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

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 subrepresen...

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Bibliographic Details
Published in:arXiv.org 2023-02
Main Authors: Isaev, A P, Krivonos, S O, Provorov, A A
Format: Article
Language:English
Subjects:
Algebra
Lie groups
Mathematical analysis
Operators (mathematics)
Parameters
Projectors
Representations
Subspaces
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2212.14761