CINXE.COM

<?xml version='1.0' encoding='iso-8859-1'?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>The Three-Fold Way (Part 2) | The n-Category Café</title> <meta name="ICBM" content="30.2893, -97.7367" /> <meta name="DC.title" content="The n-Category Café" /> <meta name="viewport" content="width=device-width, initial-scale=1"/> <meta http-equiv="Content-Style-Type" content="text/css" /> <meta http-equiv="Content-Script-Type" content="text/javascript" /> <style type="text/css" media="all" title="The n-Category Café Stylesheet">@import url("/category/styles-site.css");</style> <link rel="stylesheet" href="/category/print.css" type="text/css" media="print" /> <link rel="alternate" type="application/atom+xml" title="Atom 1.0" href="/category/atom10.xml" /> <link rel="alternate" type="application/atom+xml" title="Atom 1.0 Comments Feed" href="/category/comments.atom" /> <link rel="EditURI" type="application/rsd+xml" title="RSD" href="/category/rsd.xml" /> <link rel="shortcut icon" type="image/x-icon" href="/category/images/favicon.ico" /> <link rel="icon" type="image/x-icon" href="/category/images/favicon2.ico" /> <link rel="pgpkey" href="/category/urs.asc" title="Urs Schreiber's PGP Public Key" /> <link rel="start" href="/category/" title="Home" /> <link rel="search" href="/cgi-bin/MT-3.0/mt-search.cgi?IncludeBlogs=3&Template=category" title="Search" /> <link rel="contents" href="/category/archives.html" title="Comprehensive List of Blog Entries" /> <link rel="bookmark" href="#content" title="Main Content" /> <link rel="bookmark" href="#comments" title="Comments" /> <link rel="bookmark" href="#search" title="Search Box" /> <link rel="bookmark" href="#related" title="Related entries" /> <link rel="bookmark" href="#AccessKeyList" title="AccessKeys" /> <link rel="prev" href="/category/2010/12/division_algebras_and_quantum.html" title="The Three-Fold Way (Part 1)" /> <link rel="next" href="/category/2010/12/pictures_of_modular_curves_vi.html" title="Pictures of Modular Curves (VI)" /> <link rel="up" href="/category/2010/12/" title="December, 2010" /> <script type="text/javascript" src="/category/prototype.js"></script> <script type="text/javascript" src="/category/ie.js"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script>  </head> <body> <div class="frame"> <div id="banner"> <h1><a href="/category/" accesskey="1" title="Jump to the blog main page">The n-Category Café</a></h1> <div class="description">A group blog on math, physics and philosophy</div> </div> <h2 class="skip">Skip to the Main Content</h2> <div class="skip"> <a href="#content" accesskey="2">Enough, already! Skip to the content.</a> </div> <p style="display:none;"> <strong>Note:</strong>These pages make extensive use of the latest XHTML and CSS <a href="http://www.w3.org">Standards</a>. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently <em>only</em> supported in Mozilla. My best suggestion (and you will <em>thank</em> me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source <a href="http://www.mozilla.org">Mozilla</a> browser. </p> <div class="blog"> <div><div id='menu'> <a href='/category/2010/12/division_algebras_and_quantum.html' accesskey='p'>« The Three-Fold Way (Part 1)</a> | <a href='/category/'>Main</a> | <a href='/category/2010/12/pictures_of_modular_curves_vi.html' accesskey='n'>Pictures of Modular Curves (VI) »</a> </div> <h2 class='date'>December 7, 2010</h2> <div id='content' class='blogbody'> <h3 class='title'>The Three-Fold Way (Part 2)</h3> <h4 class='posted'>Posted by John Baez</h4> <div><a href='http://golem.ph.utexas.edu/~distler/blog/mathml.html' onclick='window.open(this.href, 'MathML', 'width=310,height=150,scrollbars=no,resizable=yes,status=no'); return false;' onkeypress='if(window.event.keyCode == 13){window.open(this.href, 'MathML', 'width=310,height=150,scrollbars=no,resizable=yes,status=no'); return false;}'><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p><a href='http://golem.ph.utexas.edu/category/2010/12/division_algebras_and_quantum.html'>Last time</a> I described some problems with real and quaternionic quantum theory — or at least, ways in which they’re peculiar compared to good old complex version of this theory.</p> <p>This time I’ll tell you about the <i>three-fold way</i>, and you’ll begin to see how real and quaternionic Hilbert spaces are lurking in complex quantum theory. </p> <p>The name ‘three-fold way’ goes back to Dyson:</p> <ul> <li>Freeman Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, <i>Jour. Math. Phys.</i> <b>3</b> (1962), 1199–1215. </li> </ul> <p>But the idea goes back much further, to a paper by Frobenius and Schur:</p> <ul> <li> F. G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen, <i>Sitzungsber. Akad. Preuss. Wiss.</i> (1906), 186–208. </li> </ul> <p>I’ll admit I haven’t read this paper, so I’m not quite sure what they did, but everyone cites this and mentions the ‘Frobenius–Schur indicator’ when discussing the fact that irreducible group representations come in three kinds. </p> <p>And that’s what I’ll explain now. As you’ll see, the trinity of ‘real’, ‘complex’ and ‘quaternionic’ goes hand-in-hand with another famous trinity: ‘orthogonal’, ‘unitary’ and ‘symplectic’!</p> <div id='more'> <div><a href='http://golem.ph.utexas.edu/~distler/blog/mathml.html' onclick='window.open(this.href, 'MathML', 'width=310,height=150,scrollbars=no,resizable=yes,status=no'); return false;' onkeypress='if(window.event.keyCode == 13){window.open(this.href, 'MathML', 'width=310,height=150,scrollbars=no,resizable=yes,status=no'); return false;}'><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>Some aspects of quantum theory become more visible when we introduce <i>symmetry</i>. This is especially true when it comes to the relation between real, complex and quaternionic quantum theory. So, instead of bare Hilbert spaces, let’s consider Hilbert spaces equipped with a representation of a group. For simplicity suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is a Lie group, where we count a discrete group as a 0-dimensional Lie group. