Skip to main content
SpringerLink
Log in

Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac–Moody (super)algebras are the most known examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. (Translator’s mistake. Should be “the”. And the author’s initials are S.A., not A.A.).

References

  1. Albuquerque, H., Barreiro, E., Benayadi, S.: Quadratic Lie superalgebras with reductive even part. J. Pure Appl. Algebra 213, 724–731 (2009)

    Article  MathSciNet  Google Scholar 

  2. Benamor, H., Benayadi, S.: Double extension of quadratic Lie superalgebras. Commun. Algebra 27(1), 67–88 (1999)

    Article  MathSciNet  Google Scholar 

  3. Benayadi, S., Bouarroudj, S.: Double extensions of Lie superalgebra in characteristic 2 with nondegenerate invariant supersymmetric bilinear form. J. Algebra 510, 141–179 (2018)

    Article  MathSciNet  Google Scholar 

  4. Błaszak, M., Szablikowski, M.: Classical R-matrix theory for bi-Hamiltonian field systems. J. Phys. A Math. Theor. 42, 404002 35pp. (2009). arXiv:0902.1511

    MathSciNet  MATH  Google Scholar 

  5. Bouarroudj, S., Grozman, P., Lebedev, A., Leites, D.: Derivations and central extensions of simple modular Lie algebras and superalgebras. arXiv:1307.1858

  6. Bouarroudj, S., Grozman, P., Leites, D.: New simple modular Lie superalgebras as generalized Cartan prolongations. Funct. Anal. Appl. 42(3), 161–168 (2008). arXiv:math.RT/0704.0130

    Article  MathSciNet  Google Scholar 

  7. Bouarroudj, S., Grozman, P., Leites, D.: Classification of finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 5, 060, 63 (2009). arXiv:math.RT/0710.5149

    MathSciNet  MATH  Google Scholar 

  8. Bouarroudj, S., Grozman, P., Leites, D.: Deforms of symmetric simple modular Lie superalgebras. arXiv:0807.3054

  9. Bouarroudj, S., Grozman, P., Lebedev, A., Leites, D., Shchepochkina, I.: New simple Lie algebras in characteristic 2. IMRN 18, 5695–5726 (2016). arXiv:1307.1551

    Article  MathSciNet  Google Scholar 

  10. Bouarroudj, S., Grozman, P., Lebedev, A., Leites, D., Shchepochkina, I.: Simple vectorial Lie algebras in characteristic 2 and their superizations. arXiv:1510.07255

  11. Bouarroudj, S., Lebedev, A., Leites, D., Shchepochkina, I.: Lie algebra deformations in characteristic 2. Math. Res. Lett. 122(2), 353–402 (2015). arXiv:1301.2781

    Article  MathSciNet  Google Scholar 

  12. Bouarroudj, S., Lebedev, A., Leites, D.: Shchepochkina I. Classification of simple Lie superalgebras in characteristic 2. arXiv:1407.1695

  13. Bouarroudj, S., Lebedev, A., Wagemann, F.: Deformations of the Lie algebra \(\mathfrak {o}(5)\) in characteristics 3 and 2. Math. Zametki 89(6), 809–924 (2011). English transl.: Math. Notes: 89: 6 (2011), 777–791; arXiv:0909.3572

    MathSciNet  MATH  Google Scholar 

  14. Bouarroudj, S., Leites, D.: Simple Lie superalgebras and and nonintegrable distributions in characteristic p. Zapiski nauchnyh seminarov POMI 331, 15–29 (2006). Reprinted in J. Math. Sci. (NY) 141 (2007) no.4, 1390–98; arXiv:math.RT/0606682

    MATH  Google Scholar 

  15. Bouarroudj, S., Leites, D.: Invariant differential operators in positive characteristic. J. Algebra 499, 281–297 (2018). arXiv:1605.09500

    Article  MathSciNet  Google Scholar 

  16. Cantarini, N., Kac, V.G.: Classification of linearly compact simple rigid superalgebras. IMRN 17, 3341–3393 (2010). arXiv:0909.3100

    MathSciNet  MATH  Google Scholar 

  17. Chapovalov, D., Chapovalov, M., Lebedev, A., Leites, D.: The classification of almost affine (hyperbolic) Lie superalgebras. J. Nonlinear Math. Phys. 17(1), 103–161 (2010). arXiv:0906.1860

    Article  MathSciNet  Google Scholar 

  18. Chebochko, N.G., Kuznetsov, M.I.: Integrable cocycles and global deformations of Lie algebra of type G 2 in characteristic 2. Commun. Algebra. 45(7), 2969–2977 (2017)

