Celebratio Mathematica — Benkart

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Works connected to José María Pérez-Izquierdo

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G. Benkart and J. M. Pérez-Izquierdo: “A quantum octonion algebra,” Trans. Am. Math. Soc. 352 : 2 (2000), pp. 935–968. To the memory of Alberto Izquierdo. MR 1637137 Zbl 0931.17011 ArXiv math/9801141 article

Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group \( U_q(\mathrm{D}_4) \) of \( \mathrm{D}_4 \), we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the \( \mathrm{q} \)-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a \( \mathrm{q} \)-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang–Baxter operator action coming from the \( \mathrm{R} \)-matrix of \( U_q(\mathrm{D}_4) \). The product in the quantum octonions is a \( U_q(\mathrm{D}_4) \)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at \( q = 1 \) new “representation theory” proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a \( \mathrm{q} \)-analogue of the 8-dimensional para-Hurwitz algebra.

G. Benkart, S. Madariaga, and J. M. Pérez-Izquierdo: “Hopf algebras with triality,” Trans. Am. Math. Soc. 365 : 2 (2013), pp. 1001–1023. MR 2995381 Zbl 1278.16032 ArXiv 1106.4302 article