Invariant tensors for simple groups
J. A. de Azcarraga, A. J. Macfarlane, A. J. Mountain, J. C. Perez Bueno
TL;DR
The paper systematically maps invariant tensors on simple Lie algebras by linking symmetric invariants to primitive Casimirs and skew-symmetric invariants to Lie algebra cocycles. It introduces a novel symmetric-tensor family $t^{(m)}$ derived from primitive $(2m-1)$-cocycles and proves their orthogonality, enabling a clean construction of primitive Casimirs from either cocycles or $t^{(m)}$ tensors. For ${\cal G}=su(n)$ it provides explicit coordinates for the $su(3)$ and $su(4)$ cocycles and develops recurrence and duality relations to generate higher cocycles while ensuring primitivity limits are respected. The results yield a structured, orthogonal basis for invariant tensors with direct implications for anomaly analysis, topological terms in gauge theories, and related physical applications.
Abstract
The forms of the invariant primitive tensors for the simple Lie algebras A_l, B_l, C_l and D_l are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A_l algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) and su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.