Two Non–*–Isomorphic *–Lie Algebra Structures on sl(2,R) and Their Physical Origins

We introduce two, Lie algebra isomorphic, real forms of $sl(2,{\mathbb C})$, i.e. two real ∗–Lie algebras, denoted respectively $sl_F (2,{\mathbb R})$ and $sl_B(2,{\mathbb R})$, such that their complexifications ($sl_F (2,{\mathbb C})$ and $sl_B(2,{\mathbb C}))$ are both isomorphic to $sl(2,{\mathbb C})$ as Lie algebras. Then we prove that $sl_B(2,{\mathbb C})$ cannot contain a real ∗–Lie sub–algebra ∗–isomorphic to $sl_F (2,{\mathbb R})$ and the same is true exchanging the indexes F and B. The meaning of the indexes B and F is explained in the last section where we show how $sl_B(2,{\mathbb R})$ (resp. $sl_F (2,{\mathbb R}))$ can be realized in terms of Bosons (resp. Fermion) operators. These realizations are known in the literature.