## Graded Lie algebras of maximal class. IV.(English)Zbl 0960.17007

In this paper the authors classify all infinite-dimensional Lie algebras $$L$$ over a field of odd characteristic where $$L$$ is graded over the positive integers $$L = \bigoplus_{i=1}^\infty L_i$$ with $$\dim (L_i) = 1$$ for all $$i {>\atop =}1$$, and $$L$$ is generated by $$L_1$$ and $$L_2$$. In addition to analogues of such algebras in characteristic 0 and to algebras that can be obtained in a natural way from certain graded algebras of the form $$L = \bigoplus_{i=1}^\infty L_i$$ generated by $$L_1$$ where $$\dim (L_1) = 2$$ and $$\dim (L_i) = 1$$ for $$i > 1$$, there is also another family of solvable algebras for all odd characteristics and another family unique to characteristic 3.
Part I, cf. Trans. Am. Math. Soc. 349, 4021-4051 (1997; Zbl 0895.17031), II, J. Algebra 229, 750-784 (2000), III (to appear)].
Reviewer: G.Brown (Boulder)

### MSC:

 17B70 Graded Lie (super)algebras 17B65 Infinite-dimensional Lie (super)algebras

Zbl 0895.17031
Full Text:

### References:

 [1] M. Avitabile , The other graded Lie algebra associated to the Nottingham group , in preparation, 1999 . · Zbl 1126.17015 [2] A. Caranti - S. Mattarei - M.F. Newman , Graded Lie algebras of maximal class , Trans. Amer. Math. Soc. 349 ( 1997 ), 4021 - 4051 . MR 1443190 | Zbl 0895.17031 · Zbl 0895.17031 [3] A. Caranti - M.F. Newman , Graded Lie algebra of maximal class II , J. Algebra , to appear. MR 1769297 | Zbl 0971.17015 · Zbl 0971.17015 [4] A. Fialowski , Classification of graded Lie algebras with two generators , Moscow Univ. Math. Bull. 38 ( 1983 ), 76 - 79 . Zbl 0533.17008 · Zbl 0533.17008 [5] G. Jurman , Graded Lie algebra of maximal class III , in preparation, 1999 . · Zbl 0940.17014 [6] A. Shalev , Simple Lie algebras and Lie algebras of maximal class , Arch. Math. ( Basel ) 63 ( 1994 ), 297 - 301 . MR 1290602 | Zbl 0803.17006 · Zbl 0803.17006 [7] A. Shalev - I. Zelmanov , Narrow Lie algebras: a coclass theory and a characterization of the Witt algebra , J. Algebra 189 ( 1997 ), 294 - 331 . MR 1438178 | Zbl 0886.17008 · Zbl 0886.17008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.