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>Rep</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Rep(G)</annotation></semantics></math> be the category where:</p> <ul> <li> An object is a finite-dimensional complex Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> equipped with a continuous unitary representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, say <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mo lspace='0em' rspace='0.16667em'>U</mo><mo stretchy='false'>(</mo><mi>H</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\rho : G \to \U(H)</annotation></semantics></math>. </li> <li> A morphism is an operator <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>T</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>T : H \to H'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>T</mi><mi>ρ</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>ρ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo><mi>T</mi></mrow><annotation encoding='application/x-tex'>T \rho(g) = \rho'(g) T</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>g \in G</annotation></semantics></math>. </li> </ul> <p>To keep the notation simple, I’ll often call a representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mo lspace='0em' rspace='0.16667em'>U</mo><mo stretchy='false'>(</mo><mi>H</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\rho : G \to \U(H)</annotation></semantics></math> simply by the name of its underlying Hilbert space, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>.</p> <p>Many of the operations that work for finite-dimensional Hilbert spaces also work for <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>Rep</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Rep(G)</annotation></semantics></math>, with the same formal properties: for example, direct sums, tensor products and duals. This lets us formulate the <b>three-fold way</b> as follows. </p> <p>If an object <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi><mo>∈</mo><mi>Rep</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H \in Rep(G)</annotation></semantics></math> is <b>irreducible</b> — not a direct sum of other representations in a nontrivial way — then there are three mutually exclusive choices:</p> <ul> <li> The representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is not isomorphic to its dual. In this case we call it <b>complex</b>. </li> <li>The representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is isomorphic to its dual and it is <b>real</b>, meaning that we can get it from a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on a real Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>H</mi> <mi>ℝ</mi></msub></mrow><annotation encoding='application/x-tex'>H_\mathbb{R}</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>H</mi> <mi>ℝ</mi></msub><mo>⊗</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'> H = H_\mathbb{R} \otimes \mathbb{C} </annotation></semantics></math> </li> <li> The representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is isomorphic to its dual and it is <b>quaternionic</b>, meaning that we can get it from a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on a quaternionic Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>H</mi> <mi>ℍ</mi></msub></mrow><annotation encoding='application/x-tex'>H_\mathbb{H}</annotation></semantics></math>: <div class='centeredfigure'> <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>H =</annotation></semantics></math> the underlying complex representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>H</mi> <mi>ℍ</mi></msub></mrow><annotation encoding='application/x-tex'>H_\mathbb{H}</annotation></semantics></math> </div> </li> </ul> <p>This is the <b>three-fold way</b>. But where do these three choices come from?</p> <p>Well, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi><mo>∈</mo><mi>Rep</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H \in Rep(G)</annotation></semantics></math> is irreducible. Then there is a 1-dimensional space of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>f</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>f : H \to H</annotation></semantics></math>, by Schur’s Lemma. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi><mo>≅</mo><msup><mi>H</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>H \cong H^*</annotation></semantics></math>, there is also a 1d space of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>T</mi><mo>:</mo><mi>H</mi><mo>→</mo><msup><mi>H</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>T : H \to H^*</annotation></semantics></math>, and thus a 1d space of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>g</mi><mo>:</mo><mi>H</mi><mo>⊗</mo><mi>H</mi><mo>→</mo><mi>ℂ</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> g : H \otimes H \to \mathbb{C} . </annotation></semantics></math> We can also think of these as bilinear maps <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>g</mi><mo>:</mo><mi>H</mi><mo>×</mo><mi>H</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'> g : H \times H \to \mathbb{C} </annotation></semantics></math> that are invariant under the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. </p> <p>But the representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi><mo>⊗</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>H \otimes H</annotation></semantics></math> is the direct sum of two others: the space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mi>H</mi></mrow><annotation encoding='application/x-tex'>S^2 H</annotation></semantics></math> of symmetric tensors, and the space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msup><mi>Λ</mi> <mn>2</mn></msup><mi>H</mi></mrow><annotation encoding='application/x-tex'>\Lambda^2 H</annotation></semantics></math> of antisymmetric tensors: <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>H</mi><mo>⊗</mo><mi>H</mi><mo>≅</mo><msup><mi>S</mi> <mn>2</mn></msup><mi>H</mi><mo>⊕</mo><msup><mi>Λ</mi> <mn>2</mn></msup><mi>H</mi></mrow><annotation encoding='application/x-tex'> H \otimes H \cong S^2 H \oplus \Lambda^2 H </annotation></semantics></math> So, either there exists a nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> that is <b>symmetric</b>: <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> g(v,w) = g(w,v) </annotation></semantics></math> or a nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> that is <b>antisymmetric</b>: <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>=</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> g(v,w) = -g(w,v) </annotation></semantics></math> One or the other, not both!