Year-end Sale % OFF 
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.
Highlights
In this paper, we demonstrate the usefulness of the g-invariant split Casimir operator b in the representation theory of Lie algebras
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 0 of two representations T and T 0
We demonstrated the usefulness of the g-invariant split Casimir operator b in the representation theory of Lie algebras
Summary
We demonstrate the usefulness of the g-invariant split Casimir operator b (see definition in Section 2) in the representation theory of Lie algebras (see [1]). 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. We try to construct invariant projectors in the representation space V ⊗2 of T ⊗2 by using only b 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 b This is not the case all invariant projectors in V ⊗2 are constructed as polynomials in C. To simplify the notation, we always write s( N ), so ( N ) and sp(2n) instead of s( N, C), so ( N, C) and sp(2n, C), respectively
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
