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It is known that under certain finite dimensionality condition the
derivation algebra of tensor product of two algebras can be
obtained in terms of the derivation algebras and the centroids of
the involved algebras. We extend this theorem to infinite
dimensional case and as an application, we determine the
derivation algebra of the fixed point algebra of the tensor
product of two algebras, with respect to the tensor product of two
finite order automorphisms. These provide the framework for
calculating the derivations of some infinite dimensional Lie
algebras.
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