Aspects of a new class of braid matrices: roots of unity and hyperelliptic $q$ for triangularity, L-algebra,link-invariants, noncommutative spaces

Abstract

Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$ corresponds to polynomial equations for $q$, the solutions ranging from roots of unity to hyperelliptic functions. The algebras of $L-$ operators are studied. As a crucial feature one obtains $2N$ central, group-like, homogenous quadratic functions of $L_{ij}$ constrained to equality among themselves by the $RLL$ equations. They are studied in detail for $N =3$ and are proportional to $I$ for the fundamental $3\times3$ representation and hence for all iterated coproducts. The implications are analysed through a detailed study of the $9\times 9$ representation for N=3. The Turaev construction for link invariants is adapted to our class. A skein relation is obtained. Noncommutative spaces associated to our class of $\hat R$ are constructed. The transfer matrix map is implemented, with the N=3 case as example, for an iterated construction of noncommutative coordinates starting from an $(N-1)$ dimensional commutative base space. Further possibilities, such as multistate statistical models, are indicated.