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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)
Keywords:
graded algebras
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