Quaternionic root systems and subgroups of the Aut(F4)

Abstract

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU (2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4), SP (2) ≈ SO(5), SO(8), SO(9), F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut(F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4 .The Weyl groups of many Lie algebras, such as, G2, SO(7), SO(8), SO(9), SU (3)XSU (3) and SP (3) × SU (2) have been constructed as the subgroups of Aut(F4) .We have also classified the other non-parabolic subgroups of Aut(F4) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e1, e2, e3 is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F4 respectively. INTRODUCTIONThere are a few celebrated Lie algebras which seem to be playing important roles in understanding the underlying symmetries of the unified theory of all interactions. The most popular ones are the exceptional Lie groups G 2 , F 4 , E 6 , E 7 and E 8 and the related groups [1]. The groups Spin7 and G 2 are proposed as the holonomy groups for the compactification of the M-theory from 11 to 4 dimensional space-time [2]. It is also well known that two orthogonal groups SO(8) and SO(9) are the little groups of the massles particles of string theories in 10-dimensions and the M-theory in 11-dimensions respectively. The fact that SO(9) can be embedded in the exceptional Lie group F 4 in a triply symmetric way and the non-compact F 4(−25) can be embedded in the Lorenz group SO(25, 1) indicates the importance of the exceptional group F 4 [3]. The largest exceptional group E 8 which had been suggested as the unified theory of the electroweak and strong interactions with three generations of lepton-quark families [4] naturally occurred as the gauge symmetry of the E 8 × E 8 heterotic string theory [5]. It has many novel mathematical aspects [6] which has not been exploited in physics. It was known that a non-compact version of E 7 manifests itself as a global symmetry of the 11-dimensional supergravity [7]. Some of its maximal subgroups show themselves as local symmetries [8]. The E 6 has been suggested as a unified theory of electroweak and strong interactions [9].The Weyl groups of these groups are also impor...

The E 8 Geometry from a Clifford Perspective

“…There is thus an abundance of root systems in 4D giving the Platonic solids, which is essentially due to the accidentalness of the spinor induction theorem, compared to arbitrary dimensions where the only root systems and Platonic solids are A n (n-simplex), B n (n-hypercube and n-hyperoctahedron) and D n . In fact the only 4D Platonic solid that is not equal or dual to a root system is the 5-cell with symmetry group A 4 , which of course could not be a root system, as its odd number (5) This systematises many case-by-case observations on the structure of the automorphism groups [17,18] (for instance, the automorphism group of the H 4 root system is 2I × 2I-in the spinor picture, it is not surprising that 2I yields both the root system and the two factors in the automorphism group), and shows that all of the 4D geometry is already contained in 3D [6] in the following sense. In terms of quaternionic representations of a 4D root system (e.g.…”

“…This gives the well-known exponents {1, 7,11,13,17,19,23, 29} a more geometric meaning as 30-fold rotations in four orthogonal eigenplanes (Fig. 9).…”

“…These are precisely the 120 root vectors of the 4D root system H 4 once reinterpreted using the 4D Euclidean metric. H 4 is exceptional and the largest noncrystallographic Coxeter group; it also has the exceptional automorphism group 2I × 2I [17,18]. However, this is trivial to see in terms of the spinor group 2I, as it must be trivially closed under left and right multiplication via group closure.…”

Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions

“…The generators in (14) generate the Coxeter group [22][23][24] which is isomorphic to the tetrahedral group of order 24. The automorphism group ( ) ≈ ( ): ≈ where is generated by the Dynkin diagram symmetry = [ , − ] * , which exchanges the first and the third simple roots and leaves the second root intact in Figure 2.…”

Catalan solids derived from three-dimensional-root systems and quaternions

“…In 4D, in addition to the Platonic Polytopes described by the symmetries of . The 24-cell is said to be selfdual because the 4 F diagram is invariant under the Dynkin-diagram symmetry [9,10].…”