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Vogel universality and beyond

A. P. Isaev

TL;DR

This work extends Vogel universality from the adjoint tensor square to Cartan powers $Y_n$ and related representations, establishing universal decompositions and split-Casimir identities that hold for all simple Lie algebras except $ rak e_8$ in certain cases. It introduces $k$-split Casimir operators, derives their eigenvalues, and constructs universal invariant projectors and dimension formulas for the resulting subrepresentations in $oxed{ m Box}igotimes Y_n$ and $oxed{ m Box}igotimes Y_n'$, with explicit treatments for $ rak{sl}_N$, $ rak{so}_N$, and the exceptional algebras. The paper delivers universal characteristic identities written in Vogel parameters, provides trace relations for higher Casimir operators, and discusses the limitations and breakdowns of universality (notably for $ rak e_8$ and for $oxed{ m Box}igotimes Y_n'$ in some exceptional cases). These results pave the way for universal colour factors in gauge theories and universal knot/Chern–Simons constructions, while also suggesting natural $q$ and $t$ deformation extensions.

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.

Vogel universality and beyond

TL;DR

This work extends Vogel universality from the adjoint tensor square to Cartan powers and related representations, establishing universal decompositions and split-Casimir identities that hold for all simple Lie algebras except in certain cases. It introduces -split Casimir operators, derives their eigenvalues, and constructs universal invariant projectors and dimension formulas for the resulting subrepresentations in and , with explicit treatments for , , and the exceptional algebras. The paper delivers universal characteristic identities written in Vogel parameters, provides trace relations for higher Casimir operators, and discusses the limitations and breakdowns of universality (notably for and for in some exceptional cases). These results pave the way for universal colour factors in gauge theories and universal knot/Chern–Simons constructions, while also suggesting natural and deformation extensions.

Abstract

For simple Lie algebras we construct characteristic identities for split (polarized) Casimir operators in representations and , where -- defining (minimal fundamental for exceptional Lie algebras) representation, -- n-Cartan powers of the adjoint representations and Y_n' -- special representations appeared in the Clebsch-Gordan decomposition of symmetric part of . By means of these characteristic identities, we derive (for all simple Lie algebras, except ) explicit formulae for invariant projectors onto irreducible subrepresentations arose in the decomposition of . These projectors and characteristic identities are written in the universal form for all simple Lie algebras (except ) in terms of Vogel parameters. Universal formulas for the dimensions of the Casimir subrepresentations appeared in the decompositions of where found.
Paper Structure (23 sections, 3 theorems, 162 equations, 4 tables)

This paper contains 23 sections, 3 theorems, 162 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Decomposition of $\rm ad^{\otimes k}$ for Lie algebras of classical series. Split Casimir operators for simple Lie algebras.
  3. Class of subrepresentations in the decomposition of $\rm ad^{\otimes k}$ for Lie algebras $\mathfrak{sl}_N$ and $\mathfrak{so}_N$
  4. $k$-split Casimir operators and their eigenvalues
  5. Traces of powers of 2-split Casimir and higher Casimir operators
  6. Beyond Vogel universality
  7. Description of irreps $Y_{n}$, $Y'_{n}$ and universal decomposition of $[1] \otimes Y_n$ and $[1] \otimes Y_n'$ for simple Lie algebras of classical series
  8. Description of $\mathfrak{sl}_N$-irreps $Y_{n}$, $Y'_{n}$ and expansion of $[1] \otimes Y_n$ and $[1] \otimes Y_n'$
  9. Description of the $\mathfrak{so}_N$-irreps $Y_{n}=[n^2]$, $Y'_{n}=[1^{2n}]$ and decomposition of $[1] \otimes Y_n$ and $[1] \otimes Y_n'$
  10. Universal expansions for $\Box \otimes Y_n$ in the case of exceptional algebras
  11. The case of $\mathfrak{g}_2$ algebra. Description of irreps $Y_{n}$ and expansion of $\Box \otimes Y_n$
  12. The case of $\mathfrak{f}_4$ algebra. Irreps $Y_{n}$ and expansion of $\Box \otimes Y_n$
  13. The case of $\mathfrak{e}_6$ algebra. Irreps $Y_{n}$ and expansion of $\Box \otimes Y_n$
  14. The case of $\mathfrak{e}_7$ algebra. Irreps $Y_{n}$ and expansion of $\Box \otimes Y_n$
  15. Lie algebra $\mathfrak{e}_8$. Irreps $Y_{n}$ and expansion of $\Box \otimes Y_n$
  16. ...and 8 more sections

Key Result

Proposition 2.1

For the Lie algebra $\mathfrak{sl}(N)$ and sufficiently large $N$, all irreps, which appear in the decomposition of ad$^{\otimes k}$, are associated to the Young diagrams $\Lambda$, with $r+s \leq 2k$ columns, for which the transposed Young diagrams have the formHere, in the square brackets, the hei where ${\sf T}$ denotes the transposition of Young diagrams $\Lambda$, numbers $\lambda_1' \geq ..

Theorems & Definitions (12)

  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.2
  • Conjecture 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.4
  • Remark 3.1
  • ...and 2 more