We describe, up to unitary equivalence, all k-tuples (A 1, A 2, . . ., A k ) of unitary operators such that \( A_i^{{{n_i}}}=I\;\mathrm{for}\;i=\overline{1,k} \) and A 1 A 2 . . . A k = λI, where the parameters (n 1, . . . , n k ) correspond to one of the extended Dynkin diagrams \( {{\tilde{D}}_4} \), \( {{\tilde{E}}_6} \), \( {{\tilde{E}}_7} \), and \( {{\tilde{E}}_8} \), and \( \lambda \in \mathbb{C} \) is a fixed root of unity.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 12, pp. 1654–1675, December, 2012.
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Livins’kyi, I.V., Radchenko, D.V. Representations of Algebras Defined by a Multiplicative Relation and Corresponding to the Extended Dynkin Graphs \( {{\tilde{D}}_4} \), \( {{\tilde{E}}_6} \), \( {{\tilde{E}}_7} \), and \( {{\tilde{E}}_8} \) . Ukr Math J 64, 1865–1892 (2013). https://doi.org/10.1007/s11253-013-0757-y
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DOI: https://doi.org/10.1007/s11253-013-0757-y