—for if we had both, the space of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi><mo>:</mo><mi>H</mi><mo>⊗</mo><mi>H</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>g : H \otimes H \to \mathbb{C}</annotation></semantics></math> would be at least two-dimensional.</p> <p>Either way, we can write <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>⟨</mo><mi>J</mi><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'> g(v,w) = \langle J v, w \rangle </annotation></semantics></math> for some function <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>J: H \to H</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> and the inner product are both invariant under the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, this function <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> must commute with the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. But note that since <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is linear in the first slot, while the inner product is not, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> must be <b>antilinear</b>, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>J</mi><mo stretchy='false'>(</mo><mi>v</mi><mi>x</mi><mo>+</mo><mi>w</mi><mi>y</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>J</mi><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo><msup><mi>x</mi> <mo>*</mo></msup><mo lspace='0.11111em' rspace='0em'>+</mo><mi>J</mi><mo stretchy='false'>(</mo><mi>w</mi><mo stretchy='false'>)</mo><msup><mi>y</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'> J(v x + w y) = J(v)x^* + J(w)y^* </annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>v,w \in V</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝕂</mi></mrow><annotation encoding='application/x-tex'>x,y \in \mathbb{K}</annotation></semantics></math>. </p> <p>The square of an antilinear operator is linear. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>J^2</annotation></semantics></math> is linear and it commutes with the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. By Schur’s Lemma, it must be a scalar multiple of the identity: <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'> J^2 = c </annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>c \in \mathbb{C}</annotation></semantics></math>. We wish to show that depending on whether <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is symmetric or antisymmetric, we can rescale <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> to achieve either <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>J^2 = 1</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>J^2 = -1</annotation></semantics></math>. To see this, first note note that depending on whether <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is symmetric or antisymmetric, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mo>±</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \pm g(v,w) = g(w,v) </annotation></semantics></math> and thus <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mo>±</mo><mo stretchy='false'>⟨</mo><mi>J</mi><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>⟩</mo><mo>=</mo><mo stretchy='false'>⟨</mo><mi>J</mi><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>⟩</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pm \langle J v, w\rangle = \langle J w, v \rangle . </annotation></semantics></math> Now choose <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>v</mi><mo>=</mo><mi>J</mi><mi>w</mi></mrow><annotation encoding='application/x-tex'>v = J w</annotation></semantics></math>. This gives <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mo>±</mo><mo stretchy='false'>⟨</mo><msup><mi>J</mi> <mn>2</mn></msup><mi>w</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>⟩</mo><mo>=</mo><mo stretchy='false'>⟨</mo><mi>J</mi><mi>w</mi><mo>,</mo><mi>J</mi><mi>w</mi><mo stretchy='false'>⟩</mo><mo>≥</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pm \langle J^2 w,w\rangle = \langle J w , J w \rangle \ge 0. </annotation></semantics></math> It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mo>±</mo><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>\pm J^2 = c</annotation></semantics></math> is positive. So, if we divide <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> by the positive square root of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>, we get a new antilinear operator — let’s again call it <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> — with <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>J^2 = \pm 1</annotation></semantics></math>.</p> <p>Rescaled this way, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> is <b>antiunitary</b>: it is an invertible antilinear operator with <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mo stretchy='false'>⟨</mo><mi>J</mi><mi>v</mi><mo>,</mo><mi>J</mi><mi>w</mi><mo stretchy='false'>⟩</mo><mo>=</mo><mo stretchy='false'>⟨</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>⟩</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> \langle J v, J w\rangle = \langle w, v \rangle . </annotation></semantics></math> Now consider the two cases:</p> <ul> <li> If <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is symmetric, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is equipped with a <b>real structure</b>: an antiunitary operator <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>.</mo></mrow><annotation encoding='application/x-tex'> J^2 = 1 . </annotation></semantics></math> It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><msub><mi>H</mi> <mi>ℝ</mi></msub><mo>=</mo><mo stretchy='false'>{</mo><mi>v</mi><mo>∈</mo><mi>H</mi><mo lspace='0.11111em'>:</mo><mspace width='0.27778em' /><mi>J</mi><mi>v</mi><mo>=</mo><mi>v</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> H_{\mathbb{R}} = \{ v \in H \colon \; J v = v \} </annotation></semantics></math> is a real Hilbert space whose complexification is <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. And since <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> commutes with the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, there is a unitary representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>H</mi> <mi>ℝ</mi></msub></mrow><annotation encoding='application/x-tex'>H_{\mathbb{R}}</annotation></semantics></math> whose complexification gets us back the representation we started with! <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>.