    Article  Google Scholar 

  19. Dzhumadildaev, A.: Central extensions and invariant forms of Cartan type Lie algebras of positive characteristic. Funkts. Anal. Prilozh. 18(4), 77–78 (1984). (in Russian); Funct. Anal. Appl. 18 (1984), no. 4, 331–332 (English translation)

    Google Scholar 

  20. Dzhumadildaev, A.: Generalized Casimir elements. Izv. Akad. Nauk SSSR Ser. Mat. 49(5), 1107–1117 (1985). (in Russian); Math. USSR Izvestiya 27 (1986), no. 2, 391–400 (English translation)

    MathSciNet  Google Scholar 

  21. Dzhumadildaev, A.: Central extensions of infinite-dimensional Lie algebras. Funkts. Anal. Prilozh. 26(4), 21–29 (1992). (in Russian); Funct. Anal. Appl. 26 (1992), no-4, 247–253 (English translation)

    MathSciNet  Google Scholar 

  22. Dzhumadildaev, A.S.: Central extensions of the Lie algebra of formal pseudodifferential operators. (Russian) Algebra i Analiz 6(1), 140–158 (1994). translation in St Derivations Petersburg. Math. J. 6 (1994), no. 1, 121–136

    MathSciNet  Google Scholar 

  23. Dzhumadildaev, A.S., Kostrikin, A.I.: Deformations of the Lie algebra W 1(m). Trudy Mat. Inst. steklov 1148, 141–155 (1978). (Russian) English translation: Proceedings of the Steklov Institute of Mathematics, v. 148 (1980) 143–158

    Google Scholar 

  24. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons, p. 592. Springer, Berlin (2007)

    MATH  Google Scholar 

  25. Farnsteiner, R.: The associative forms of the graded Cartan type Lie algebras. Trans. Amer. Math. Soc. 295, 417–427 (1986)

    Article  MathSciNet  Google Scholar 

  26. Favre, G., Santharoubane, L.J.: Symmetric, invariant, non-degenerate bilinear form on a Lie algebra. J. Algebra 105, 451–464 (1987)

    Article  MathSciNet  Google Scholar 

  27. Fulton, W., Harris, J.: Representation Theory. A First Course. Graduate Texts in Mathematics, 129. Readings in Mathematics, p. xvi+551. Springer, New York (1991)

    Google Scholar 

  28. Garibaldi, S.R., Premet, A.A.: Vanishing of trace forms in low characteristics. Algebra Number Theory 3(5), 543–566 (2009). arXiv:0712.3764

    Article  MathSciNet  Google Scholar 

  29. Grozman, P. http://www.equaonline.com/math/SuperLie

  30. Grozman, P., Leites, D.: Defining relations associated with the principal \(\mathfrak {s}\mathfrak {l}(2)\)-subalgebras. In: Dobrushin, R., Minlos, R., Shubin, M., Vershik, A. (eds.) Contemporary Mathematical Physics (F. A. Berezin memorial volume), Amer. Math. Soc. Transl. Ser. 2, vol. 175, Amer. Math. Soc., Providence, RI, pp. 57–67 (1996). arXiv:math-ph/0510013

  31. Grozman, P., Leites, D.: Structures of G(2) type and nonintegrable distributions in characteristic p. Lett. Math. Phys. 74(3), 229–262 (2005). arXiv:math.RT/0509400

    Article  MathSciNet  Google Scholar 

  32. Grozman, P., Leites, D., Shchepochkina, I.: Lie superalgebras of string theories. Acta Mathematica Vietnamica 26(1), 27–63 (2001). arXiv:hep-th/9702120

    MathSciNet  MATH  Google Scholar 

  33. Hoyt, C., Serganova, V.: Classification of finite-growth general Kac-Moody superalgebras. Comm. Algebra 35(3), 851–874 (2007)

    Article  MathSciNet  Google Scholar 

  34. Kac, V.G.: Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated. Math. USSR-Izvestiya 8(4), 801–835 (1974)

    Article  Google Scholar 

  35. Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edition, p xxii+400. Cambridge UP, Cambridge (1995)

    Google Scholar 

  36. Kac, V.: Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. 139(1), 1–55 (1998)

    Article  MathSciNet  Google Scholar 

  37. Kac, V.: Classification of supersymmetries. In: Proceedings of the International Congress of Mathematicians, v. I (Beijing, 2002), Higher Ed. Press, Beijing, pp. 319–344 (2002). arXiv:math-ph/0302016

  38. Kac, V., Van de Leur, J., et al.: On Classification of Superconformal Algebras. In: Gates, S.J. (ed.) Strings-88, pp. 77–106. World Scientific, Singapore (1989)

