On the structure of ternary Clifford algebras and their irreducible representations (Q2077413)

The author studies the structure of the algebra generated over \(\mathbb{C}\) by generators \(e_1,\dots,e_n\) whose third powers are equal to 1 and \(e_k e_\ell=\omega e_\ell e_k\) for each \(k<\ell\) where \(\omega=e^{\frac{2\pi i}{3}}\). He concludes that when \(n\) is even, this algebra is isomorphic to \(M_{\frac{n}{2}}(\mathbb{C})\), and when it is odd, it is isomorphic the direct sum of three complex matrix algebras of degree \(\frac{n-1}{2}\). Reviewer's note: The results reported in the paper appeared earlier in a much greater generality, see for instance [\textit{M. Vela}, Commun. Algebra 30, No. 4, 1995--2021 (2002; Zbl 1011.16030)].