Since their introduction a century ago, subgroups of SU(2) and simple Lie groups have evolved almost separately. In operator algebras no geometrical structure of subfactors has been previously found. The proponent has found the natural link between subfactors, the subgroups of quantum SU(2) and the classical and quantum Lie groups, showing that the information for building a simple Lie group comes naturally from the fusion structure on representations of a quantum subgroup of SU(2). The bridge between these two areas of research is a new crystallographic property of homology theory, wherein for instance six term exact sequences correspond to regular hexagons. These methods yield a natural elementary construction of a canonical basis of the quantum enveloping algebra of the semisimple Lie groups. The link found between quantum subgroups and root lattices extends beyond SU(2) and the classical Lie groups. The proponent found new unimodular root systems in weight lattices associated to general quantum subgroups, which are not connected to any known structures. The proponent considers the development of the higher analogs of simple Lie groups corresponding to the new root systems a priority. These are likely to be essentially new mathematical objects with natural many-to-one laws, which have potential applications in constructive QFT in a physical (3 or 4) number of dimensions, while the usual binary laws produce naturally 2-dimensional field theories. The project is centered around the construction, classification and study of the properties and manifestations of the quantum subgroups of Lie groups. The quantum deformations of the semisimple Lie groups have, when the quantization parameter is a root of unity, subgroups which are the analogs of the finite subgroups of the classical Lie groups. The proponent has introduced this structure over the years, starting with the classification of the algebraic structure of small index subfactors in the noncommutative Galois theory for operator algebras. Other manifestations of these structures appear in topological quantum field theory, where they provide boundary extensions of numerical invariants for 3-manifolds , conformal field theory, and modular invariants. The quantum subgroups of SU(2), SU(3) and SU(4) are now classified by the proponent, and show that the quantum world is very different and apparently nearly unrelated to the classical world, with a markedly simpler situation for the exceptional quantum subgroups than for the corresponding classical subgroups. The project introduces geometrical structures associated to quantum subgroups, with the quantum subgroups of SU(2) producing the roots weights and canonical bases for the simple Lie groups, while the other quantum subgroups give raise to essentially new generalized root systems in weight lattices. It is hoped that the new structures produced by the project could play a role in constructing models of quantum field theory in a physical number of dimensions.