<br /><br /> </li> <li> If <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is antisymmetric, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is equipped with a <b>quaternionic structure</b>: an antiunitary operator <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mn>1</mn><mo>.</mo></mrow><annotation encoding='application/x-tex'> J^2 = -1 . </annotation></semantics></math> Whenever a complex Hilbert space has such a structure on it, we can make it into a quaternionic Hilbert space in exactly one way such that multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> is what it was before and multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> is the operator <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math>. Let’s call this quaternionic Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>H</mi> <mi>ℍ</mi></msub></mrow><annotation encoding='application/x-tex'>H_\mathbb{H}</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> commutes with the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, there is a unitary representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>H</mi> <mi>ℍ</mi></msub></mrow><annotation encoding='application/x-tex'>H_\mathbb{H}</annotation></semantics></math> whose underlying complex representation is the one we started with! </li> </ul> <p>In both these cases, <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is <b>nondegenerate</b>, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mo>∀</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mspace width='0.27778em' /><mspace width='0.27778em' /><mi>g</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn><mspace width='1em' /><mo>⇒</mo><mspace width='1em' /><mi>w</mi><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding='application/x-tex'> \forall v \in V \;\; g(v,w) = 0 \quad \implies \quad w = 0 .</annotation></semantics></math> The reason is that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>⟨</mo><mi>J</mi><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy='false'>⟩</mo></mrow><annotation encoding='application/x-tex'>g(v,w) = \langle J v, w \rangle</annotation></semantics></math>, and the inner product is nondegenerate, while <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> is one-to-one. </p> <p>We can compare real versus quaternionic representations using either <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math>. In the following statements, we don’t need the representation to be irreducible:</p> <ul> <li> Given a complex Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>, a nondegenerate symmetric bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi><mo>:</mo><mi>H</mi><mo>×</mo><mi>H</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>g : H \times H \to \mathbb{C}</annotation></semantics></math> is called an <b>orthogonal structure</b> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. So, a representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mo lspace='0em' rspace='0.16667em'>U</mo><mo stretchy='false'>(</mo><mi>H</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\rho : G \to \U(H)</annotation></semantics></math> is real iff it preserves some orthogonal structure <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. It’s also easy to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math> is real iff there is a real structure <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>J : H \to H</annotation></semantics></math> that commutes with the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. <br /><br /> </li> <li> Similarly, given a complex Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>, a nondegenerate skew-symmetric bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi><mo>:</mo><mi>H</mi><mo>×</mo><mi>H</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>g : H \times H \to \mathbb{C}</annotation></semantics></math> is called a <b>symplectic structure</b> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. So, a representation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mo lspace='0em' rspace='0.16667em'>U</mo><mo stretchy='false'>(</mo><mi>H</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\rho : G \to \U(H)</annotation></semantics></math> is quaternionic iff it preserves some symplectic structure <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>. It’s also easy to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math> is quaternionic iff there is a quaternionic structure <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>J</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>J : H \to H</annotation></semantics></math> that commutes with the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. </li> </ul> <p>This pattern is so cool that I can’t resist summarizing it in a cute little chart, suitable for printing out and putting in your wallet:</p> <div class='centeredfigure'> <b>THE THREEFOLD WAY</b> </div> <math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mrow><mtable displaystyle='true' columnalign='right left right left right left right left right left' columnspacing='0em'><mtr><mtd><mi>complex</mi></mtd> <mtd><mspace width='1em' /></mtd> <mtd><mi>H</mi><mo>≇</mo><msup><mi>H</mi> <mo>*</mo></msup></mtd> <mtd><mspace width='1em' /></mtd> <mtd><mi>unitary</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>real</mi></mtd> <mtd></mtd> <mtd><mi>H</mi><mo>≅</mo><msup><mi>H</mi> <mo>*</mo></msup></mtd> <mtd></mtd> <mtd><mi>orthogonal</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>quaternionic</mi></mtd> <mtd></mtd> <mtd><mi>H</mi><mo>≅</mo><msup><mi>H</mi> <mo>*</mo></msup></mtd> <mtd></mtd> <mtd><mi>symplectic</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>J</mi> <mn>2</mn></msup><mo>=</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} complex & \quad & H \ncong H^* & \quad & unitary \\ & & \\ real & & H \cong H^* & &orthogonal \\ & &J^2 = 1 \\ \\ quaternionic & & H \cong H^* & & symplectic \\ & &J^2 = -1 \end{aligned} </annotation></semantics></math> <p>For more on how this pattern pervades mathematics, see Dyson’s paper on the three-fold way, and also Arnold’s paper on mathematical ‘trinities’, which you can easily get online:</p> <ul> <li> Vladimir I. Arnold, <a href='http://www.neverendingbooks.org/DATA/ArnoldTrinities.pdf'>Symplectization, complexification and mathematical trinities</a>, in <i>The Arnoldfest: Proceedings of a Conference in Honour of V.I. Arnold for His Sixtieth Birthday</i>, edited by E. Bierstone, B. Khesin, A. Khovanskii and J. E. Marsden, AMS, Providence, Rhode Island, 1999. </li> </ul> <p>Also see Arnold’s paper on <a href='http://math.ucr.edu/home/baez/Polymath.pdf'>‘polymathematics’</a> and <a href='http://www.neverendingbooks.org/index.php/arnolds-trinities-version-20.html'>Lieven le Bruyn’s discussion</a> of this paper. Arnold does a pretty good job of grabbing the reader’s attention at the start here:</p> <blockquote> <p>All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as <acronym title='National Æronautics and Space Administration'>NASA</acronym>).