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Given a Heisenberg Lie algebra of dimension $2n+1$ with generators $X_i$, how can I calculate the Casimir function of the Heiseneberg Lie algebra ?

Daniele Tampieri
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user134982
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    I think the usual definition of a Casimir element requires the Lie algebra to be semisimple, no? The Heisenberg algebra is nilpotent, so... – Jyrki Lahtonen Jan 25 '19 at 22:43

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A Casimir operator can also be defined for non-semisimple Lie algebras:

Definition: Let $\mathfrak{g}$ be a complex Lie algebra und $U(\mathfrak{g})$ be its universal enveloping algebra. An element $C\in U(\mathfrak{g})$ is called a Casimir operator, if $[C,X]=0$ for all $X\in U(\mathfrak{g})$. In other words, $C$ is in the center of the universal enveloping algebra.

For the $3$-dimensional Heisenberg Lie algebra, with basis $(e_1,e_2,e_3)$ and $[e_1,e_2]=e_3$, we have that $C=e_3$ is a Casimir operator. The $2n+1$-dimensional Heisenberg can be done similarly, see here:

Reference: Casimir operators for non-semisimple Lie algebras.

Dietrich Burde
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  • thank you Dietrich Burde . I find a reference that gives the Casimir function for the 3-dimensional Heisenberg Lie algebra is C = x + y and the Casimir function for the 2n+1-dimensional Heisenberg Lie algebra is Ci=xi+yi . – user134982 Jan 26 '19 at 21:29
  • Reference: https://ejpam.com/index.php/ejpam/article/viewFile/1515/278 – user134982 Jan 26 '19 at 21:33
  • see the end of this article and in the references – user134982 Jan 26 '19 at 21:37
  • I am looking for how does the calculation to find C= x+y and Ci = xi+yi – user134982 Jan 26 '19 at 21:47
  • No idea why they call them Casimir operators... these are not operators by the way. Maybe because it looks more fashionable than "central elements of the universal enveloping algebra"? – YCor Jan 27 '19 at 04:10
  • @YCor Several words are in use, Casimir element, Casimir operator, Casimir invariant, see for example the German Wikipedia page. – Dietrich Burde Jan 27 '19 at 09:18
  • @YCor They are called operators because there is an isomorphism between the universal enveloping algebra and the left-invariant differential operators on the corresponding Lie group. – horropie Dec 25 '21 at 15:49