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, are studied considering $N2×N2(N=3,4,…)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(R̂2=I)$ 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, grouplike, homogenous quadratic functions of $Lij$ 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 $3×3$ representation and hence for all iterated coproducts. The implications are analyzed through a detailed study of the $9×9$ 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 $R̂$ 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.