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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.

17B70 Graded Lie (super)algebras
16W50 Graded rings and modules (associative rings and algebras)
graded algebras
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