Weight lattice and the first fundamental group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T00:58:47Z http://mathoverflow.net/feeds/question/87015 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group Weight lattice and the first fundamental group math3.14159 2012-01-30T09:22:25Z 2012-02-23T23:05:39Z <p>Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of $\frak{g}$. It is known that the kernel of exponential map $exp : \frak{t} \to$ $T$ <strong>is the lattice of all integral weights of $\frak{g}$</strong>, i.e. weihts $\lambda \in (it)^*$ such that $\lambda(H)\in 2\pi i\mathbb{Z},$ whenever $exp H= I$ for $H\in\frak{t}$.</p> <p>I have the following questions: </p> <p>1) What is the <strong>relation</strong> between <strong>the first fundamental group</strong> $\pi_{1}(G)$ of $G$ with the <strong>integral lattice</strong> described above? I am trying to find any good references about this fact, but it seems difficult. </p> <p>2) How we can use the fibration $T\to G$$\to G/T$ to compute $\pi_{1}(G/T)$? (<strong>answered</strong>)</p> <p>3) What we can say about <strong>the second homotopy group</strong> $\pi_{2}(G)$? (<strong>answered</strong>)</p> <p>4) Is it true, that if $G$ is <strong>semisimple</strong>, then $\pi_{1}(G)$ is finite? (<strong>answered</strong>)</p> <p>Thank you!</p> http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group/87018#87018 Answer by Jesper Grodal for Weight lattice and the first fundamental group Jesper Grodal 2012-01-30T10:27:49Z 2012-01-30T10:27:49Z <p>A good reference for 1) is Bourbaki: Lie groups and Lie algebras Chapter 9. See in particular Section 4.6. </p> <p>In particular it follows that 2) $\pi_1(G/T) = 0$ and that 4) $\pi_1(G)$ is finite if and only if $G$ is semisimple. </p> <p>Concerning 3) $\pi_2(G) = 0$ always, which is a theorem of Cartan. I don't recall Cartan's proof, but it follows from Bott's analysis of the cell structure of G/T, and can also be proved using that $H^*(G)$ is a Hopf algebra (See Browder: Torsion in H-spaces. Ann. of Math. (2) 74 1961 24–51.).</p> http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group/87021#87021 Answer by math3.14159 for Weight lattice and the first fundamental group math3.14159 2012-01-30T11:45:59Z 2012-01-30T11:45:59Z <p>One can also check Daniel Bump's book:</p> <p>Lie Groups (Graduate Text in Mathematics, Springer 2004) </p> <p>I found some nice proofs of the above arguments at pages 146-181. </p> <p><a href="http://books.google.gr/books?id=wCE8dInDw5gC&amp;pg=PA464&amp;lpg=PA464&amp;dq=Daniel+Bump+Lie+groups&amp;source=bl&amp;ots=4SCqDYYguR&amp;sig=gPcTZEDPW5xz1skCOkswKIO1Msk&amp;hl=el&amp;sa=X&amp;ei=ZIEmT5_CN5Gd-waFpajKCA&amp;ved=0CEUQ6AEwBA#v=onepage&amp;q=Daniel%2520Bump%2520Lie%2520groups&amp;f=false" rel="nofollow">link text</a></p> <p>Thanks</p> http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group/87477#87477 Answer by Jim Humphreys for Weight lattice and the first fundamental group Jim Humphreys 2012-02-03T20:52:59Z 2012-02-04T14:10:13Z <p>Jesper Grodal's reference to Bourbaki is a reasonable one for these questions, including 1). There are also two volumes in the Springer GTM series which treat many aspects of compact Lie groups, including the book by Bump indicated and the earlier text GTM 98 (1985) by Brocker &amp; Tom Dieck on <em>Representations of Compact Lie Groups</em> where V.7 treats the fundamental group and related matters thoroughly. Naturally the notation differs somewhat in such sources, but the answers to the questions raised here are all standard and arise from early work of Cartan, Weyl, and others. In general, the topology of a semisimple Lie group depends just on the topology of a maximal compact subgroup. </p> <p>In the setting of abstract root systems, motivated by the theory of semisimple Lie groups over the complex field and their Lie algebras or by compact semisimple Lie groups, the notion of "fundamental group" focuses on the quotient of the abstract weight lattice by the abstract root lattice. In semisimple groups or Lie algebras, the actual weight lattice (or character group) of a maximal torus can vary from the root lattice to the full abstract weight lattice, but the quotients in any case are finite and easily computable for each simple type. Moreover, the abstract fundamental group for a given root system is realized internally as the center of a simply connected group. </p> http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group/87532#87532 Answer by Thomas Nikolaus for Weight lattice and the first fundamental group Thomas Nikolaus 2012-02-04T15:15:42Z 2012-02-04T15:20:47Z <p>The fundamental group $\pi_1(G)$ is isomorphic to the quotient of the integral lattice by the inverse roots. </p> <p>A nice exposition (also including answers to the other questions) is in Chapter V (7) of "Broecker , tom Dieck - Representations of compact Lie groups (Springer 1985)."</p> http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group/89285#89285 Answer by Claudio Gorodski for Weight lattice and the first fundamental group Claudio Gorodski 2012-02-23T14:28:15Z 2012-02-23T14:28:15Z <p>Question 3 has also been answered in <a href="http://mathoverflow.net/questions/66099/h-2-of-a-simply-connected-lie-group-vanishes/66373/" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/87015/weight-lattice-and-the-first-fundamental-group/89343#89343 Answer by Renato G Bettiol for Weight lattice and the first fundamental group Renato G Bettiol 2012-02-23T23:05:39Z 2012-02-23T23:05:39Z <p>Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found <a href="http://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups/8961#8961" rel="nofollow">here</a>. A nice observation made there is that one can use the same arguments to show that <strong>$\pi_3(G)$ is torsion-free!</strong> (which I guess was not in your list of facts about homotopy of compact Lie groups).</p>