{"id":352,"date":"2007-03-19T15:03:15","date_gmt":"2007-03-19T15:03:15","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/03\/19\/the-mapping-of-the-e8-lie-group-minor-update\/"},"modified":"2007-03-19T15:03:15","modified_gmt":"2007-03-19T15:03:15","slug":"the-mapping-of-the-e8-lie-group-minor-update","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/03\/19\/the-mapping-of-the-e8-lie-group-minor-update\/","title":{"rendered":"The Mapping of the E8 Lie Group (Minor Update)"},"content":{"rendered":"

I’ve been getting tons of mail from people in response to the announcement of the mapping of
\nthe E8<\/sub> Lie group, asking what a Lie group is, what E8<\/sub> is, and why the mapping of E8<\/sub> is such a big deal?<\/p>\n

<\/p>\n

Let me start by saying that this is way outside of my area of expertise. So I fully expect that I’ll manage to screw something<\/em> up as I try to figure it out and explain it – so do follow the comments, where I’m sure people who know this better than I do will correct whatever errors I make.<\/p>\n

Let’s start with the easy part. What’s a Lie group? Informally, it’s a group whose objects
\nform a manifold, and whose group operation is a continuous function. We can break that down a bit, to make it a little bit clearer.<\/p>\n

A group is a set of objects\/values with a single binary operator that has a certain set of basic properties: associativity, existence of inverse, existence of identity. It’s one of the simplest constructions of abstract algebra. What’s really fascinating about it is that that simple construction – the set plus one operation will a simple set of properties – defines the entire concept of symmetry.<\/p>\n

Groups don’t normally require any structure on their members beyond what’s required to make the group operator work properly. You can define a group whose values are a set of points, a set of numbers, a set of coins – very nearly anything you want.<\/p>\n

But there are certain structured sets of values that we care about, which you can
\nuse as the objects for a group. One of those is a topological space. A topological space is
\njust a collection of objects which have a kind of nearness\/adjacency relationship between
\nthe objects in the collection. So a group on a topological space is interesting, because what it does is define symmetry on a set of values that preserves the nearness\/adjacency relationships
\nof the objects in the space.<\/p>\n

Even more interesting, we can define a particular kind of topological space: a manifold<\/em>, which is a sort of “smooth” topological space: a manifold is a topological space where the structure of the nearness\/adjacency relations makes every small finite region of the space appear to be Euclidean.<\/p>\n

So a Lie group is a group whose objects form a manifold, and whose group operations preserve
\nthe manifold structure of the nearness\/adjacency relations.<\/p>\n

Moving on – what’s E8<\/sub>? <\/p>\n

\"e8plane2a.jpg\"<\/p>\n

Many lie groups are based on topological spaces that whose values are representable as some collection of matrices or groups. E8<\/sub> is one of those – it’s a group based on something called a root system<\/em>. The root system for E8<\/sub> consists of a set of 8-dimensional vectors, which fall into two families. One family consists of all 8-dimensional vectors, with
\n2 unit-length elements, and 6 0-length elements; things like (1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0, 0, 1), etc. The other family consists of all of the 8-dimensional
\nvectors whose elements are all either +1\/2 or -1\/2, where the sum of all of the elements are even. So (1\/2, 1\/2, 1\/2, 1\/2, -1\/2, -1\/2, -1\/2, -1\/2) is a member of the root system, since the sum of those elements is 0; (1\/2, 1\/2, -1\/2, 1\/2, 1\/2, -1\/2, 1\/2, -1\/2) is not<\/em> an element of the root, system, since it’s sum is 1. The beautiful image over to the right is the image of the root system of E8<\/sub>.<\/p>\n

The E8<\/sub> Lie group is based on that root system – it’s a massive structure with one complex dimension<\/em> (complex as in complex numbers – it’s value in each dimension is a complex number) for each of the members of the root system. So its a manifold with 248<\/em> complex dimensions, or 496 real dimensions.<\/p>\n

There are two reasons that having mapped E8<\/sub> is so important. The practical one is that E8<\/sub> has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8<\/sub>. E8<\/sub> seems to be part of the structure of our universe. <\/p>\n

The other reason is just that the complete mapping of E8<\/sub> is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes<\/em> to store the map of E8<\/sub>. If you were to write it out on paper in 6-point print (that’s really small print<\/em>), you’d need a piece of paper bigger than the island of Manhattan. This thing is huge<\/em>.<\/p>\n

Update: <\/b> For those who claim that mathematicians have no sense of himor, I heard via Gooseana<\/a> that the title of the formal presentation where they’ll be talking about the E8<\/sub> map is: “The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness”.<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve been getting tons of mail from people in response to the announcement of the mapping of the E8 Lie group, asking what a Lie group is, what E8 is, and why the mapping of E8 is such a big deal?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-352","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-5G","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/352","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=352"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/352\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=352"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}