On new simple Lie algebras of Shen Guangyu (Q909767)

The author calculates the invariant filtrations and the associated graded Lie algebras of the simple Lie algebras \({\bar \Sigma}\), \({\tilde \Sigma}\) and \(\Sigma^*\) (of characteristic \(p>3)\) which were constructed in [Chin. Ann. Math., Ser. B 4, 328-346 (1983; Zbl 0507.17007)] by the reviewer and were shown to be of generalized Cartan type H. By direct computations he is able to show that if \({\mathfrak n}=(n_ 1,...,n_ r,n_{r+1},...,n_{2r})\), then \(\{\{n_ 1,n_{r+1}\},...,\{n_ r,n_{2r}\}\}\) is the complete set of invariants of the Cartan type Lie algebra H(2r,\({\mathfrak n})\). A similar result for \(K(2r+1,{\mathfrak n})\) is also obtained. Combining the above results he obtains sets of invariants for the Lie algebras, \({\bar \Sigma}\), \({\tilde \Sigma}\) and \(\Sigma^*\) which refine certain results of the reviewer (loc. cit.). Some corrections to that paper are also made.