Minimal varieties of graded Lie algebras of exponential growth and the special Lie algebra [formula omitted

Abstract

The Lie algebra sl2=sl2(K) of 2×2 traceless matrices over a field K has only three non-trivial G-gradings when G is a group, the ones induced by G=Z2, Z2×Z2 and Z. Here we prove that when char(K)=0, the variety varG(sl2) of G-graded Lie algebras generated by sl2, is a minimal variety of exponential growth, and in case G=Z2×Z2 or Z, varG(sl2) has almost polynomial growth.