Sharp estimnew series of simple Lie algebras of characteristic 3

S. M. Skryabin

Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 389-406

Résumé

Three new series of simple finite-dimensional Lie algebras over a field of characteristic 3 are constructed. For explicit realization, these algebras can be represented as a sum of a Lie algebra of general or special Cartan type and of certain of its tensor modules. The algebras have gradings of depth 2 or 4, with a classical zero component; and the author gives a characterization for them in the class of all graded Lie algebras. In order to prove nonisomorphism of these algebras with the Lie algebras of Cartan type, the subalgebras containing a certain invariant set are studied. With a view of systematizing the known examples of simple finite-dimensional Lie algebras of characteristic 3, analogous realizations of two previously known series are also presented.

Bibliographie

[1] Melikyan G. M., “O prostykh algebrakh Li kharakteristiki $5$”, UMN, 35:1 (1980), 203–204 | MR | Zbl

[2] Melikyan G. M., Prostykh neprivodimye $2$-graduirovannye algebry Li s komponentoi $L_0=W_1+k$, Dep. v VINITI. No 1688-82, 1982

[3] Kuznetsov M. I., “Melikyan algebras as Lie algebras of the type $G_2$”, Comm. in Algebra, 19 (1991), 1281–1312 | DOI | MR | Zbl

[4] Kaplansky I., “Lie algebras of characteristic $p$”, Trans. Amer. Math. Soc., 89 (1958), 149–183 | DOI | MR | Zbl

[5] Brown G., “Lie algebras of characteristic three with nondegenerate Killing form”, Trans. Amer. Math. Soc., 137 (1969), 259–268 | DOI | MR | Zbl

[6] Kostrikin A. I., “Parametricheskoe semeistvo prostykh algebr Li”, Izv. AN SSSR. Ser. matem., 34 (1970), 744–756 | MR | Zbl

[7] Veisfeiler B. Yu., Kats V. G., “Eksponentsialy v algebrakh Li kharakteristiki p”, Izv. AN SSSR. Ser. matem., 35 (1971), 762–788 | MR | Zbl

[8] Frank M., “A new simple Lie algebra of characteristic three”, Proc. Amer. Math. Soc., 38 (1973), 43–46 | DOI | MR | Zbl

[9] Ermolaev Yu. B., “Ob odnom semeistve prostykh algebr Li nad polem kharakteristiki $3$”, Pyatyi vsesoyuzn. simp. po teorii kolets, algebr i modulei, Tezisy soobschenii (Novosibirsk, 1982 g.), Novosibirsk, 1982, 52–53

[10] Brown G., “Properties of a $29$ dimensional simple Lie algebra of characteristic three”, Math. Ann., 261 (1982), 487–492 | DOI | MR | Zbl

[11] Brown G., “A class of simple Lie algebras of characteristic three”, Proc. Amer. Math. Soc., 107 (1989), 901–905 | DOI | MR | Zbl

[12] Kuznetsov M. I., “Klassifikatsiya prostykh graduirovannykh algebr Li s nepoluprostoi komponentoi $L_0$”, Matem. sb., 180:2 (1989), 147–158 | Zbl

[13] Chan Nam Zung, “O dvukh klassakh prostykh algebr Li nad polem kharakteristiki $3$”, Vestnik Moskov. un-ta. Ser. $1$. Matematika, mekhanika, 1992 (to appear)

[14] Kostrikin A. I., Shafarevich I. R., “Graduirovannye algebry Li konechnoi kharakteristiki”, Izv. AN SSSR. Ser. matem., 33 (1969), 251–322 | MR | Zbl

[15] Skryabin S. M., Algebry Li differentsirovanii kommutativnykh kolets: obobscheniya algebr Li kartanovskogo tipa, Dep. v VINITI. No 4405-V87, 1987

[16] Dzhumadildaev A. S., “O kogomologiyakh modulyarnykh algebr Li”, Matem. sb., 119 (1982), 132–149 | MR | Zbl

[17] Kats V. G., “O klassifikatsii prostykh algebr Li nad polem s nenulevoi kharakteristikoi”, Izv. AN SSSR. Ser. matem., 34 (1970), 385–408 | MR | Zbl

[18] Blattner R. J., “Induced and produced representations of Lie algebras”, Trans. Amer. Math. Soc., 144 (1969), 457–474 | DOI | MR

[19] Kuznetsov M. I., “Usechennye indutsirovannye moduli nad tranzitivnymi algebrami Li kharakteristiki $p$”, Izv. AN SSSR. Ser. matem., 53 (1989), 557–589 | Zbl

[20] Krylyuk Ya. S., Algebry kartanovskogo tipa: predstavleniya i prodolzheniya, Dis. ...kand. fiz. -mat. nauk, MGU, M., 1978