symmetry S S Article Split Casimir Operator and Universal Formulation of the Simple Lie Algebras Alexey Isaev 1,2,3,† and Sergey Krivonos 1,*,†   Citation: Isaev, A.; Krivonos, S. Split Casimir Operator and Universal Formulation of the Simple Lie Algebras. Symmetry 2021, 13, 1046. https://doi.org/10.3390/sym13061046 Academic Editor: Michal Hnatiˇ c Received: 18 May 2021 Accepted: 7 June 2021 Published: 9 June 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia; isaevap@theor.jinr.ru 2 St. Petersburg Department of Steklov Mathematical Institute of RAS, Fontanka 27, 191023 St. Petersburg, Russia 3 Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia * Correspondence: krivonos@theor.jinr.ru; Tel.: +7-(49621)-6-33-94 † These authors contributed equally to this work. Abstract: We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T ⊗2 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. Keywords: split (polarized) Casimir operators; simple Lie algebras; adjoint representations; Vogel parameters 1. Introduction In this paper, we demonstrate the usefulness of the g-invariant split Casimir operator  C (see definition in Section 2) in the representation theory of Lie algebras (see also [1]). Namely, for all simple Lie algebras g, explicit formulas can be found for invariant projectors onto irreducible representations that appear in the expansion of the tensor product T ⊗ T ′ of two representations T and T ′ . In particular, these invariant projectors are constructed in terms of the g-invariant operator  C. It is natural to find invariant projectors in terms of g-invariant operators, which in general are images of special elements of the so-called centralizer algebra. In the paper, we consider a very particular problem of constructing invariant projectors in representation spaces of T ⊗2 , where T ≡ ad is the adjoint representation but for all simple Lie algebras g. Our approach is closely related to the one outlined in [1,2]. In [2], such invariant projectors were obtained in terms of several special invariant operators and the calculations were performed using a peculiar diagram technique. In our approach, we try to construct invariant projectors in the representation space V ⊗2 of T ⊗2 by using only one g-invariant operator, which is the split Casimir operator  C. It turns out (see [3]) that for all simple Lie algebras g in the defining representations all invariant projectors in V ⊗2 are constructed as polynomials in  C. This is not the case for the adjoint representation, i.e., not for all algebras g the invariant projectors in V ⊗2 ad are constructed as polynomials of only one operator  C ad ≡ ad ⊗2  C. Namely, in the case of sℓ( N) and so(8) algebras there are additional g-invariant operators that are independent of  C ad and act, respectively, in the anti-symmetrized and symmetrized parts of the space V ⊗2 ad . In [3], we constructed such additional operators explicitly. Our study of the split Casimir operator  C was motivated by the works [4–7], and by the idea that the knowledge of the characteristic identities for  C ad turns out to be a key point for understanding the so-called universal formulation of the simple Lie algebras [8] Symmetry 2021, 13, 1046. https://doi.org/10.3390/sym13061046 https://www.mdpi.com/journal/symmetry