A Note on the Height of Transitive Depth-One Graded Lie Algebras Generated by Their Local Parts

Abstract

For a transitive depth-one graded Lie algebra over a field of characteristic greater than two, a limit on the degree of the highest gradation space is determined.

Keywords

Share and Cite:

Gregory, T. (2014) A Note on the Height of Transitive Depth-One Graded Lie Algebras Generated by Their Local Parts. Advances in Pure Mathematics, 4, 242-243. doi: 10.4236/apm.2014.46032.

Let the characteristic of the base field be greater than two. In [1] , V.G. Kac defined the local part of a graded Lie algebra to be which we will assume to be finite dimen- sional. We will refer to as the height of.

Lemma 1. If is a transitive graded Lie algebra which is generated by its finite- dimensional local part, then the height of is less than or equal to, where

Proof. If are any elements of (or or, respectively) and is an element of then we have by the Leibniz rule that

Now, is an element of, (or or, respectively) which is zero, since is assumed to be greater than two, and the depth of is one. Because of the commutativity of, we have that. Consequently, by the transitivity of, we have for greater than or equal to.

Note that the height of the Jacobson-Witt algebra is.

Because the gradation spaces of positive degree of transitive graded Lie algebras are contained in Cartan prolongations, those gradation spaces will be finite dimensional whenever the local part of the transitive Lie algebra is finite dimensional. Thus, when the gradation degrees of a transitive graded Lie algebra with a finite- dimensional local part are bounded, the Lie algebra is itself finite dimensional.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Kac, V.G. (1968) Simple Irreducible Graded Lie Algebras of Finite Growth. Mathematics of the USSR-Izvestiya, 2, 1271-1312 (English), Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 32, 1323-1367 (Russian).