</p> </blockquote> <p>But he gets into some very deep waters!</p> <p>Next time, I’ll say more about what the three-fold way means for quantum physics.</p> </div> <span class='posted'>Posted at December 7, 2010 1:00 AM UTC </span> </div> <p class='trackback-url'>TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2320</p> <h2 class='comments-head' id='related'>Some Related Entries</h2> <div id='pane'> <form method='get' action='/cgi-bin/MT-3.0/mt-search.cgi'> <fieldset class='search'> <input type='hidden' name='IncludeBlogs' value='3' /> <input type='hidden' name='Template' value='category' /> <label for='search' accesskey='4'>Search for other entries:</label><br /> <input id='search' name='search' size='25' /><br /> <input type='submit' value='Search' /> </fieldset> </form> </div> <ul class='blogbody related'> <li><a href='/category/2010/12/division_algebras_and_quantum.html'>The Three-Fold Way (Part 1)</a> — <i>Dec 03, 2010</i></li> <li><a href='/category/2010/12/solers_theorem.html'>Solèr’s Theorem</a> — <i>Dec 01, 2010</i></li> <li><a href='/category/2010/11/stateobservable_duality_part_3.html'>State-Observable Duality (Part 3)</a> — <i>Nov 29, 2010</i></li> <li><a href='/category/2010/11/stateobservable_duality_part_2.html'>State-Observable Duality (Part 2)</a> — <i>Nov 27, 2010</i></li> <li><a href='/category/2010/11/stateobservable_duality_part_1.html'>State-Observable Duality (Part 1)</a> — <i>Nov 25, 2010</i></li> <li><a href='/category/2009/02/the_cocktail_party_version.html'>The Cocktail Party Version</a> — <i>Feb 04, 2009</i></li> <li><a href='/category/2008/10/the_nature_of_time.html'>The Nature of Time</a> — <i>Oct 13, 2008</i></li> <li><a href='/category/2008/07/categories_logic_and_foundatio_1.html'>Categories, Logic and Foundations of Physics in Oxford</a> — <i>Jul 25, 2008</i></li> </ul> </div> <h2 class="comments-head" id="comments">17 Comments & 0 Trackbacks</h2> <div class="blogbody"> <div class="comments-body" id="c035961"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>There’s a problem with the ‘polymathematics’ link. The <a>http://math.ucr.edu/home/baez/Polymath.pdf</a> after the “(” works.</p></div> <div class="comments-post">Posted by: Peter Morgan on December 7, 2010 1:25 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035961" title="URL for comment by Peter Morgan [December 7, 2010 1:25 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35961" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Peter Morgan [December 7, 2010 1:25 PM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c035963"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>Fixed.</p></div> <div class="comments-post">Posted by: <a title="http://www.kent.ac.uk/secl/philosophy/staff/corfield.html" href="http://www.kent.ac.uk/secl/philosophy/staff/corfield.html" rel="nofollow">David Corfield</a> on December 7, 2010 2:01 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035963" title="URL for comment by David Corfield [December 7, 2010 2:01 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35963" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by David Corfield [December 7, 2010 2:01 PM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c035965"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>Thanks, David!</p></div> <div class="comments-post">Posted by: <a title="http://math.ucr.edu/home/baez/" href="http://math.ucr.edu/home/baez/" rel="nofollow">John Baez</a> on December 7, 2010 2:41 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035965" title="URL for comment by John Baez [December 7, 2010 2:41 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35965" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by John Baez [December 7, 2010 2:41 PM]">Reply to this</a> </div> </div> </div> </div> <div class="comments-body" id="c035966"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>A little bit of advertisement for the nLab, somewhat related to the role of symmetry in quantum physics:</p> <p><a href="http://ncatlab.org/nlab/show/topological+group">topological group</a> explains what an unitary representation of a topological group on a Hilbert space is, and </p> <p><a href="http://ncatlab.org/nlab/show/gauge+group">gauge group</a> explains the concepts of local and global internal symmetry groups.</p> </div> <div class="comments-post">Posted by: Tim van Beek on December 7, 2010 2:50 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035966" title="URL for comment by Tim van Beek [December 7, 2010 2:50 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35966" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Tim van Beek [December 7, 2010 2:50 PM]">Reply to this</a> </div> </div> <div class="comments-body" id="c035973"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>It’s now realized that Dyson’s 3-fold way is part of a 10-fold way (name due to Zirnbauer), with 10=8+2 (8 coming from real K-theory and 2 from complex K-theory). So…if you’ll allude to Dyson, perhaps one should say why things aren’t more general in your 3-fold way.</p></div> <div class="comments-post">Posted by: matt on December 7, 2010 10:15 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035973" title="URL for comment by matt [December 7, 2010 10:15 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35973" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by matt [December 7, 2010 10:15 PM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c035975"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>In previous posts I listed various theorems that pick out <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ℝ</mi><mo>,</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}, \mathbb{C}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{H}</annotation></semantics></math> as special — go <a href="http://golem.ph.utexas.edu/category/2010/12/division_algebras_and_quantum.html">here</a> and follow the links to read those posts.