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Gradings of non-graded Hamiltonian Lie algebras

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

A. Caranti  and
S. Mattarei
A. Caranti
Affiliation:
Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38050 Povo (Trento), Italy, e-mail: caranti@science.unitn.it, mattarei@science.unitn.it
S. Mattarei
Affiliation:
Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38050 Povo (Trento), Italy, e-mail: caranti@science.unitn.it, mattarei@science.unitn.it
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Abstract

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A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; ω2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n; ω2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).

Keywords

Modular Lie algebrasgraded Lie algebrasderivationscentral extensionsloop algebras.

MSC classification

Secondary: 17B50: Modular Lie (super)algebras 17B70: Graded Lie (super)algebras 17B56: Cohomology of Lie (super)algebras 17B65: Infinite-dimensional Lie (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 3 , December 2005 , pp. 399 - 440
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Albert, A. A. and Frank, M. S., ‘Simple Lie algebras of characteristic p’, Univ. e Politec. Torino. Rend. Sem. Mat. 14 (1954–1955), 117–139.Google Scholar
[2]Avitabile, M., Some loop algebras of Hamiltonian Lie algebras (Ph.D. Thesis, Trento, 11 1999).Google Scholar
[3]Avitabile, M., ‘Some loop algebras of Hamiltonian Lie algebras’, Internat. J. Algebra Comput. (4) 12 (2002), 535–567.CrossRefGoogle Scholar
[4]Avitabile, M. and Jurman, G., ‘Diamonds in thin Lie algebras’, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) (3) 4 (2001), 597–608.Google Scholar
[5]Avitabile, M. and Mattarei, S., ‘Thin Lie algebras with diamonds of finite and infinite type’, J. Algebra 293 (2005), 34–64.CrossRefGoogle Scholar
[6]Avitabile, M., ‘Thin loop algebras of Albert-Zassenhaus algebras’, preprint.Google Scholar
[7]Benkart, G., Kostrikin, A. I. and Kuznetsov, M. I., ‘Finite-dimensional simple Lie algebras with a nonsingular derivation’, J. Algebra (3) 171 (1995), 894–916.CrossRefGoogle Scholar
[8]Benkart, G. and Osborn, J. M., ‘Toral rank one Lie algebras’, J. Algebra (1) 115 (1988), 238–250.CrossRefGoogle Scholar
[9]Benkart, G. M., Gregory, T. B., Osborn, J. M., Strade, H. and Wilson, R. L., ‘Isomorphism classes of Hamiltonian Lie algebras’, in: Lie algebras, Madison 1987, Lecture Notes in Math. 1373 (Springer, Berlin, 1989) pp. 42–57.CrossRefGoogle Scholar
[10]Benkart, G. M. and Moody, R. V., ‘Derivations, central extensions, and affine Lie algebras’, Algebras Groups Geom. (4) 3 (1986), 456–492.Google Scholar
[11]Block, R., ‘New simple Lie algebras of prime characteristic’, Trans. Amer. Math. Soc. 89 (1958), 421–449.CrossRefGoogle Scholar
[12]Block, R. E., ‘Determination of the differentiably simple rings with a minimal ideal.’, Ann. of Math. (2) 90 (1969), 433–459.CrossRefGoogle Scholar
[13]Block, R. E. and Wilson, R. Lee, ‘The simple Lie p-algebras of rank two’, Ann. of Math. (2) (1) 115 (1982), 93–168.CrossRefGoogle Scholar
[14]Brandl, R., ‘The Dilworth number of subgroup lattices’, Arch. Math. (Basel) (6) 50 (1988), 502–510.CrossRefGoogle Scholar
[15]Brandl, R., Caranti, A. and Scoppola, C. M., ‘Metabelian thin p-groups’, Quart. J. Math. Oxford Ser. (2) (170) 43 (1992), 157–173.CrossRefGoogle Scholar
[16]Caranti, A., ‘Presenting the graded Lie algebra associated to the Nottingham group’, J. Algebra (1) 198 (1997), 266–289.CrossRefGoogle Scholar
