Vogel universality and beyond

TL;DR

This paper develops universal characteristic identities and explicit projectors for simple Lie algebras, enabling a unified description of their representation decompositions using Vogel parameters, except for e8_8.

Contribution

It introduces universal characteristic identities and projectors for simple Lie algebra representations, extending the Vogel parametrization to a broad class of decompositions.

Findings

Derived explicit formulas for invariant projectors in Lie algebra representations.

Established universal identities valid for all simple Lie algebras except e8_8.

Calculated dimensions of Casimir subrepresentations in tensor product decompositions.

Abstract

For simple Lie algebras we construct characteristic identities for split (polarized) Casimir operators in representations $T \otimes Y_n$ and $T \otimes Y_n'$, where $T$ -- defining (minimal fundamental for exceptional Lie algebras) representation, $Y_n$ -- n-Cartan powers of the adjoint representations $ad = Y_1$ and Y_n' -- special representations appeared in the Clebsch-Gordan decomposition of symmetric part of $ad^{\otimes n}$. By means of these characteristic identities, we derive (for all simple Lie algebras, except $\mathfrak{e}_8$) explicit formulae for invariant projectors onto irreducible subrepresentations arose in the decomposition of $T \otimes Y_n$. These projectors and characteristic identities are written in the universal form for all simple Lie algebras (except $\mathfrak{e}_8$) in terms of Vogel parameters. Universal formulas for the dimensions of the Casimir subrepresentations appeared in the decompositions of $T \otimes Y_n$ where found.