# Examples of prime ideals for Lie algebras

I am looking for examples of prime ideals for Lie algebras. In particular, I am interested in examples involving the Lie algebra given by the commutator of endomorphisms of a complex vector space and the Lie algebra of functions on the phase space $$\mathbb{R}^{2n}=(x^1,\ldots,x^n,p_1,\ldots,p_n)$$ given by $$[f,g]=\sum_i\frac{\partial f}{\partial x^i}\frac{\partial g}{\partial p_i}- \frac{\partial g}{\partial x^i}\frac{\partial f}{\partial p_i},$$ where $$f$$ and $$g$$ are real or complex-valued functions on the phase space.

My interest has originated in physics, where the time derivative of a physical "observable" $$X$$ can generally be written (except for multiplicative constants) as $$\frac{d X}{d t} = [H,X],$$ where $$H$$ is the Hamiltonian operator in quantum mechanics or the Hamiltonian function on classical phase space. If the Hamiltonian and $$X$$ are not in the prime ideal at some initial time, then $$X$$ will never be.

I have made some searching on the internet, but I have found only two papers that are rather abstract with no examples at all.

• According to Theorem 5 of projecteuclid.org/download/pdf_1/euclid.hmj/1206136843, maximal ideals of codimension $\ge 1$ are prime. Also, maximal ideals of perfect Lie algebras are prime. That gives tons of examples, although maybe not in the case you're interested in. – Torsten Schoeneberg Feb 8 at 5:49
• This is one of the "two papers that are rather abstract with no examples at all" that I mentioned in the question. If there are "tons of examples", could you give me just one? – jobe Feb 8 at 9:56
• Take $A=$ your favourite simple Lie algebra (maybe $\mathfrak{sl}_2$, maybe $\mathfrak{sp}_{328}$), and $B=$ any Lie algebra you like. Then $0 \oplus B$ is a prime ideal in $A \oplus B$. – Torsten Schoeneberg Feb 8 at 18:21
• The definition of prime ideal for Lie algebras needs ideals, not just elements (at least in one of the factors), see first page of that paper. Otherwise no ideal $I$ except the entire Lie algebra could ever be prime, since for $x \notin I$, $[x,x]=0\in I$. – Torsten Schoeneberg Feb 8 at 19:29
• A fancier example is $\pmatrix{0&0&*\\0&0&*\\0&0&0} \unlhd \pmatrix{a&b&*\\c&-a&*\\0&0&0}$. – Torsten Schoeneberg Feb 8 at 23:19