What groups are Lie groups?

What is a simple explanation of E8 and the other Lie Groups?

• Answer:

  I first apologise about my lack of knowledge, and hence comprehensive approach in this field. But i will try my best to give you an understandable and interesting answer.First of all.....E8 is a "Lie group", that is an abstract concept of symmetry, that is found in a "ball" of 240 points in 8 dimensional space. (there are other "balls" but this is the better know one, called 4).A Lie group is a "group" that is at the same time also a "Differentiable Manifold".A "Differentiable Manifold" is a "manifold" that is "smooth" enough to enable mathematicians to apply Calculus on it. (As opposed to un-differentiable manifolds)A "Manifold" is a space that satisfy properties of Euclidean space when the space is microscopically viewed from the neighbourhood of each point. (As opposed to non-Euclidean space such as curved space) A Lie group is an important part of the Classification of finite simple groups which took about 200 years to be complete just in 2012 and had only been recently proved to be completed, and have a proof so long that almost no body can understand the entire proof. Efforts have been trying to publish the entire proof (which will take about 5000 pages), and to find new ways to present it.  The classification is important because it is just like formulating a "periodic table" to describe all the fundamental "elements"  that makes up the "universe" of Finite groups. Secondly, The E8 is a very interesting thing for one single reason. It is the biggest member of the three members, in the "trinities".The "trinities" are a set of 3 things, that appear in various different fields of mathematics, in different "manifestations".For example, in the field of numbers (A) real numbers (B) complex numbers (C) quarternionsis related to the(A) tetrahedron (B) cube (C) dodecahedronin the field of symmetries 3D Platonic solids.which is related to the Lie Groups(A)E6  (B)E7   (C)E8Please refer to http://www.neverendingbooks.org/arnolds-trinities  for more examples of trinities.Another note-worthy things is the the Bigger member of the trinity often "contain" information that makes up other two "smaller" members. For example, in the trinity of (A) tetrahedron (B) cube (C) dodecahedronby using lines to join certain vertices in certain ways, one regular Dodecahedron contains 5 unique regular Cubes and 10 unique regular Tetrahedrons, that are overlapping.Therefore you can say in a way, as you hold this 8 dimensional ball of 240 points in you hand, you are in some ways also holding a good deal of other Lie Groups, namely E7, E6, F4, G2.because of these "trinities", (1) E8 is related to the Leech lattice, which is the densiest way to pack spheres in a "very beautiful" way, that is only possible at 24 dimensions, based on the "very beautiful" fact that the first 24 squares (ie. 1,4,9,16,25,36.....) sum to the square of 70 (ie. 4900) .(2) E8 is related to the Monster group, which is the largest of the 26 (or 27) sporadic groups in the Classification of Finite Simple Groups, as well as the only one that "contains" the greatest number of other groups (ie. 19 out of 27 have been contained)(3) E8 is related to certain branches of string theory as a fact(3.1)  and E8 might be related to every particle and force as a controversial theory, named as "Exceptionally Simply Theory of Everything", formulated by Garrett Lisi, that is trying to give a "full map of all forces and particles" in subatomic physics and string theory, which according to its proponents have "successfully predicted the Higgs particle's existence before it's discovery".(3.2) E8 had predicted how electrons behave in very cold conditions https://www.newscientist.com/article/dn18356-most-beautiful-math-structure-appears-in-lab-for-first-time/ Lastly, this is how you can relate to E8 from your everyday life.(4.1) E8 is "contained" in the number 6.According in one of the interesting articles written by John Baez (i forgot the original discoverer whom he was quoting) ,the "Extended Dynkin diagram" of E8, which are strings of 6 dots, 3 dots and 2 dots joined together at 120 degrees to each other with the first member overlapping so that there will be only 9 dots altogether.   is exactly describes how you can divide 6 into 3 and 2 and 1.Therefore if you draw a diagram how 6 can be divided by all smaller numbers, you are drawing exactly the diagram that describes symmetries of E8.(4.2)according to McKay's correspondence, E8 can be "simplified" into a dodecahedron, which is found in the twelve pentagons of typical football.So next time when you write the number 6, or when you see a football, remember to "see the universe in a grain of sand", to see E8 from a number 6 and a football, and hence all other related topic of mathematics and string theory.