On the Initial Subalgebra of a Graded Lie Algebra

Affiliation(s)

Department of Mathematics, The Ohio State University at Mansfield, Mansfield, Ohio, USA.

Department of Mathematics, The Ohio State University at Mansfield, Mansfield, Ohio, USA.

ABSTRACT

We show that each irreducible, transitive finite-dimensional graded Lie algebra over a field of prime characteristic p contains an initial subalgebra in which the p^{th} power of the adjoint transformation associated with any element in the lowest gradation space is zero.

We show that each irreducible, transitive finite-dimensional graded Lie algebra over a field of prime characteristic p contains an initial subalgebra in which the p

Cite this paper

Gregory, T. (2014) On the Initial Subalgebra of a Graded Lie Algebra.*Advances in Pure Mathematics*, **4**, 513-517. doi: 10.4236/apm.2014.49058.

Gregory, T. (2014) On the Initial Subalgebra of a Graded Lie Algebra.

References

[1] Kostrikin, A.I. and Shafarevich, I.R. (1969) Graded Lie Algebras of Finite Growth. Mathematics of the USSR-Izvestiya, 3, 237-304. (English)

[2] Premet, A. and Strade, H. (2006) Classification of Finite-Dimensional Simple Lie Algebras in Prime Characteristics. arXiv: math/0601380v2 [math. RA].

[3] Skryabin, S.M. (1992) New Series of Simple Lie Algebras of Characteristic 3. Russian Academy of Sciences Sbornik Mathematics, 70, 389-406. (English)

[4] Benkart, G.M., Kostrikin, A.I. and Kuznetsov, M.I. (1996) The Simple Graded Lie Algebras of Characteristic Three with Classical Reductive Component L0. Communications in Algebra, 24, 223-234.

http://dx.doi.org/10.1080/00927879608825563

[5] Benkart, G.M., Gregory, T.B. and Kuznetsov, M.I. (1998) On Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component. In: Ferrar, J.C. and Harada, K., Eds., The Monsteer and Lie Algebras, Vol. 7, Ohio State University Mathematical Research Publications, 149-164.

[6] Gregory, T.B. and Kuznetsov, M.I. (2004) On Depth-Three Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component. Communications in Algebra, 32, 3339-3371.

http://dx.doi.org/10.1081/AGB-120039401

[7] Gregory, T.B. and Kuznetsov, M.I. On Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component. (In Preparation)

[8] Weisfeiler, B.J. (1978) On the Structure of the Minimal Ideal of Some Graded Lie Algebras in Characteristic p > 0. Journal of Algebra, 53, 344-361. http://dx.doi.org/10.1016/0021-8693(78)90280-6

[9] Benkart, G.M. and Gregory, T.B. (1989) Graded Lie Algebras with Classical Reductive Null Component. Mathematische Annalen, 285, 85-98. http://wdx.doi.org/10.1007/BF01442673

[10] Jacobson, N. (1962) Lie Algebras, Tracts in Mathematics. Vol. 10, Interscience, New York.

[1] Kostrikin, A.I. and Shafarevich, I.R. (1969) Graded Lie Algebras of Finite Growth. Mathematics of the USSR-Izvestiya, 3, 237-304. (English)

[2] Premet, A. and Strade, H. (2006) Classification of Finite-Dimensional Simple Lie Algebras in Prime Characteristics. arXiv: math/0601380v2 [math. RA].

[3] Skryabin, S.M. (1992) New Series of Simple Lie Algebras of Characteristic 3. Russian Academy of Sciences Sbornik Mathematics, 70, 389-406. (English)

[4] Benkart, G.M., Kostrikin, A.I. and Kuznetsov, M.I. (1996) The Simple Graded Lie Algebras of Characteristic Three with Classical Reductive Component L0. Communications in Algebra, 24, 223-234.

http://dx.doi.org/10.1080/00927879608825563

[5] Benkart, G.M., Gregory, T.B. and Kuznetsov, M.I. (1998) On Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component. In: Ferrar, J.C. and Harada, K., Eds., The Monsteer and Lie Algebras, Vol. 7, Ohio State University Mathematical Research Publications, 149-164.

[6] Gregory, T.B. and Kuznetsov, M.I. (2004) On Depth-Three Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component. Communications in Algebra, 32, 3339-3371.

http://dx.doi.org/10.1081/AGB-120039401

[7] Gregory, T.B. and Kuznetsov, M.I. On Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component. (In Preparation)

[8] Weisfeiler, B.J. (1978) On the Structure of the Minimal Ideal of Some Graded Lie Algebras in Characteristic p > 0. Journal of Algebra, 53, 344-361. http://dx.doi.org/10.1016/0021-8693(78)90280-6

[9] Benkart, G.M. and Gregory, T.B. (1989) Graded Lie Algebras with Classical Reductive Null Component. Mathematische Annalen, 285, 85-98. http://wdx.doi.org/10.1007/BF01442673

[10] Jacobson, N. (1962) Lie Algebras, Tracts in Mathematics. Vol. 10, Interscience, New York.