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It is a well known result that the fixed point
subalgebra of a finite dimensional complex simple Lie algebra
under a finite order auto-morphism is a reductive Lie subalgebra
and an abelian subalgebra. We consider this for the class of
extended affine Lie algebras and are able to show that the fixed
point subalgebra of an extended affine Lie algebra under a finite
order auto-morphism (which satisfies certain natural properties)
is a sum of extended affine Lie algebras (up to existence of some
isolated root spaces), an abelian sublagebra and a subspace which
is contained in the centralizer of the core.
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