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Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters

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arxiv 2102.08258 v2 pith:QRU6MGCM submitted 2021-02-16 math-ph hep-thmath.MPmath.QAmath.RT

A.P.Isaev , S.O.Krivonos (Bogoliubov Laboratory of Theoretical Physics , JINR) This is my paper · ORCID
classification math-phhep-thmath.MPmath.QAmath.RT
keywords algebrassimpleadjointcasimircharacteristicdefiningidentitiesprojectors
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We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{\otimes 2} in two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang-Baxter equations. In the case when T is the adjoint representation, these projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Torus knots in adjoint representation and Vogel's universality

    hep-th 2025-06 conditional novelty 6.0 of 10

    Universal adjoint invariants for T[4,n] torus knots are constructed via Vogel's universality, completing the T[4,n] case after previous T[2,n] and T[3,n] results.

  2. Macdonald deformation of Vogel's universality and link hyperpolynomials

    hep-th 2025-05 conditional novelty 6.0 of 10

    For the adjoint square in ADE Lie algebras, products of Macdonald dimensions with deformed Littlewood-Richardson coefficients are universal, yielding universal formulas for T[2,2n] link hyperpolynomials.

  3. Towards the {\tau}-function of the quantum groups

    hep-th 2025-08 conditional novelty 5.0 of 10

    An explicit bilinear identity and tau-function construction is given for U_q(sl3) that works for arbitrary elements of the quantum group, not only special 'group-like' ones.

  4. Vogel's universality and Macdonald dimensions

    hep-th 2025-07 conditional novelty 4.0 of 10

    The paper gives a single rational formula for adjoint Macdonald dimensions that unifies the simply laced Lie algebras A_n, D_n, E6, E7, E8, plus explicit mixed-root-system dimension formulas.

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