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arXiv · 2212.14761

Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters

A. P. Isaev·S. O. Krivonos·A. A. Provorov
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Abstract

We constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations $\mathsf{ad}$ and deduced a certain class of subrepresentations in $\mathsf{ad}^{\otimes 3}$. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in $\mathsf{ad}^{\otimes 3}$ were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in $\mathsf{ad}^{\otimes 3}$ for simple Lie algebras in terms of the Vogel parameters.

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A. P. Isaev, S. O. Krivonos, A. A. Provorov. 2022-12-30. Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters. https://doi.org/10.1142/s0217751x23500379

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