5
$\begingroup$

For a semi-simple compact Lie group $G$ with center $Z(G)$, one can characterize the preimage of $Z(G)$ in the Cartan subalgebra under the exponential map as the nodes of the Stiefel diagram (see for instance V.7.16 of "Representation of compact Lie groups", by Bröcker and tom Dieck).

Is there any generalization of this result, for instance for non-compact Lie groups, or for classes of infinite dimensional Lie groups?

Update: A look at $SL(2,\mathbb{R})$ shows that the preimage of the center consists in elements of the form

$ \left(\begin{array}{cc} a & b \\ -(k^2\pi^2 + a^2)/b & -a \end{array}\right) \;, \quad a \in \mathbb{R} \;, \quad b \in \mathbb{R}^\ast \;, \quad k \in \mathbb{N} $

One complication in this case is that the Cartan subalgebras are not all conjugate, and it looks indeed that the intersection with $\log Z(G)$ depends on the choice of Cartan subalgebra.

$\endgroup$
  • 1
    $\begingroup$ One big obstacle to getting a reasonable generalization is the fact that for non-compact (say semisimple) real or complex Lie groups, the exponential map may fail to be surjective. There's a lot of literature about such issues, touching also on the infinite dimensional case. Have you looked at small rank cases involving matrix groups? $\endgroup$ – Jim Humphreys Nov 16 '13 at 14:47
  • $\begingroup$ @Jim Humphreys That's right... but one might still hope to characterize those elements which do exponentiate to the center. I updated my question to describe the case of $SL(2,\mathbb{R})$. $\endgroup$ – Samuel Monnier Nov 16 '13 at 23:06
  • 1
    $\begingroup$ @JimHumphreys This is probably useless to the OP by now but just for the record (and because it is beautiful): There is this nice theorem of Hochschild asserting that for connected finite-dimensional Lie groups the center always lies in the image of the exponential map: "The center of an analytic group G is contained in an abelian analytic subgroup of G." (The Structure of Lie Groups, Holden-Day, 1965, page 189, Theorem 1.2) $\endgroup$ – Yannick Voglaire Mar 25 '16 at 4:20
  • $\begingroup$ @Yannick: Thanks for the reference, though it doesn't help directly with the concrete question raised here. Hochschild studied carefully the general foundations for analytic groups and Lie groups, but without getting into the detailed study of special classes such as semisimple Lie groups (which probably need case-by-study at least initially).. $\endgroup$ – Jim Humphreys Mar 27 '16 at 19:50
2
$\begingroup$

I have only some simple remarks here, valid also for non-compact Lie groups. Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A first remark is that for $X\in Z(\mathfrak{g})$ and $Y\in \mathfrak{g}$, $\exp (X)$ and $\exp(Y)$ commute. Hence we have $\exp(Z(\mathfrak{g}))\subseteq Z(G)$. However, $\exp$ need not be injective, even if $G$ is simply connected. Also, $\exp$ need not be surjective (but there is a classification of all simple Lie groups with surjective exponential). There are some special cases, where we can say which elements do not exponentiate to the centre of $G$. As an example, the following is true:

Lemma: Let $G$ be a real or complex connected Lie group whose Lie algebra $\mathfrak{g}$ is linear and centerfree. If $X ∈\mathfrak{g}$ is nilpotent and $\exp(X) \in Z(G)$, then $X =0$.

Proof: By assumption $\exp({\rm ad} X)Y=Y$ for all $Y\in \mathfrak{g}$, so that $[X,Y]=0$ for all $Y\in \mathfrak{g}$. It follows $X=0$ because $Z(\mathfrak{g})=0$.

$\endgroup$
  • $\begingroup$ Thanks. But I expect that it should be possible to say more. Another simple fact is that if a Lie algebra element X exponentiates to the center, then it's whole adjoint orbit does. What I'm after is a characterization of these orbits. The lemma above is a first step... $\endgroup$ – Samuel Monnier Nov 19 '13 at 20:06
  • $\begingroup$ "(but there is a classification of all simple Lie groups with surjective exponential". A pointer to where that can be found would be very helpful to me! $\endgroup$ – Benjamin Apr 14 '16 at 21:00
  • $\begingroup$ @Benjamin: Michael Wuestner, The classification of all simple Lie groups with surjective exponential map, Journal of Lie Theory Vol. 15, No. 1, pp. 269–278 (2005) emis.de/journals/JLT/vol.15_no.1/19.html $\endgroup$ – Yannick Voglaire Nov 10 '16 at 14:22

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.