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

Various properties of a class of braid matrices, presented before, arestudied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for twosubclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2=I)$ corresponds to polynomial equations for $q$, the solutions ranging fromroots of unity to hyperelliptic functions. The algebras of $L-$ operators arestudied. As a crucial feature one obtains $2N$ central, group-like, homogenousquadratic 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 iteratedcoproducts. The implications are analysed through a detailed study of the$9\times 9$ representation for N=3. The Turaev construction for link invariantsis adapted to our class. A skein relation is obtained. Noncommutative spacesassociated to our class of $\hat R$ are constructed. The transfer matrix map isimplemented, with the N=3 case as example, for an iterated construction ofnoncommutative coordinates starting from an $(N-1)$ dimensional commutativebase space. Further possibilities, such as multistate statistical models, areindicated.Comment: 34 pages, pape