I am now trying to understand my new set of Lie algebra. I am deforming the \mathfrak{so}(3) $\mathfrak{so}(3)$ algebra using the ordinary boson operator. The new algebra is 4-dimensional i.e. 4 generators.
As we all know, for \mathfrak{so}(3), $\mathfrak{so}(3)$, we have 3 generators satisfying [J_{i}, $[J_{i}, J_{j}]= i\varepsilon_{ijk}J_{k}i\varepsilon_{ijk}J_{k}$. In addition to that we have a single independent Casimir. Symbolically, the Casimir is
C = J^{2}
$C=J^{2}{1} + J^{2}1}+J^{2}{2} + J^{2}_{3}.2}+J^{2}_{3}$.
Also this is the center of the \mathfrak{so}(3) $\mathfrak{so}(3)$ algebra which commute with J_{i}, i=1,2,3$J_{i}, i=1,2,3$.
For my new algebra, I have found that there are two type of Casimir which are independent with each other. Despite knowing that I have two-dimensional center(is this term correct?), what other information can I get from the Casimir?
Thanks in advance N.B. I am a theoretical physicist with physics background and no formal pure maths knowledge.
