Lie Subgroups of the 2D Torus Group

This Demonstration draws the one-dimensional Lie group plotted as a subgroup embedded (or immersed) in the two-dimensional torus group . The slope is set by the direction of the velocity, and the subgroup is plotted from up to a maximum value set by the "length" value; this latter is the product of the "multiplier" value and the length of the velocity, so you can watch the thread grow by putting the velocity near (left-bottom corner) and dragging it to longer and longer lengths.


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Consider the one-dimensional Lie group plotted as a subgroup embedded (or immersed) in the two-dimensional torus group , also an Abelian Lie group. The slope of the one-dimensional group is , and if the slope is a rational number of the form where and are relatively prime (have no common factors), then is compact: it is a closed loop wound around the torus and joins onto itself (comes back to the group identity) when . If, however, is irrational, is not a loop; it is noncompact and indeed isomorphic to the one-dimensional Lie group . One can readily understand that this is true from the conditions for a closed loop: there are angles , such that for some integer and also for another integer , whence , contradicting the irrationality of . Moreover, the subgroup is dense in the torus; if it were not, then there would some open ball (open in the group topology of ) such that . But, since for any , is a group; we then see that for every group member . This means that, for example, the intersections of with the loop must be
1. distinct (otherwise closes on itself, gainsaying the irrationality of ); that is, there are countably infinitely many of them, and
2. spaced by at least , which is impossible, because the loop is compact, thus has the Bolzano–Weierstrass property, and the necessary limit point of the set of intersections means that the spacings between intersections has a greatest lower bound of zero.
A theorem due to Cartan shows that any closed subgroup of a Lie group is itself a Lie group: this is the situation we have for when is rational and is closed in the Lie group . In this case, the topology that makes a Lie group is the relative topology inherited by from , and is a topological embedding within . However, when is irrational, is not closed in , and the topology that must be given to it to make it a Lie group is the group topology as defined in §2.3 of [1], and in this case, this topology is different from the relative topology. is no longer a topological embedding in when is irrational, but is instead an immersion in . Some authors [2] use the terminology "virtual Lie subgroup" for a Lie subgroup that fails to be a topological embedding; however, it is important to realize there is nothing "virtual" about the Lie-hood of such a subgroup when the latter is given the right topology, as in §2.3 of [1].
This example is an excellent illustration of the Lie correspondence, as discussed in [3]. [1] also proves the Lie correspondence by a similar method to that used in [3].
[1] W. Rossmann, Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford: Oxford University Press, 2003.
[2] V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, Foundations of Lie Theory and Lie Transformation Groups, Berlin: Springer-Verlag, 1997, §2.3 "Virtual Lie Subgroups," as well as Theorem 5.4.
[3] R. Vance, "The Lie Correspondence: Lie's Analogies with Galois Theory," Chap. 9 in Wet Savanna Animals: An Introduction to Lie Theory through Path Geometry (May 14, 2014).
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