- #1

redtree

- 322

- 13

- TL;DR Summary
- Can the Lie algebra of ##\frak{so}(3)## (or ##\frak{su}(2)##) be formulated without complexification utilizing the Cartan subalgebra?

All of the formulations of the Lie algebra of ##\frak{so}(3)## (or ##\frak{su}(2)##) utilizing raising/lowering operators that I have seen in the literature involve complexification to ##\frak{su}(2) + i \frak{su}(2) \cong \frak{sl}(2,\mathbb{C})##. I have found explicit derivations in a particular representation, but none from the Cartan subalgebra.

Can one derive a formulation of the Lie algebra of ##\frak{so}(3)## utilizing the Cartan subalgebra and root vectors without complexification? If so, where can I find it?

Can one derive a formulation of the Lie algebra of ##\frak{so}(3)## utilizing the Cartan subalgebra and root vectors without complexification? If so, where can I find it?