  39. Kaplansky, I.: Graded Lie algebras I, II, preprints, Univ. Chicago, Chicago, Ill. see http://www1.osu.cz/zusmanovich/links/files/kaplansky/ (1975)

  40. Kirillov, S.A.: Sandwich subalgebras in Cartan type Lie algebras. Russian Math. (Iz. VUZ) 36(4), 16–23 (1992)

    MathSciNet  MATH  Google Scholar 

  41. Kiselev, A.V., Krutov, A.O.: Non-abelian Lie algebroids over jet spaces. J. Nonlin. Math. Phys. 121(12), 188–213 (2014). arXiv:1305.4598

    Article  MathSciNet  Google Scholar 

  42. Konstein, S.E., Stekolshchik, R.: Klein operator and the number of independent traces and supertraces on the superalgebra of observables of rational Calogero model based on the root system. J Nonlinear Math. Phys. 20, 295–308 (2013)

    Article  MathSciNet  Google Scholar 

  43. Konstein, S.E., Tyutin, I.: The number of independent traces and supertraces on symplectic reflection algebras. J. Nonlinear Math. Phys. 20, 295–308 (2013)

    Article  MathSciNet  Google Scholar 

  44. Kostrikin, A.I., Dzhumadildaev, A.S.: Modular Lie Algebras: New Trends. In: Bahturin, Y. (ed.) Algebra Proceedings of the International Algebraic Conference on the Occasion of the 90Th Birthday of A.G. Kurosh (May, 1998. Moscow), de Gruyter, Berlin, pp. 181–203 (2000)

  45. Krutov, A., Lebedev, A.: On gradings modulo 2 of simple Lie algebras in characteristic 2. arXiv:1711.00638

  46. Kuznetsov, M.I., Chebochko, N.G.: Deformations of classical Lie algebras. Sb. Math. 191(7-8), 1171–1190 (2000)

    Article  MathSciNet  Google Scholar 

  47. Leites, D.: New Lie superalgebras and mechanics. Soviet Math. Doklady 18(5), 1277–1280 (1977)

    MATH  Google Scholar 

  48. Leites, D. (ed.): Seminar on supersymmetries (v. 1: Algebra and Calculus on supermanifolds. MCCME, Moscow (2011). (in Russian)

    Google Scholar 

  49. Leites, D., Saveliev, M., Serganova, V.: Embeddings of \(\mathfrak {o}\mathfrak {s}\mathfrak {p} (N|2)\) and Completely Integrable Systems. In: Markov, M., Man’ko, V. (eds.) Proceedings of the International Conference Group-Theoretical Methods in Physics, Yurmala, May, 1985. Nauka, Moscow, 1986, 377–394 MR 89h:17042 (English translation: VNU Sci Press), pp. 255–297 (1987)

  50. Leites, D., Sergeev, A.: Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices. Theor. and Math. Physics 123(2), 205–236 (2000). (Russian), 582–609; arXiv:math.RT/0509528

    Article  Google Scholar 

  51. Leites, D., Shchepochkina, I.: The classification of simple Lie superalgebras of vector fields. Preprint MPIM-2003-28 http://www.mpim-bonn.mpg.de/preblob/2178 For a short version, see Toward classification of simple vectorial Lie superalgebras. Seminar on Supermanifolds, Reports of Stockholm University, 31/1988-14, 235–278. In: Nahm, W., Chau, L. (eds.) Differential geometric methods in theoretical physics (Davis, CA, 1988), NATO Adv. Sci. Inst. Ser. B Phys., 245, Plenum, New York, pp. 633–651 (1990)

  52. Leites, D., Shchepochkina, I.: The Howe duality and Lie superalgebras. In: Duplij, S., Wess, J. (eds.) Noncommutative structures in mathematics and physics, Proc. NATO Advanced Research Workshop, Kiev, pp. 93–112. Kluwer (2001). arXiv:math.RT/0202181

    Chapter  Google Scholar 

  53. Leites, D., Shchepochkina, I.: How should an antibracket be quantized?. Theoret. Math. Phys. 126(3), 281–306 (2001). arXiv:math-ph/0510048

    Article  MathSciNet  Google Scholar 

  54. Manin, Y.: Gauge field theory and complex geometry. 2nd edition. Grundlehren der Mathematischen Wissenschaften, 289, p xii+346. Springer, Berlin (1997)

    Book  Google Scholar 

  55. Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant. Ann. Scient. É,c. Norm. Sup., 4 série 18, 553–561 (1985)

    MATH  Google Scholar 

  56. Nahm, W., Scheunert, M.: On the structure of simple pseudo Lie algebras and their invariant bilinear forms. J of Mathematical Physics 17(6) 1976. https://doi.org/10.1063/1.522999

