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    Fine gradings of complex simple Lie algebras and Finite Root Systems

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    A GG-grading on a complex semisimple Lie algebra LL, where GG is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems, we define a finite root system RR to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to RR a semisimple Lie algebra L(R)L(R) together with a quasi-good grading on it. Thus one can construct nice basis of L(R)L(R) by means of finite root systems. We classify finite maximal abelian subgroups TT in \Aut(L) for complex simple Lie algebras LL such that the grading induced by the action of TT on LL is quasi-good, and show that the set of roots of TT in LL is always a finite root system. There are five series of such finite maximal abelian subgroups, which occur only if LL is a classical simple Lie algebra
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