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Zap. Nauchn. Sem. POMI, 2001, Volume 281, Pages 60–104 (Mi znsl1489)  

This article is cited in 18 scientific papers (total in 18 papers)

Do it yourself structure constants for Lie algebras of types $\mathrm E_l$

N. A. Vavilov

Saint-Petersburg State University

Abstract: We compare two algorithms to compute structure table of the Lie algebras of type $\mathrm E_l$ with respect to a Chevalley base: the usual inductive algorithm and an algorithm based on the use of Frenkel–Kac cocycle. It turns out that Frenkel–Kac algorithm is several dozen times faster but with the ‘natural’ choice of the bilinear form and the sign function it does not give the result in a positive Chevalley base. We show how to modify the sign function to get the right choice of structure constants. Cohen, Griess and Lisser obtained similar result by varying the bilinear form. We recall the hyperbolic realisation of root systems of type $\mathrm E_l$ which dramatically simplify calculations as compared with the usual Euclidean realisation. We give Matematica definitions which realise root systems and implement both the inductive and Frenkel–Kac algorithms. Using these definitions one can compute the whole structure table for $\mathrm E_8$ within a quarter of an hour at a home computer. At the end of the paper we reproduce tables of roots according to HeightLex and the resulting tables of structure constants.

Full text: PDF file (2607 kB)

English version:
Journal of Mathematical Sciences (New York), 2004, 120:4, 1513–1548

Bibliographic databases:

UDC: 513.6
Received: 05.06.2001
Language: English

Citation: N. A. Vavilov, “Do it yourself structure constants for Lie algebras of types $\mathrm E_l$”, Problems in the theory of representations of algebras and groups. Part 8, Zap. Nauchn. Sem. POMI, 281, POMI, St. Petersburg, 2001, 60–104; J. Math. Sci. (N. Y.), 120:4 (2004), 1513–1548

Citation in format AMSBIB
\Bibitem{Vav01}
\by N.~A.~Vavilov
\paper Do it yourself structure constants for Lie algebras of types~$\mathrm E_l$
\inbook Problems in the theory of representations of algebras and groups. Part~8
\serial Zap. Nauchn. Sem. POMI
\yr 2001
\vol 281
\pages 60--104
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1489}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1875718}
\zmath{https://zbmath.org/?q=an:1062.17004}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 120
\issue 4
\pages 1513--1548
\crossref{https://doi.org/10.1023/B:JOTH.0000017882.04464.97}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$ i $\mathrm E_7$”, Algebra i analiz, 16:4 (2004), 54–87  mathnet  mathscinet  zmath; N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-proof of structure theorem for Chevaller groups of type $\mathrm E_6$ and $\mathrm E_7$”, St. Petersburg Math. J., 16:4 (2005), 649–672  crossref
    2. N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Gruppa Shevalle tipa $\mathrm E_6$ v 27-mernom predstavlenii”, Voprosy teorii predstavlenii algebr i grupp. 14, Zap. nauchn. sem. POMI, 338, POMI, SPb., 2006, 5–68  mathnet  mathscinet  zmath  elib; N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation”, J. Math. Sci. (N. Y.), 145:1 (2007), 4697–4736  crossref  elib
    3. N. A. Vavilov, N. P. Kharchev, “Orbity stabilizatora podsistem”, Voprosy teorii predstavlenii algebr i grupp. 14, Zap. nauchn. sem. POMI, 338, POMI, SPb., 2006, 98–124  mathnet  mathscinet  zmath  elib; N. A. Vavilov, N. P. Kharchev, “Orbits of subsystem stabilisers”, J. Math. Sci. (N. Y.), 145:1 (2007), 4751–4764  crossref
    4. N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Stroenie grupp Shevalle: Dokazatelstvo iz Knigi”, Voprosy teorii predstavlenii algebr i grupp. 13, Zap. nauchn. sem. POMI, 330, POMI, SPb., 2006, 36–76  mathnet  mathscinet  zmath  elib; N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book”, J. Math. Sci. (N. Y.), 140:5 (2007), 626–645  crossref  elib
    5. N. A. Vavilov, “Kak uvidet znaki strukturnykh konstant?”, Algebra i analiz, 19:4 (2007), 34–68  mathnet  mathscinet  zmath; N. A. Vavilov, “Can one see the signs of structure constants?”, St. Petersburg Math. J., 19:4 (2008), 519–543  crossref  isi
    6. Vavilov N., “An A(3)-proof of structure theorems for Chevalley groups of types E-6 and E-7”, International Journal of Algebra and Computation, 17:5–6 (2007), 1283–1298  crossref  mathscinet  zmath  isi
    7. N. Vavilov, “Vesovye elementy grupp Shevalle”, Algebra i analiz, 20:1 (2008), 34–85  mathnet  mathscinet  zmath  elib; N. Vavilov, “Weight elements of Chevalley groups”, St. Petersburg Math. J., 20:1 (2009), 23–57  crossref  isi
    8. N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya gruppy Shevalle tipa $\mathrm F_4$”, Algebra i analiz, 20:4 (2008), 27–63  mathnet  mathscinet  zmath  elib; N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$”, St. Petersburg Math. J., 20:4 (2009), 527–551  crossref  isi
    9. N. A. Vavilov, “Numerologiya kvadratnykh uravnenii”, Algebra i analiz, 20:5 (2008), 9–40  mathnet  mathscinet  zmath; N. A. Vavilov, “Numerology of square equations”, St. Petersburg Math. J., 20:5 (2009), 687–707  crossref  isi
    10. N. Vavilov, A. Luzgarev, A. Stepanov, “Calculations in exceptional groups over rings”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XVII, Zap. nauchn. sem. POMI, 373, POMI, SPb., 2009, 48–72  mathnet; J. Math. Sci. (N. Y.), 168:3 (2010), 334–348  crossref
    11. N. A. Vavilov, “Some more exceptional numerology”, J. Math. Sci. (N. Y.), 171:3 (2010), 317–321  mathnet  crossref
    12. Distler J., Garibaldi S., “There is No “Theory of Everything” Inside E-8”, Comm Math Phys, 298:2 (2010), 419–436  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    14. N. A. Vavilov, A. Yu. Luzgarev, “Gruppa Shevalle tipa $\mathrm E_7$ v 56-mernom predstavlenii”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 5–99  mathnet; N. A. Vavilov, A. Yu. Luzgarev, “Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251  crossref
    15. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. II”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 242–264  mathnet; I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. II”, J. Math. Sci. (N. Y.), 180:3 (2012), 338–350  crossref
    16. I. M. Pevzner, “Geometriya kornevykh elementov v gruppakh tipa $\mathrm E_6$”, Algebra i analiz, 23:3 (2011), 261–309  mathnet  mathscinet  zmath  elib; I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  crossref  isi  elib
    17. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. I”, Algebra i analiz, 23:5 (2011), 155–198  mathnet  mathscinet  elib; I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. I”, St. Petersburg Math. J., 23:5 (2012), 891–919  crossref  isi  elib
    18. N. A. Vavilov, “$\mathrm A_3$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$$\mathrm E_7$. II. Osnovnaya lemma”, Algebra i analiz, 23:6 (2011), 1–31  mathnet  mathscinet  elib; N. A. Vavilov, “An $\mathrm A_3$-proof of the structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$. II. The main lemma”, St. Petersburg Math. J., 23:6 (2012), 921–942  crossref  isi  elib
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