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Chapter 1
Introduction
In 1974, Arthur Young, at the Institute for the Study of Consciousness in Berkeley, California, asked me as his research associate to work out the symmetry group of a toy tetrahedron. This turned out to be
4, the symmetric-4 group, which is the set of all 24 permutations of four objects (for example, the four vertices of the tetrahedron). The rotations of the tetrahedron corresponded to the 12-element normal subgroup of
4, called
4, the alternating-4 group, or the tetrahedral group. The
4 group also includes reflections of the tetrahedron and is called the octahedral group because it corresponds to all the rotations of the octahedron (or the cube, which is its dual).
These group structures entail the basic themes of this book: symmetry, rotation, reflection, permutation, subgroup, duality, commutativity and non-commutativity, cosets, mappings, representations, and character tables. Underlying it all is the interplay between geometry and algebra.
Moreover, the tetrahedral (
) and octahedral (
) groups correpond to two of the ADE Coxeter graphs, E
6 and E
7, via the McKay correspondence groups
and
, which are double covers of
and
.
The ostensible connection between group structures and consciousness (a very controversial topic) was via Arthur Eddingtonās fascination with symmetry groups as fundamental to physics. In his book, The Philosophy of Physical Science, Eddington [1939] wrote:
āThe recognition that physical knowledge is structural knowledge abolishes all dualism of consciousness and matter. Dualism depends on the belief that we find in the external world something of a nature incommensurable with what we find in consciousness; but all that physical science reveals to us in the external world is group-structure, and group-structure is also to be found in consciousness. When we take a structure of sensations in a particular consciousness and describe it in physical terms as part of the structure of an external world, it is still a structure of sensations. It would be entirely pointless to invent something else for it to be a structure of.ā
Since 1939 the importance of group theory in physics has grown by leaps and bounds, and is now at the very center of advances in unified field theory and other areas of theoretical physics. Indeed, prior theoretical advances such as Newtonian mechanics and Maxwellās electromagnetic theory are viewed from the standpoint of symmetry groups.
By far the deepest theoretical advance afforded by the group-theory approach is the set of ADE Coxeter graphs, which originally classified the most important (and useful) finite reflection groups, and in the form of Dynkin diagrams classified the most important (and useful) Lie groups. This work of the 1930s and 1940s has, in the last several decades evolved into the ADE classification of twenty-some mathematical categories, due to the work of many other mathematicians. The Russian mathematician, V. I. Arnold, as one of the most active and perceptive theoreticians in this process, wrote in his book, Catastrophe Theory [Arnold, 1986]:
āAt first glance, functions, quivers, caustics, wave fronts and regular polyhedra have no connection with each other. But in fact, corresponding objects bear the same label not just by chance: for example, from the icosahedron one can construct the function x2 + y3 + z5, and from it the diagram E8, and also the caustic and wave front of the same name.
āTo easily checked properties of one of a set of associated objects correspond properties of the others which need not be evident at all. Thus the relations between all the A, D, E-classifications can be used for the simultaneous study of all simple objects, in spite of the fact that the origin of many of these relations (for example, of the connections between functions and quivers) remains an unexplained manifestation of the mysterious unity of all things.ā
It is especially striking that the objects classified by the ADE graphs are of great utility in the advance of unified field theory afforded by superstring theory and its generalization to M-theory. The list of ADE classified categories (to be described in this book) should make this plain.
Lie algebras (and Lie groups): gauge group theory;
KacāMoody (infinite-d) algebras;
Coxeter (reflection) groups, also called Weyl groups...