[17]Caranti, A., ‘Thin groups of prime-power order and thin Lie algebras: an addendum’, Quart. J. Math. Oxford Ser. (2) (196) 49 (1998), 445–450.CrossRefGoogle Scholar
[18]Caranti, A., ‘Loop algebras of Zassenhaus algebras in characteristic three’, Israel J. Math. 110 (1999), 61–73.CrossRefGoogle Scholar
[19]Caranti, A. and Jurman, G., ‘Quotients of maximal class of thin Lie algebras. The odd characteristic case’, Comm. Algebra (12) 27 (1999), 5741–5748.CrossRefGoogle Scholar
[20]Caranti, A. and Mattarei, S., ‘Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class’, J. Austral. Math. Soc. Ser. A (2) 67 (1999), 157–184 Group theory.CrossRefGoogle Scholar
[21]Caranti, A. and Mattarei, S., ‘Nottingham Lie algebras with diamonds of finite type’, Internat. J. Algebra Comput. (1) 14 (2004), 35–67.CrossRefGoogle Scholar
[22]Caranti, A., Mattarei, S. and Newman, M. F., ‘Graded Lie algebras of maximal class’, Trans. Amer. Math. Soc. (10) 349 (1997), 4021–4051.CrossRefGoogle Scholar
[23]Caranti, A., Mattarei, S., Newman, M. F. and Scoppola, C. M., ‘Thin groups of prime-power order and thin Lie algebras’, Quart. J. Math. Oxford Ser. (2) (187) 47 (1996), 279–296.CrossRefGoogle Scholar
[24]Caranti, A. and Newman, M. F., ‘Graded Lie algebras of maximal class. II’, J. Algebra (2) 229 (2000), 750–784.CrossRefGoogle Scholar
[25]Carrara, C., ‘(Finite) presentations of the Albert-Frank-Shalev Lie algebras’, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) (2) 4 (2001), 391–427.Google Scholar
[26]Dzhumadil'daev, A. S., ‘Central extensions and invariant forms of Lie algebras of positive characteristic of Cartan types’, Funktsional. Anal. i Prilozhen, (4) 18 (1984), 77–78.Google Scholar
[27]Dzhumadil'daev, A. S., ‘Central extensions of the Zassenhaus algebra and their irreducible representations’, Mat. Sb. (N.S.) (4) 126 (168) (1985), 473–489, 592.Google Scholar
[28]Farnsteiner, R., ‘The associative forms of the graded Cartan type Lie algebras’, Trans. Amer. Math. Soc. (1) 295 (1986), 417–427.CrossRefGoogle Scholar
[29]Farnsteiner, R., ‘Central extensions and invariant forms of graded Lie algebras’, Algebras Groups Geom. (4) 3 (1986), 431–455.Google Scholar
[30]Farnsteiner, R., ‘Dual space derivations and H2 (L, F) of modular Lie algebras’, Canad. J. Math. (5) 39 (1987), 1078–1106.CrossRefGoogle Scholar
[31]Jurman, G., ‘Quotients of maximal class of thin Lie algebras. The case of characteristic two’, Comm. Algebra (12) 27 (1999), 5749–5789.CrossRefGoogle Scholar
[32]Jurman, G., ‘A family of simple Lie algebras in characteristic two’, J. Algebra (2) 271 (2004), 454–481.CrossRefGoogle Scholar
[33]Jurman, G., ‘Graded Lie algebras of maximal class. III’, J. Algebra (2) 284 (2005), 435–461.CrossRefGoogle Scholar
[34]Jurman, G. and Young, D. S., ‘Quotients of maximal class of thin Lie algebras in characteristic two: errata and addendum’, preprint.Google Scholar
[35]Kac, V. G., ‘A description of the filtered Lie algebras with which graded Lie algebras of Cartan type are associated’, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 800–834.Google Scholar
[36]Klaas, G., Leedham-Green, C. R. and Plesken, W., Linear pro-p-groups of finite width, Lecture Notes in Math. 1674 (Springer, Berlin, 1997).CrossRefGoogle Scholar
[37]Kostrikin, A. I., ‘The beginnings of modular Lie algebra theory’, in: Group theory, algebra, and number theory (Saarbrücken, 1993) (de Gruyter, Berlin, 1996) pp. 13–52.CrossRefGoogle Scholar
[38]Kostrikin, A. I. and Kuznetsov, M. I., ‘On the structure of modular Lie algebras associated with analytic pro-p-groups’, Dokl. Akad. Nauk (5) 339 (1994), 591–593.Google Scholar
[39]Kostrikin, A. I., ‘Finite-dimensional Lie algebras with a nonsingular derivation’, in: Algebra and analysis (Kazan, 1994) (de Gruyter, Berlin, 1996) pp. 81–90.Google Scholar
