Compact Lie groups: Euler constructions and generalized Dyson conjecture
HTML articles powered by AMS MathViewer
- by S. L. Cacciatori, F. Dalla Piazza and A. Scotti PDF
- Trans. Amer. Math. Soc. 369 (2017), 4709-4724 Request permission
Abstract:
A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the class of all compact connected Lie groups. We present a general method for realizing their generalized Euler parameterization starting from any symmetrically embedded Lie group. Our construction is based on a detailed analysis of the geometry of these groups. As a byproduct this gives rise to an interesting connection with certain Dyson integrals. In particular, we obtain a geometry based proof of a Macdonald conjecture regarding the Dyson integrals correspondent to the root systems associated to all irreducible symmetric spaces. As an application of our general method we explicitly parameterize all groups of the class of simple, simply connected compact Lie groups. We provide a table giving all necessary ingredients for all such Euler parameterizations.References
- Shôrô Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. MR 153782
- Sergio L. Cacciatori, Francesco Dalla Piazza, and Antonio Scotti, $E_7$ groups from octonionic magic square, Adv. Theor. Math. Phys. 15 (2011), no. 6, 1605–1654. MR 2989810, DOI 10.4310/ATMP.2011.v15.n6.a2
- Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, and Antonio Scotti, Mapping the geometry of the $E_6$ group, J. Math. Phys. 49 (2008), no. 1, 012107, 19. MR 2385254, DOI 10.1063/1.2830522
- Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, and Antonio Scotti, Mapping the geometry of the $F_4$ group, Adv. Theor. Math. Phys. 12 (2008), no. 4, 889–944. MR 2420906, DOI 10.4310/ATMP.2008.v12.n4.a6
- Sergio L. Cacciatori, Bianca L. Cerchiai, Alberto Della Vedova, Giovanni Ortenzi, and Antonio Scotti, Euler angles for $G_2$, J. Math. Phys. 46 (2005), no. 8, 083512, 17. MR 2165858, DOI 10.1063/1.1993549
- S. L. Cacciatori and B. L. Cerchiai, Exceptional groups, symmetric spaces and applications to supergravity, in Group Theory: Classes, Representations and Connections, and Applications, Charles Danelling, editor, Nova Science Publisher, 2009 (2010). ISBN: 978-1-60876-175-3
- Sergio L. Cacciatori, A simple parametrization for $G_2$, J. Math. Phys. 46 (2005), no. 8, 083520, 6. MR 2166426, DOI 10.1063/1.2009627
- Stefano Bertini, Sergio L. Cacciatori, and Bianca L. Cerchiai, On the Euler angles for $\textrm {SU}(N)$, J. Math. Phys. 47 (2006), no. 4, 043510, 13. MR 2226347, DOI 10.1063/1.2190898
- Daniel Bump, Lie groups, 2nd ed., Graduate Texts in Mathematics, vol. 225, Springer, New York, 2013. MR 3136522, DOI 10.1007/978-1-4614-8024-2
- Peter J. Forrester and S. Ole Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 4, 489–534. MR 2434345, DOI 10.1090/S0273-0979-08-01221-4
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Czes Kosniowski, A first course in algebraic topology, Cambridge University Press, Cambridge-New York, 1980. MR 586943, DOI 10.1017/CBO9780511569296
- I. G. Macdonald, The volume of a compact Lie group, Invent. Math. 56 (1980), no. 2, 93–95. MR 558859, DOI 10.1007/BF01392542
- I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988–1007. MR 674768, DOI 10.1137/0513070
- E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1–18. MR 1010152, DOI 10.1007/BF01388841
- Gestur Ólafsson and Angela Pasquale, The $\textrm {Cos}^\lambda$ and $\textrm {Sin}^\lambda$ transforms as intertwining operators between generalized principal series representations of $\textrm {SL}(n+1,\Bbb K)$, Adv. Math. 229 (2012), no. 1, 267–293. MR 2854176, DOI 10.1016/j.aim.2011.08.015
- Todd Tilma and E. C. G. Sudarshan, Generalized Euler angle parametrization for $\textrm {SU}(N)$, J. Phys. A 35 (2002), no. 48, 10467–10501. MR 1947318, DOI 10.1088/0305-4470/35/48/316
- Todd Tilma and E. C. G. Sudarshan, Generalized Euler angle parameterization for $\textrm {U}(N)$ with applications to $\textrm {SU}(N)$ coset volume measures, J. Geom. Phys. 52 (2004), no. 3, 263–283. MR 2099153, DOI 10.1016/j.geomphys.2004.03.003
Additional Information
- S. L. Cacciatori
- Affiliation: Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy – and – INFN, via Celoria 16, 20133 Milano, Italy
- MR Author ID: 635454
- Email: sergio.cacciatori@uninsubria.it
- F. Dalla Piazza
- Affiliation: Dipartimento di Matematica, Università “La Sapienza”, Piazzale A. Moro 2, I-00185, Roma, Italy
- MR Author ID: 812707
- Email: f.dallapiazza@gmail.com
- A. Scotti
- Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy
- MR Author ID: 291967
- Email: antonio.scotti@gmail.com
- Received by editor(s): May 19, 2014
- Received by editor(s) in revised form: June 22, 2015, June 25, 2015, and July 20, 2015
- Published electronically: January 9, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4709-4724
- MSC (2010): Primary 22C05, 22E15, 22E46
- DOI: https://doi.org/10.1090/tran/6795
- MathSciNet review: 3632547