The main aim of this paper is to describe finite Z-gradings of infinite dimensional simple Lie algebras. Here, a finite Z-grading of an algebra A is a decomposition A = Åi = -nnAi such that AiAj Í Ai+j where Ai = 0 for |i| > n.
The results on Z-gradings of simple Lie algebras are applied numerously in various branches of mathematics. For instance, Kac, and Vinberg employed the classification of Z-gradings of the finite-dimensional simple complex Lie algebras given by Kantor to study nilpotent orbits of connected linear groups. In differential geometry a classification of gradings of real simple Lie algebras leads to a classification of certain classes of affine symmetric spaces such as Riemannian spaces, quaternionic symmetric spaces, pseudo-hermitian spaces of Ke -type, etc. Also, the study of gradings is relevant for different classes of non-associative algebras, e.g., Jordan algebras and pairs, conservative algebras, generalized Jordan triple systems, and structurable algebras.
If L is a finite dimensional simple Lie algebra over a field F of complex or real numbers, the classification of Z-gradings which are necessarily finite in this case, is given in terms of partitions of fundamental root systems, a tool which is not available in the infinite dimensional case. Our approach to this classification problem is based on Zelmanov's Classification Theorem. According to this theorem it suffices to describe the gradings of the Lie algebras K¢ = [K,K]/Center([K,K]) where K = K(R,*) is the Lie algebra of skew-symmetric elements of a simple associative algebra with involution (R,*). The main result of the paper reads that any grading
| (1) |
| (2) |
In general, the length of grading (1) of K¢ may be less than the length of the corresponding grading (2) of R. Having classified the gradings of algebras K¢ we study the supports of these gradings and show that in fact the difference between the supports of ( 1) and (2) is small. Namely, we prove that m £ 2n, and R±i = 0 for any n < i < m.
If m = n, then the grading of K¢ is special in the sense of Zelmanov. He proved that any grading of a simple finite dimensional Lie algebra of types An, Cn is special, and that the algebras of types Bn, Dn have exceptional gradings. To give a generalization of this result for infinite dimensional algebras we extend the notion of symplectic involution to the infinite dimensional case, and prove that any grading of the Lie algebra K¢ is special if and only if either * is of the second kind or * is a symplectic involution. This is true modulo some known exceptions in low dimensions.