I'm studying Lie theory from Brian C. Hall's "Lie Groups, Lie Algebras, and Representations," in which he focuses on matrix Lie groups (defined as sets of matrices) rather than general Lie groups (defined as smooth manifolds). He proves that all matrix Lie groups are also general Lie groups, but that the converse doesn't hold: not all Lie groups can be represented as matrix Lie groups. He even gives two examples, though his examples seem fairly obscure to me. Hence my question:

Is there any example of a Lie group that cannot be represented as a matrix Lie group and also has an application in physics?

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    $\begingroup$ Maybe $\text{Mp}(2,\mathbb R)$? According to Wikipedia, “Symplectic spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology.“ $\endgroup$ – G. Smith Dec 16 '19 at 23:14

In this answer we will assume that (i) matrix Lie groups consist of finite-dimensional matrices, and more generally (ii) only consider finite-dimensional Lie groups.

Examples of Lie groups with no non-trivial finite-dimensional representations:

  1. The continuous Heisenberg Lie group, whose corresponding Heisenberg Lie algebra form the CCR. (Here we implicitly assume that the identity operator from the CCR is represented by the identity matrix. It follows that the CCR does not have finite-dimensional representations.) This is e.g. used in quantum mechanics.

  2. The metalinear group $ML(n,\mathbb{R})$. (Although it has finite-dimensional projective representations.)

  3. The metaplectic group $Mp(2n,\mathbb{R})$. This is e.g. used in the metaplectic correction/Maslov index. See also this related Phys.SE post.

  • $\begingroup$ I'm confused about the first example, as the Heisenberg group is defined (at least in the Wiki link above) as a matrix Lie algebra. Is that a mistake or am I missing something? $\endgroup$ – WillG Dec 18 '19 at 2:10
  • $\begingroup$ You are missing the parenthesis in the first example. $\endgroup$ – Qmechanic Dec 18 '19 at 2:13
  • $\begingroup$ I still don't understand... is the group the Heisenberg group, or some other group related to the CCR? Why is CCR relevant? $\endgroup$ – WillG Dec 18 '19 at 2:20
  • $\begingroup$ I updated the answer. $\endgroup$ – Qmechanic Dec 18 '19 at 2:35
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    $\begingroup$ Ok, now I see. Now I'm confused as to why the Heisenberg algebra $\mathfrak h$ (defined as $3\times 3$ matrices) is useful when studying the CCR, as the correspondence only seems to hold if we identify $i\hbar 1$ (for CCR) with $Z\in\mathfrak h$, which is certainly not a multiple of $I\in\mathfrak h$. $Z$ defined here: en.wikipedia.org/wiki/Heisenberg_group#Lie_algebra_2 $\endgroup$ – WillG Dec 18 '19 at 3:18

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