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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.

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# What does it mean by atwo-dimensionalcentertheorder of aLiealgebraCasimirinvariants

I have found a am now trying to understand my new set of Lie algebrathat . I am deforming the \mathfrak{so}(3) algebra using the ordinary boson operator. The new algebra is 4-dimensional i.e. with 4 generators.

As we all know, for \mathfrak{so}(3), we have 3 generators satisfying [J_{i}, J_{j}]= i\varepsilon_{ijk}J_{k}. In addition to that I we have also found two type of Casimir invariant that is a single independent Casimir.

Can anyone explain to me what does it mean having oneSymbolically, two or more dimensional center? What are their significantthe Casimir is

C = J^{2}{1} + J^{2}{2} + J^{2}_{3}.All I understand

Also this is that the center of the \mathfrak{so}(3) algebra which commute with all other set in the 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.

It is much helpfulDespite knowing that I have two-dimensional center(is this term correct?), if someone what other information can give a basic explicit example. I get from the Casimir?

Thanks in advance N.B. I am a theoretical physicist with physics background and no formal pure maths knowledge.

Post Closed as "not a real question" by Andreas Thom, Steven Landsburg, Yemon Choi, Bruce Westbury, Mariano Suárez-Alvarez
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