't Hooft symbol

The t Hooft symbol is a collection of numbers which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft. It is used in the construction of the BPST instanton.

Definition

is the 't Hooft symbol:
Where and are instances of the Kronecker delta, and is the Levi–Civita symbol.
In other words, they are defined by
where the latter are the anti-self-dual 't Hooft symbols.

Matrix form

In matrix form, the 't Hooft symbols are
and their anti-self-duals are the following:

Properties

They satisfy the self-duality and the anti-self-duality properties:
Some other properties are
The same holds for except for
and
Obviously due to different duality properties.
Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.