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We develop the notion of multi-loop algebras and study their derivations algebras.
Multi-loop algebras are natural generalizations of loop algebras and are determined by n com-
muting finite order automorphisms and a Laurent polynomials in n variables as the coordinate
algebra. In this article, we introduce extended multi-loop algebras by extending the finite number
of automorphisms to a family (possibly infinitely many) of automorphisms and also using coordi-
nate algebras in a family (possibly infinitely many) of variables. Also, we develop a result of the
derivations algebra of the fixed point algebra of the tensor product of two algebras with respect
to the tensor product of two finite order automorphisms. Indeed, we prove this theorem for a
family (possibly infinitely many) of automorphisms instead of one automorphism. Consequently,
we specify the derivations algebras of some extended multi-loop algebras.
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