Construction and Classification Results for Commuting Squares of Finite Dimensional *-Algebras

In this dissertation, we present new constructions of commuting squares, and we investigate finiteness and isolation results for these objects. We also give applications to the classification of complex Hadamard matrices and to Hopf algebras.

In the first part, we recall the notion of commuting squares which were introduced by Popa and arise naturally as invariants in Jones' theory of subfactors. We review some of the main known examples of commuting squares such as those constructed from finite groups and from complex Hadamard matrices. We also recall Nicoara's notion of defect which gives an upper bound for the number of continuous deformations in the space of commuting squares. Finally, we prove new formulas that lead to computations of defects.

In the second part, we prove a finiteness result for circulant core Hadamard matrices (and thus, for their associated commuting squares). We show that the number of such matrices is finite when the order of the matrix is p+1 with p a fixed prime number. We then discuss concrete examples of these matrices of small orders.

In the third part, we give an explicit construction of multi-parametric analytic families of commuting squares obtained as deformations of group commuting squares. In the particular case of cyclic groups of non-prime orders, this gives multi-parametric families of complex Hadamard matrices containing the Fourier matrix. This result expands on the work of Nicoara and White. We then give bounds on the number of parameters in any family stemming from our construction method. We also discuss other parametric families containing the Fourier matrix, some of which include our families as (equivalent) sub-families.

In the last part, we construct a new class of commuting squares which we call bismash commuting squares. They are obtained from bismash product Hopf algebras based on exact factorizations of finite groups, L. We then investigate the defect of a bismash commuting square which leads us to the conjecture that the defect of the commuting square is equal to the defect of the group L. We prove this conjecture when L is the direct or semidirect product of two proper subgroups.