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Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.

External Links:

GAP

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2nd item.

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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..

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ISSN 1751-8113, Document, Link, MathReview (Brian Simanek)

Referenced by:

§18.36(vi), §18.36(vi).

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Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt

Referenced by:

§18.39(i).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

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Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics. Texts and Monographs in Physics, Springer.

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Referenced by:

§33.22(iv).

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