    Article  MathSciNet  Google Scholar 

  57. Pinczon, G., Ushirobira, R.: New applications of graded Lie algebras to Lie algebras, generalized Lie algebras, and cohomology. J. Lie Theory 17(3), 633–667 (2007). arXiv:math/0507387

    MathSciNet  MATH  Google Scholar 

  58. Poletaeva E.: A spinor-like representation of the contact superconformal algebra K ′(4). J. Math.Phys. 42, 526–540 (2001). arXiv:hep-th/0011100

    Article  MathSciNet  Google Scholar 

  59. Serganova, V.: Automorphisms of simple Lie superalgebras. Math. USSR-Izvestiya 24(3), 539–551 (1985)

    Article  Google Scholar 

  60. Shchepochkina, I.M.: Five simple exceptional Lie superalgebras of vector fields. Funct. Anal. Appl. 33(3), 208–219 (1999)

    Article  MathSciNet  Google Scholar 

  61. Shchepochkina, I.: The five exceptional simple Lie superalgebras of vector fields and their fourteen regradings. Represent. Theory 3, 373–415 (1999)

    Article  MathSciNet  Google Scholar 

  62. Shchepochkina, I.: Maximal subalgebras of matrix Lie superalgebras. In: Duval, C., Guieu, L., Ovsienko, V. (eds.) The orbit method in geometry and physics. In honor of A. A. Kirillov. Papers from the International Conference held in Marseille, December 4–8, 2000.Progress in Mathematics, 213. Birkhäuser Boston, Inc., Boston, MA, pp. 445–472 (2003). arXiv:1311.4131

  63. Shchepochkina I.: How to realize Lie algebras by vector fields. Theor. Mat. Fiz. 147(3), 821–838 (2006). arXiv:math.RT/0509472

    Article  MathSciNet  Google Scholar 

  64. Skryabin S.: Normal shapes of Hamiltonian and contact forms over algebras of divided powers. VINITI deposition 8504-B86 (in Russian)

  65. Skryabin, S.: Classification of Hamiltonian forms over divided power algebras. Math. USSR-Sbornik 69(1), 121–141 (1991)

    Article  MathSciNet  Google Scholar 

  66. Strade, H.: Simple Lie algebras over fields of positive characteristic. I. Structure theory. de Gruyter Expositions in Mathematics, v. 38. Walter de Gruyter & Co., Berlin, (2004) viii+540 pp

  67. Strade H., Farnsteiner R.: Modular Lie algebras and their representations. Marcel Dekker, 1988. viii+301pp

  68. Tyurin, A.A.: Classification of deformations of aFootnote

    (Translator’s mistake. Should be “the”. And the author’s initials are S.A., not A.A.).

    special Lie algebra of Cartan type. Math. notes of the Acad. of Sci. of the USSR 24(6), 948–954 (1978)

    Article  Google Scholar 

  69. Tyutin I.: The general form of the ∗-commutator on the Grassmann algebra. Theor. Math. Phys. 128, 1271–1292 (2001). arXiv:hep-th/0101068

    Article  MathSciNet  Google Scholar 

  70. Wilson, R.L.: Simple Lie algebras of type S. J. Algebra 62(2), 292–298 (1980)

    Article  MathSciNet  Google Scholar 

  71. Zusmanovich P.: Non-existence of invariant symmetric forms on generalized Jacobson–Witt algebras revisited. Comm. Algebra 39, 548–554 (2011). arXiv:0902.0038

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Science and Mathematics, New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates

    Sofiane Bouarroudj & Dimitry Leites

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland

    Andrey Krutov

  3. Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden

    Dimitry Leites

  4. Independent University of Moscow, Bolshoj Vlasievsky per, dom 11, RU-119 002, Moscow, Russia

    Andrey Krutov & Irina Shchepochkina

Authors
  1. Sofiane Bouarroudj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrey Krutov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dimitry Leites
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Irina Shchepochkina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sofiane Bouarroudj.

Additional information

Presented by: Valentin Ovsienko

To Alexandre Kirillov-père, our teacher

S.B. and A.K. were partly supported by the grant AD 065 NYUAD. A.K. was partly supported by WCMCS post-doctoral fellowship. A part of this research was done while A.K. was visiting NYUAD; the financial support and warm atmosphere of this institute are gratefully acknowledged. We are thankful to J. Bernstein, P. Grozman, S. Skryabin, P. Zusmanovich, and especially A. Lebedev, for help. For the possibility to conduct difficult computations of this research we are grateful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouarroudj, S., Krutov, A., Leites, D. et al. Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras. Algebr Represent Theor 21, 897–941 (2018). https://doi.org/10.1007/s10468-018-9802-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-018-9802-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation