tag:blogger.com,1999:blog-68591010106571528422024-02-20T01:19:15.435-08:00badchemicalsUnknownnoreply@blogger.comBlogger1125tag:blogger.com,1999:blog-6859101010657152842.post-83016035693296825602011-01-18T20:53:00.000-08:002011-01-20T12:30:19.888-08:00/E8 Root System<br />
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">What is E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">?</span></span></h1>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">There actually are 4 different but related things called E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> is first of all the largest exceptional root system, which is a set of vectors in an 8-dimensional real vector space satisfying certain properties. Root systems were classified by Wilhelm Killing in the 1890s. He found 4 infinite classes of Lie algebras, labelled A</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">n</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, B</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">n</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, C</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">n</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, and D</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">n</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, where n=1,2,3.... He also found 5 more </span></span><em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">exceptional</span></span></em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> ones: G</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">2</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, F</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">4</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">6</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">7</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, and E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">The E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> root system consists of all vectors (called roots) (a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">1</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">2</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">3</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">4</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">5</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">6</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">7</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">,a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">) where all a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">i</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> are integers or all a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">i</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> are integers plus 1/2, the sum is an even integer, and sum of the squares is 2. An example with all integers is (-1,0,1,0,0,0,0,0) (there are 112 of these) and an example with half-integers is (1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2) (there are 128 of these). E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> has 240 roots. The 8 refers to the fact that there are 8 coordinates. See </span></span><a href="http://aimath.org/E8/mcmullen.html"><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">a</span></span><span class="Apple-style-span" style="color: #666666;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> picture of the E</span></span><sub><span class="Apple-style-span" style="color: #666666;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #666666;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> root system</span></span></a><span class="Apple-style-span" style="color: #999999;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">Secondly E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> refers to the </span></span><em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">root lattice</span></span></em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> obtained by taking all sums (with integral coefficients) of the vectors in the root system. It consists of all vectors above with all a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">i</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> integers, or all a</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">i</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> integers plus 1/2, and whose sum is even. The integers of squared length 2 are precisely the roots. This lattice, sometimes called the "8-dimensional diamond lattice", has a number of remarkable properties. It gives most efficient sphere-packing in 8 dimensions, and is also the unique</span></span><em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">even, unimodular</span></span></em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> lattice in 8 dimensions. This latter property makes it important in </span></span><a href="http://aimath.org/E8/e8andphysics.html"><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">string theory</span></span></a><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">Next E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> is a semisimple Lie algebra. A Lie algebra is a vector space, equipped with an operation called the Lie bracket. A simple example is the set of all 2 by 2 matrices. This is a 4-dimensional vector space. The Lie bracket operation is [X,Y]=XY-YX.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> is a 248-dimensional Lie algebra. Start with the 8 coordinates above, and add a coordinate for each of the 240 roots of the E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> root system. This vector space has an operation on it, called the </span></span><em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">Lie bracket</span></span></em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">: if X,Y are in the Lie algebra so is the Lie bracket [X,Y]. This is like multiplication, except that it is not commutative. Unlike the example of 2 x 2 matrices, it is very hard to write down the formula for the Lie bracket on E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">This is a complex Lie algebra, i.e. the coordinates are complex numbers. Associated to this Lie algebra is a (complex) Lie group, also called E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">. This complex group has (complex) dimension 248. The E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> Lie algebra and group were studied by Elie Cartan in 1894.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">Finally E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> is one of three real forms of the the complex Lie group E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">. Each of these three real forms has </span></span><em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">real</span></span></em><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> dimension 248. The group which we are referring to in this web site is the split real form of E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">.</span></span></div>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: large;">Geometric description of the split real form of E</span></span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: large;">8</span></span></span></sub></h3>
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<span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">Consider 16x16 real matrices X satisfying two conditions. First of all X is a rotation matrix, i.e. its rows and columns are orthonormal. Secondly assume X</span></span><sup><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">2</span></span></sup><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">=-I. The set of all such matrices V</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">0</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> is a geometric object (a "real algebraic variety"), and it is 56-dimensional. There is a natural way to add a single circle to this to make a 57-dimensional variety V. (V=Spin(16)/SU(8), and is circle bundle over V</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">0</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">, to anyone keeping score). Finally E</span></span><sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">8</span></span></sub><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> is a group of symmetries of V.</span></span><br />
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<span class="Apple-style-span" style="color: #eeeeee; font-family: 'Courier New', Courier, monospace;">related:</span></div>
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<span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"></span><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="color: #674ea7;"> <a href="http://www.sciencedaily.com/releases/2010/01/100118232345.htm"><span class="Apple-style-span" style="color: #674ea7;">E8 Symmetry and Golden Ratio Seen in Spins of Ultracold Electrons</span></a></span></span><br />
<span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="color: #674ea7;"> <a href="http://www.aimath.org/E8/e8andphysics.html"><span class="Apple-style-span" style="color: #674ea7;">E8 in String Theory</span></a></span></span><br />
<span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="color: #674ea7;"> <a href="http://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything"><span class="Apple-style-span" style="color: #674ea7;">E8 Theory proposal for UFT</span></a></span></span><br />
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<span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-family: Times;"><span class="Apple-style-span" style="color: #eeeeee;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">the above article is reposted from:</span></span><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> </span><a href="http://aimath.org/E8/e8.html"><span class="Apple-style-span" style="color: #674ea7;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">http://aimath.org/E8/e8.html</span></span></a></span></span></div>
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<span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="color: #eeeeee;">back to</span> <a href="http://astralquilting.blogspot.com/2011/01/hidden-symmetry.html"><span class="Apple-style-span" style="color: #674ea7;">hidden symmetry</span></a></span></div>
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<br /></div>Unknownnoreply@blogger.com