[40]Kostrikin, A. I. and Ŝafareviĉ, I. R., ‘Graded Lie algebras of finite characteristic’, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 251–322.Google Scholar
[41]Kuznetsov, M. I., ‘Truncated induced modules over transitive Lie algebras of characteristic p’, Izv. Akad. Nauk SSSR Ser. Mat. (3) 53 (1989), 557–589, 671.Google Scholar
[42]Leedham-Green, C. R., ‘The structure of finite p-groups’, J. London Math. Soc. (2) (1) 50 (1994), 49–67.CrossRefGoogle Scholar
[43]Leedham-Green, C. R. and McKay, S., ‘On the classification of p-groups of maximal class’, Quart. J. Math. Oxford Ser. (2) (139) 35 (1984), 293–304.CrossRefGoogle Scholar
[44]Leedham-Green, C. R. and McKay, S., The structure of groups of prime power order, London Math. Soc. Monogr. 27, Oxford Science Publications (Oxford University Press, Oxford, 2002).CrossRefGoogle Scholar
[45]Leedham-Green, C. R. and Newman, M. F., ‘Space groups and groups of prime-power order. I’, Arch. Math. (Basel) (3) 35 (1980), 193–202.CrossRefGoogle Scholar
[46]Lucas, È., ‘Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier’, Bull. Soc. Math. France 6 (1878), 49–54.CrossRefGoogle Scholar
[47]Mattarei, S., ‘Some thin pro-p-groups’, J. Algebra (1) 220 (1999), 56–72.CrossRefGoogle Scholar
[48]Seligman, G. B., Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40 (Springer, New York, 1967).CrossRefGoogle Scholar
[49]Serconek, S. and Wilson, R. L., ‘Classification of forms of restricted simple Lie algebras of Cartan type’, Comm. Algebra (6) 19 (1991), 1603–1628.CrossRefGoogle Scholar
[50]Shalev, A., ‘Simple Lie algebras and Lie algebras of maximal class’, Arch. Math. (Basel) (4) 63 (1994), 297–301.CrossRefGoogle Scholar
[51]Shalev, A., ‘The structure of finite p-groups: effective proof of the coclass conjectures’, Invent. Math. (2) 115 (1994), 315–345.CrossRefGoogle Scholar
[52]Shalev, A. and Zelmanov, E. I., ‘Pro-p groups of finite coclass’, Math. Proc. Cambridge Philos. Soc. (3) 111 (1992), 417–421.CrossRefGoogle Scholar
[53]Skryabin, S., ‘Toral rank one simple Lie algebras of low characteristics’, J. Algebra (2) 200 (1998), 650–700.CrossRefGoogle Scholar
[54]Skryabin, S. M., ‘Classification of Hamiltonian forms over algebras of divided powers’, Mat. Sb. (1) 181 (1990), 114–133.Google Scholar
[55]Strade, H., ‘Representations of the (p2 – 1)-dimensional Lie algebras of R. E. Block’, Canad. J. Math. (3) 43 (1991), 580–616.CrossRefGoogle Scholar
[56]Strade, H., ‘The classification of the simple modular Lie algebras. VI. Solving the final case’, Trans. Amer. Math. Soc. (7) 350 (1998), 2553–2628.CrossRefGoogle Scholar
[57]Strade, H., Simple Lie algebras over fields of positive characteristic. I, volume 38 of de Gruyter Expositions in Mathematics, Structure theory (Walter de Gruyter & Co., Berlin, 2004).CrossRefGoogle Scholar
[58]Strade, H. and Farnsteiner, R., Modular Lie algebras and their representations, volume 116 of Monographs and Textbooks in Pure and Applied Mathematics (Marcel Dekker Inc., New York, 1988).Google Scholar
[59]van der Kallen, W. L. J., Infinitesimally central extensions of Chevalley groups, Lecture Notes in Mathematics, Vol. 356 (Springer, Berlin, 1973).CrossRefGoogle Scholar
[60]Waterhouse, W. C., ‘Automorphisms and twisted forms of generalized Witt Lie algebras’, Trans. Amer. Math. Soc. (1) 327 (1991), 185–200.CrossRefGoogle Scholar
[61]Wilson, R. L., ‘Simple Lie algebras of type S’, J. Algebra (2) 62 (1980), 292–298.CrossRefGoogle Scholar
[62]Zusmanovich, P., ‘Central extensions of current algebras’, Trans. Amer. Math. Soc. (1) 334 (1992), 143–152.CrossRefGoogle Scholar