1

Given a unitary matrix $U$ and given a direct sum of Lie algebras all being subalgebras of the Lie algebra of skew-Hermitian operators, e.g. $\mathfrak{su}(n) \oplus \mathfrak{su}(n)$. How do can I check whether or not there exists $g$ in the Lie algebra s.t. $U = e^g$.

For a irreducible rep, their representation is given by the tensor product of reps of the constituents of the direct sum. But I am not interested in irreducible reps only, but rather in the (non-)existence of a rep that includes an arbitrary unitary matrix.

user18722294
  • 81
  • I don't understand what you are asking. – Torsten Schoeneberg Feb 06 '25 at 15:28
  • So for example, given $ \mathfrak{su}(n) \oplus \mathfrak{su}(n) $ as a Lie algebra and we call the associated Lie group $G = SU(n) \times SU(n)$. How can I check whether an arbitrary unitary is in $G$? – user18722294 Feb 06 '25 at 15:31
  • "an arbitrary unitary" what? – Torsten Schoeneberg Feb 06 '25 at 15:36
  • sorry, an arbitary unitary matrix – user18722294 Feb 06 '25 at 15:37
  • If $G$ is given as a matrix Lie group, you check whether your matrix is in it. If $G$ is not given as a matrix group, the question makes no sense. Are you maybe asking which representations of $G$ include a given matrix? That seems an awkward question though because it's not even stable under isomorphism, i.e. one representation might contain a specific explicit matrix, and another one might not, yet they are isomorphic. – Torsten Schoeneberg Feb 06 '25 at 15:46
  • Just to check my understanding, if the Lie algebra is a direct sum of say $\mathfrak{su}$, is there a unique matrix Lie group associated to it? In my case, the Lie algebra is fixed and the unitary matrix for which I want to know whether I can generate it with elements of the Lie algebra via the exponential map. So I want to know whether there exists a representation of $G$ that includes a given matrix. – user18722294 Feb 06 '25 at 15:50
  • The matrix exponential is a well-defined map, but for that you have to fix your Lie algebra as a set of matrices. And again, that might be done differently. There is a standard matrix definition for $\mathfrak{su}_n$ and I assume that one is meant, but already when you take a direct sum of those, are we identifying that with something inside a bigger matrix (Lie) algebra? – Torsten Schoeneberg Feb 06 '25 at 17:14
  • What my skepticism boils down to is that if you throw any isomorphism, even automorphisms, on a certain matrix Lie group, you'll get a new one. Asking whether a specific matrix is included in something is highly sensitive to these operations, so one cannot argue with anything "up to isomorphism". – Torsten Schoeneberg Feb 06 '25 at 17:16
  • That being said, questions about the image of the exponential map have been asked a-plenty here. See e.g. https://math.stackexchange.com/q/3859971/96384, https://math.stackexchange.com/q/643216/96384, https://math.stackexchange.com/q/191228/96384 and many links from there. – Torsten Schoeneberg Feb 06 '25 at 17:20
  • @TorstenSchoeneberg thanks for your elaborations. I see that the question is not stable under isomorphism. But is it possible to determine despite the existence of one representation s.t. a given unitary matrix is or is not possibly $e^{ig}$ for some representation of $g \in \mathfrak{g}$? If we restrict ourselves to the compact setting of direct sums consisting out of $\mathfrak{su}, \mathfrak{sp}, \mathfrak{so}$, is there a finite amount of reps one could brute force go through? I am looking for an efficient method, but is it in principle possible? – user18722294 Feb 07 '25 at 09:55

0 Answers0