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Consider the two-dimensional complex Lie algebra defined by $[X,Y]=iY$. Are there non-trivial representations where $X$ and $Y$ are Hermitian (possibly unbounded)? By the way, the above is the unique two-dimensional non-abelian Lie algebra up to isomorphism.

Note such representations cannot be finite-dimensional, in short because $Y$ raises and lowers the eigenvalues of $X$ by $i$, so $X$ cannot be Hermitian. More precisely, if $X$ has eigenvector $v$ with eigenvalue $x \in \mathbb{R}$, then $XYv=(iY+YX)v=(i+x)Yv$, so either $Yv=0$ (and by extension $Y=0$) or else $X$ has an eigenvector $Yv$ with eigenvalue $x+i \not\in \mathbb{R}$, a contradiction.

Alternatively, re-defining $X \mapsto iX, Y \mapsto iY$, I am interested in reps of $[X,Y]=Y$ with $X,Y$ anti-Hermitian.

EmmyNoether
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  • Do you implicitly mean to ask for repns beyond those of restrictions of repns of $\mathfrak{sl}_2$? – paul garrett Mar 15 '21 at 22:03
  • I don't mean to dismiss reps arising from $\mathfrak{sl}_2$; are there $\mathfrak{sl}_2$ reps where the analog of my $X,Y$ above are Hermitian? – EmmyNoether Mar 15 '21 at 22:06
  • A unitary repn of $SU(2)$ would give skew-hermitian repns of (suitable constant multiples of... maybe this is your issue?) the Lie algebra... – paul garrett Mar 15 '21 at 22:15
  • I'm probably confused, but would the reps you're referring to be infinite-dimensional, and if not, does my argument in the OP fail? – EmmyNoether Mar 15 '21 at 22:19
  • The (irred) repns of $SU(2)$ are finite-dimensional... But/and there are (yes, infinite-dimensional) repns of the same Lie algebra, attached to unitary repns of (the split real form of $SL_2$), $SL_2(\mathbb R)$, which have complementary features... – paul garrett Mar 15 '21 at 22:32
  • Thanks, I follow now. So the short answer to OP is "yes," and these reps can be derived from well-known infinite-dimensional unitary reps of e.g. $so(1,1)$ or $sl_2$. I may try to write up a nice answer when I get a chance. – EmmyNoether Mar 15 '21 at 23:31
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    (Just to be sure, typo: it'd be so(2,1), not so(1,1)...) – paul garrett Mar 15 '21 at 23:36

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