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 $3\times 3$ 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.

arXiv comment34 pages, paper

References in corpus (6)

• Dimensional crossover in dipolar magnetic layers
• General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_q$-matrices
• Canonical factorization and diagonalization of Baxterized braid matrices: Explicit constructions and applications
• Representations of U_q(sl(N)) at Roots of Unity
• Representations of $SO(5{)}_q$ and Non-Minimal $q$-Deformation
• Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups $G{L}_q(N),S{O}_q(N)$ and $S{p}_q(N)$

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