</p> <p>How about explaining to me more precisely how the 3-fold way fits into some “10-fold way”? </p></div> <div class="comments-post">Posted by: <a title="http://math.ucr.edu/home/baez/" href="http://math.ucr.edu/home/baez/" rel="nofollow">John Baez</a> on December 8, 2010 5:21 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035975" title="URL for comment by John Baez [December 8, 2010 5:21 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35975" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by John Baez [December 8, 2010 5:21 AM]">Reply to this</a> </div> </div> <div class="comments-body" id="c035986"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>Zirnbauer’s paper is</p> <ul><li>Martin R. Zirnbauer, “<a href="http://jmp.aip.org/resource/1/jmapaq/v37/i10/p4986_s1">Riemannian symmetric superspaces and their origin in random‐matrix theory</a>” <i>J. Math. Phys.</i> <b>37,</b> 4986 (1996). <a href="http://arxiv.org/abs/math-ph/9808012">arXiv:math-ph/9808012</a>.</li></ul> <p>His ten-fold way stems from Cartan’s classification of <a href="http://en.wikipedia.org/wiki/Symmetric_space">symmetric spaces</a>.</p> <p>Shinsei Ryu and colleagues have written several papers which I don’t pretend to understand, apparently connecting Zirnbauer’s classification to the condensed-matter physics of topological insulators and to <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>-theory. Here is one in an open-access journal:</p> <ul><li> S. Ryu <i>et al.</i> “<a href="http://iopscience.iop.org/1367-2630/12/6/065010">Topological insulators and superconductors: tenfold way and dimensional hierarchy</a>” <i>New J. Phys.</i> <b>12,</b> 065010 (2010).</li></ul> </div> <div class="comments-post">Posted by: <a title="http://www.sunclipse.org" href="http://www.sunclipse.org" rel="nofollow">Blake Stacey</a> on December 8, 2010 4:34 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c035986" title="URL for comment by Blake Stacey [December 8, 2010 4:34 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=35986" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Blake Stacey [December 8, 2010 4:34 PM]">Reply to this</a> </div> </div> <div class="comments-body" id="c036002"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>The refs given by Blake are good ones. See also recent papers by Kitaev. The 10-fold way is a classification of Hamiltonians with various anti-unitary symmetries. For example, time reversal is such a symmetry, and depending whether it squares to -1 or +1 you get either Dyson’s symplectic or orthogonal ensembles. However, there are other possible anti-unitary symmetries one can impose too. Another way to think of it is that each of the classes in the 10-fold way is a linear vector space of Hamiltonians, such that if A, B, and C are in that space, then so is the double commutator [A,[B,C]]. (Of course, one could have other spaces which are closed under this double commutator by taking direct sums, but other than direct sums, these spaces are, I think, all that there is).</p></div> <div class="comments-post">Posted by: matt on December 9, 2010 4:18 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036002" title="URL for comment by matt [December 9, 2010 4:18 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36002" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by matt [December 9, 2010 4:18 AM]">Reply to this</a> </div> </div> <div class="comments-body" id="c036004"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>Thanks, Matt and Blake! Clearly I have some reading to do. Luckily I sort of understand Cartan’s classification of symmetric spaces and its relation to Lie triple systems, which sounds like what <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[A,[B,C]]</annotation></semantics></math> is all about.</p> <p>In a nutshell: you get a symmetric space whenever you have a Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> with an automomorphism that squares to 1. This lets you give <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math>-grading <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>L</mi><mo>=</mo><msub><mi>L</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>L</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>L = L_0 \oplus L_1</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>L</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>L_k</annotation></semantics></math> is the space on which that automorphism acts as multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mi>k</mi></msup></mrow><annotation encoding='application/x-tex'>(-1)^k</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is the group with Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is the subgroup with Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>L_0</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>G</mi><mo stretchy='false'>/</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G/H</annotation></semantics></math> is your symmetric space.</p> <p>But the bracket in <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> also yields a trilinear operation</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo>,</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>,</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>:</mo><msub><mi>L</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>L</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>L</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>L</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>[[-,-],-] : L_1 \times L_1 \times L_1 \to L_1</annotation></semantics></math></p> <p>and this makes <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>L_1</annotation></semantics></math> into something called a <a href="http://eom.springer.de/l/l130040.htm">Lie triple system</a>. To build your symmetric space all you need is the Lie triple system. So, there’s an analogy:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display='block'><semantics><mrow><mi>Lie</mi><mspace width='0.16667em' /><mi>algebra</mi><mspace width='0.16667em' /><mo>:</mo><mspace width='0.16667em' /><mi>Lie</mi><mi>group</mi><mspace width='0.16667em' /><mo>:</mo><mo>:</mo><mspace width='0.16667em' /><mi>Lie</mi><mspace width='0.16667em' /><mi>triple</mi><mspace width='0.16667em' /><mi>system</mi><mspace width='0.16667em' /><mo>:</mo><mspace width='0.16667em' /><mi>symmetric</mi><mspace width='0.16667em' /><mi>space</mi></mrow><annotation encoding='application/x-tex'> Lie \, algebra \, : \, Lie group \, :: \, Lie \, triple \, system \, :\, symmetric \, space </annotation></semantics></math></p> <p>But Zirnbauer seems to be using ‘symmetric superspaces’, or (bad pun) ‘supersymmetric spaces’. Everything should work in a similar way here, and I bet Victor Kac, after classifying simple Lie supergroups, went ahead and generalized Cartan’s classification of symmetric spaces!