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.

  • $\begingroup$ 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. $\endgroup$ – Torsten Schoeneberg Feb 8 at 5:49
  • $\begingroup$ 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? $\endgroup$ – jobe Feb 8 at 9:56
  • $\begingroup$ 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 $A \oplus B$. $\endgroup$ – Torsten Schoeneberg Feb 8 at 18:21
  • 1
    $\begingroup$ 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 $x \notin I$, $[x,x]=0\in I$. $\endgroup$ – Torsten Schoeneberg Feb 8 at 19:29
  • 1
    $\begingroup$ A fancier example is $\pmatrix{0&0&*\\0&0&*\\0&0&0} \unlhd \pmatrix{a&b&*\\c&-a&*\\0&0&0}$. $\endgroup$ – Torsten Schoeneberg Feb 8 at 23:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.