Exceptional groups and elementary particle structures

Elementary Abelian Hopf Galois Structures and Polynomial Formal Groups

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galoi...

Particle Structures in Elementary Cellular Automaton Rule 146

Stochastic particle-like persistent structures are found in the class 3 elementary cellular automaton rule number 146. These particles arise as boundaries separating regions with black cells occupying sites at spacetime points Hx, tL of constant parity x⊕ t. The particles execute random walks and annihilate in pairs, with particle density decaying with time in a power-law fashion. It is shown t...

Lie’s, Tracks, and Exceptional Groups

Exceptional Lie groups

Now, in the present book, we describe simply connected compact exceptional simple Lie groups G2, F4, E6, E7, E8, in very elementary way. The contents are given as follows. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms σ of G, and determine the group structures of the fixed points subgroup G by σ. Note that they co...

Symmetric Spaces of Exceptional Groups

We adress the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1+2 cases for G2 and F4 respectively are easily explained from the octonionic nature of these groups. The 4+3+2 cases on the E6,7,8 series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in 5, 4 and 3 spacetime dimensions.