</p></div> <div class="comments-post">Posted by: <a title="http://math.ucr.edu/home/baez/" href="http://math.ucr.edu/home/baez/" rel="nofollow">John Baez</a> on December 9, 2010 4:40 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036004" title="URL for comment by John Baez [December 9, 2010 4:40 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36004" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by John Baez [December 9, 2010 4:40 AM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c036006"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>In case any ultra-observant readers out there noticed the switch from Matt’s <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[A,[B,C]]</annotation></semantics></math> to my <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo>,</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>,</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[[-,-],-]</annotation></semantics></math>, I did that because the standard definition of Lie triple system uses the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo>,</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>,</mo><mo lspace='0.11111em' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[[-,-],-]</annotation></semantics></math>, which is antisymmetric in the first two arguments. It’s no big deal but it affects the definition.</p></div> <div class="comments-post">Posted by: <a title="http://math.ucr.edu/home/baez/" href="http://math.ucr.edu/home/baez/" rel="nofollow">John Baez</a> on December 9, 2010 6:55 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036006" title="URL for comment by John Baez [December 9, 2010 6:55 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36006" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by John Baez [December 9, 2010 6:55 AM]">Reply to this</a> </div> </div> </div> </div> <div class="comments-body" id="c036009"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><blockquote> <p>where we count a discrete group as a 0-dimensional Lie group.</p> </blockquote> <p>Huh. What does this add? Is it reasonable to consider the product of an n-dimensional Lie group and a discrete group to be another n-dimensional Lie group?</p> </div> <div class="comments-post">Posted by: Aaron Denney on December 9, 2010 8:01 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036009" title="URL for comment by Aaron Denney [December 9, 2010 8:01 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36009" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Aaron Denney [December 9, 2010 8:01 PM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c036010"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class="mathlogo" src="https://golem.ph.utexas.edu/~distler/blog/images/MathML.png" alt="MathML-enabled post (click for more details)." title="MathML-enabled post (click for details)." /></a></div> <blockquote> <blockquote> <p>where we count a discrete group as a 0-dimensional Lie group.</p> </blockquote> <p>Huh. What does this add? Is it reasonable to consider the product of an n-dimensional Lie group and a discrete group to be another n-dimensional Lie group?</p> <p></p></blockquote> <p>Yes. What else would you do?</p> <div class="comments-post">Posted by: Eugene Lerman on December 9, 2010 8:38 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036010" title="URL for comment by Eugene Lerman [December 9, 2010 8:38 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36010" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Eugene Lerman [December 9, 2010 8:38 PM]">Reply to this</a> </div> </div> <div class="comments-body" id="c036011"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>Isn’t that basically what a Lie group with more than one component is?</p></div> <div class="comments-post">Posted by: Tim Silverman on December 9, 2010 9:49 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036011" title="URL for comment by Tim Silverman [December 9, 2010 9:49 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36011" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Tim Silverman [December 9, 2010 9:49 PM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c036012"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <blockquote> <p>Isn’t that basically what a Lie group with more than one component is?</p> </blockquote> <p>In case this is not meant as a rhetorical question: the answer is Yes.</p> </div> <div class="comments-post">Posted by: <a title="http://nlab.mathforge.org/nlab/show/Urs%20Schreiber" href="http://nlab.mathforge.org/nlab/show/Urs%20Schreiber" rel="nofollow">Urs Schreiber</a> on December 9, 2010 11:58 PM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036012" title="URL for comment by Urs Schreiber [December 9, 2010 11:58 PM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36012" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by Urs Schreiber [December 9, 2010 11:58 PM]">Reply to this</a> </div> </div> </div> <div class="comments-body" id="c036015"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><div><a href="http://golem.ph.utexas.edu/~distler/blog/mathml.html"><img class='mathlogo' src='https://golem.ph.utexas.edu/~distler/blog/images/MathML.png' alt='MathML-enabled post (click for more details).' title='MathML-enabled post (click for details).' /></a></div> <p>John wrote:</p> <blockquote> <p>… where we count a discrete group as a 0-dimensional Lie group.</p> </blockquote> <p>Aaron wrote:</p> <blockquote> <p>Huh. What does this add?