Split Casimir Operator of the Lie Algebra so(2r) in Spinor Representations, Colour Factors and Yang-Baxter Equation

TL;DR

This paper derives characteristic identities for the split Casimir operator of so(2r) in spinor representations, enabling explicit computation of invariant projectors, colour factors in gauge theories, and a new Yang-Baxter solution.

Contribution

It introduces new characteristic identities and explicit projectors for the split Casimir operator in spinor representations of so(2r), and derives a novel Yang-Baxter solution.

Findings

Explicit formulas for projectors onto invariant subspaces.

Derived colour factors for ladder Feynman diagrams.

Presented a new solution to the Yang-Baxter equation.

Abstract

In this paper, we derive characteristic identities for the split Casimir operator of the Lie algebra $so(2r)$ in tensor products of spinor representations of the same and opposite chiralities. Using these identities, we explicitly construct projectors onto invariant subspaces of this operator and compute their traces. The results obtained allow us to derive explicit expressions for the colour factors of ladder Feynman diagrams in gauge theories with gauge group $Spin(2r)$. In addition, we obtain a new form of a solution to the Yang-Baxter equation that is invariant under the action of the Lie algebra $so(2r)$ in spinor representations.