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Realization of graded-simple algebras as loop algebras. (English) Zbl 1157.17009
Let $$A$$ be an algebra, not necessarily associative with a set $$\sigma_1,\ldots,\sigma_n$$ of commutating automorphisms of finite orders $$m_1,\ldots,m_n$$. Suppose that the base field $$k$$ of characteristic zero contains primitive roots $$\xi_1,\ldots,\xi_n$$ of degrees $$m_1,\ldots,m_n$$. Then $$A$$ admits a $${\mathbb Z}^{n}$$-grading $$A=\oplus A^{(l_1,\ldots,l_n)}$$ where $$A^{(l_1,\ldots,l_n)}$$ is the set of all elements $$u$$ such that $$\sigma_i(u)=\xi_i^{l_i} u$$. The multiloop algebra $M(A,\sigma_1,\ldots,\sigma_n) = \oplus_{(l_1,\ldots,l_n)\in {\mathbb Z}^{n}} \left(A^{(l_1,\ldots,l_n)}\otimes z_1^{l_1} \cdots z_n^{l_n}\subset A\otimes k\left[z_1^{\pm 1},\ldots,z_1^{\pm 1}\right]\right).$ Necessary and sufficient conditions are found under which a $${\mathbb Z}^{n}$$-graded algebra is isomorphic to a multiloop algebra. A criterion is given under which two multiloop algebras are graded isomorphic, up to an automorphism of the grading group.

##### MSC:
 17B70 Graded Lie (super)algebras 16W50 Graded rings and modules (associative rings and algebras)