</p> </blockquote> <p>It’s just a clarification. A discrete group <i>is</i> a 0-dimensional Lie group, because a discrete topological space is a manifold. (See extra fine print below.) But people in Lie theory often avoid counting discrete groups as Lie groups. I just wanted to reassure the reader that I’m not avoiding those.</p> <p>So, for example, when people speak of the classification of simple Lie groups, they don’t include the <i>finite</i> simple groups — much less any <i>infinite</i> discrete simple groups. Indeed, you’ll see that in the study of <a href="http://en.wikipedia.org/wiki/Simple_Lie_group">simple Lie groups</a>, they’re defined in a rather convoluted way, not as Lie groups lacking nontrivial normal subgroups (the obvious thing you’d guess at first), but as ‘connected non-abelian Lie groups which do not have nontrivial connected normal subgroups.’ There’s a very good reason for this: it makes the classification manageable and nice. One thing it does is eliminate 0-dimensional Lie groups from consideration. But beware: <a href="http://en.wikipedia.org/wiki/Simple_Lie_group#Comments_on_the_definition">different people use different definitions of simple Lie group</a>, depending on what they want.</p> <p>So: I just wanted to say <i>relax: I’m not eliminating discrete groups from consideration</i>. There are interesting discrete symmetries in quantum theory, notably for systems like crystals or highly symmetric molecules like buckyballs. The results discussed here apply to those symmetries.</p> <div class='centeredfigure'> <img src='http://upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Buckminsterfullerene-perspective-3D-balls.png/300px-Buckminsterfullerene-perspective-3D-balls.png' alt='' /> </div> <p>In fact, all the results I mentioned in this particular blog entry apply not just to Lie groups but to arbitrary topological groups! But I thought physicists would enjoy the phrase ‘Lie group’ more than ‘topological group’. I wasn’t trying for maximum generality, just something easy to read.</p> <p><b>Extra fine print:</b> some people do not consider uncountable discrete topological spaces to be 0-dimensional manifolds, because <a href="http://en.wikipedia.org/wiki/Topological_manifold#Compactness_and_countability_axioms">they want every manifold to be second-countable</a>. </p> <blockquote> <p>Is it reasonable to consider the product of an <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-dimensional Lie group and a discrete group to be another <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-dimensional Lie group?</p> </blockquote> <p>Yes: as Tim pointed out, every Lie group with <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> components is the product of a connected Lie group and an <math xmlns="http://www.w3.org/1998/Math/MathML" display='inline'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math>-element discrete group.</p></div> <div class="comments-post">Posted by: <a title="http://math.ucr.edu/home/baez/" href="http://math.ucr.edu/home/baez/" rel="nofollow">John Baez</a> on December 10, 2010 1:08 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036015" title="URL for comment by John Baez [December 10, 2010 1:08 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36015" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by John Baez [December 10, 2010 1:08 AM]">Reply to this</a> </div> </div> </div> <div class="comments-body" id="c036083"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>Hi John.</p> <p>You defined irreducible as not having proper direct summands - but as you know that’s indecomposable. Irreducible means no proper subobjects, which is why we can apply Schur’s Lemma.</p></div> <div class="comments-post">Posted by: andrew hubery on December 14, 2010 9:42 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036083" title="URL for comment by andrew hubery [December 14, 2010 9:42 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36083" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by andrew hubery [December 14, 2010 9:42 AM]">Reply to this</a> </div> </div> <div class="comments-nest-box"> <div class="comments-body" id="c036088"> <h3 class="title">Re: The Three-Fold Way (Part 2)</h3> <div><p>Hi! I’m talking about finite-dimensional continuous unitary representations of Lie groups. These are indecomposable iff they’re irreducible, so my definition is equivalent to the usual one in this context.</p> <p>You might consider it perverse to use a nonstandard definition even in a context where it’s equivalent to the usual one. However, I’d like physicists to understand and enjoy what I’m saying. Many physicists know and love “irreducible” representations but have never heard about “indecomposable” representations — in part because quantum physics focuses on representations of the sort I’m considering, where there’s no difference. To make such physicists feel at home, I wanted to use the word “irreducible”. </p> <p>Google says:</p> <p>+”irreducible representation” +physics — 124,000 results</p> <p>+”indecomposable representation” +physics — 2,100 results</p></div> <div class="comments-post">Posted by: <a title="http://math.ucr.edu/home/baez/" href="http://math.ucr.edu/home/baez/" rel="nofollow">John Baez</a> on December 14, 2010 10:43 AM | <a href="/category/2010/12/the_threefold_way_part_2.html#c036088" title="URL for comment by John Baez [December 14, 2010 10:43 AM]">Permalink</a> | <a href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321;parent_id=36088" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}" title="Respond to comment by John Baez [December 14, 2010 10:43 AM]">Reply to this</a> </div> </div> </div> </div> <p class="newpost"><a class="comments-post" href="/cgi-bin/MT-3.0/sxp-comments.fcgi?entry_id=2321" onclick="OpenComments(this.href); this.blur(); return false;" onkeypress="if(window.event.keyCode == 13){OpenComments(this.href); this.blur(); return false;}">Post a New Comment</a></p> </div> <div id="footer"> <h2>Access Keys:</h2> <dl id="AccessKeyList"> <dt>0</dt><dd><a href="/category/accessibility.html" accesskey="0">Accessibility Statement</a></dd> <dt>1</dt><dd>Main Page</dd> <dt>2</dt><dd>Skip to Content</dd> <dt>3</dt><dd>List of Posts</dd> <dt>4</dt><dd>Search</dd> <dt>p</dt><dd>Previous (individual/monthly archive page)</dd> <dt>n</dt><dd>Next (individual/monthly archive page)</dd> </dl> <a href="/category/archives.html" accesskey="3"></a> </div> </div> </body> </html>