Let Cℓd13\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{d}^{\frac{1}{3}}$$\end{document} be the complex ternary Clifford algebra on d generators as defined in Cerejeiras and Vajiac (Adv Appl Clifford Algebras 31:13, 2021). The main objective of this work is to investigate algebraic structure of these algebras and present a new Structure Theorem. In particular, it is shown that when d is even, the algebras are central simple (CSA/C\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}$$\end{document}), and when d is odd (d>1)\documentclass[12pt]{minimal}
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\begin{document}$$(d>1)$$\end{document} then the algebras are direct sums of three simple ideals, each being isomorphic to a ternary Clifford algebra Cℓd-113\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{d-1}^{\frac{1}{3}}$$\end{document}. In the latter case, each simple ideal is generated by a central primitive idempotent as, for each d, there are exactly three central primitive orthogonal idempotents which decompose the algebra unit. A formula is given for computing these idempotents for each odd d(d>1)\documentclass[12pt]{minimal}
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\begin{document}$$(d>1)$$\end{document}. We conclude that Cℓ2k13≅Mat(3k,C)\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{2k}^{\frac{1}{3}} \cong \mathop {\mathrm{Mat}}\nolimits (3^k,{\mathbb {C}})$$\end{document} and Cℓ2k+113≅3Mat(3k,C)\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{2k+1}^{\frac{1}{3}} \cong {}^3 \mathop {\mathrm{Mat}}\nolimits (3^k,{\mathbb {C}})$$\end{document}. We describe an algorithm for finding a complete set of 3k\documentclass[12pt]{minimal}
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\begin{document}$$3^k$$\end{document} orthogonal primitive idempotents in Cℓ2k13\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{2k}^{\frac{1}{3}}$$\end{document}. Each such idempotent generates a minimal left (or right) ideal which carries an irreducible faithful representation of Cℓ2k13\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{2k}^{\frac{1}{3}}$$\end{document}. This allows us then to find irreducible representations of Cℓ2k+113\documentclass[12pt]{minimal}
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\begin{document}$$C \ell _{2k+1}^{\frac{1}{3}}$$\end{document} in minimal left (or right) ideals. This paper is a continuation of Abl amowicz (Adv Appl Clifford Algebras 31: 62, 2021).
