Compilation of relations for the antisymmetric tensors defined by the Lie algebra cocycles of $su(n)$

TL;DR

This paper compiles and derives new identities involving invariant antisymmetric tensors in $su(n)$, crucial for understanding Lie algebra cohomology and Casimir operators, with detailed algebraic relations and tensor formulas.

Contribution

It provides a comprehensive collection of new and existing results on Omega tensors in $su(n)$, including formulas for their squares and algebraic relations, enhancing the understanding of Lie algebra cohomology.

Findings

Formulas for the squares of all Omega tensors in $su(n)$

Derived identities for antisymmetric tensor relations

Detailed algebraic methods for tensor derivations

Abstract

This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of $su(n)$, and that play an essential role in the optimal definition of Racah-Casimir operators of $su(n)$. Since the Omega tensors occur naturally within the algebra of totally antisymmetrised products of $\lambda$-matrices of $su(n)$, relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of $su(n)$. Various key derivations are given to illustrate the methods employed.