Higher Dimensional Unitary Braid Matrices: Construction, Associated Structures and Entanglements

Abstract

We construct $(2n{)}^2\times (2n{)}^2$ unitary braid matrices $\hat{R}$ for $n\ge 2$ generalizing the class known for $n=1$. A set of $(2n)\times (2n)$ matrices $(I,J,K,L)$ are defined. $\hat{R}$ is expressed in terms of their tensor products (such as $K\otimes J$), leading to a canonical formulation for all $n$. Complex projectors $P_{\pm }$ provide a basis for our real, unitary $\hat{R}$. Baxterization is obtained. Diagonalizations and block-diagonalizations are presented. The loss of braid property when $\hat{R}$ $(n>1)$ is block-diagonalized in terms of $\hat{R}$ $(n=1)$ is pointed out and explained. For odd dimension $(2n+1{)}^2\times (2n+1{)}^2$, a previously constructed braid matrix is complexified to obtain unitarity. $\hat{R}\mathrm{LL}$- and $\hat{R}\mathrm{TT}$-algebras, chain Hamiltonians, potentials for factorizable $S$-matrices, complex non-commutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements.

26 pages, 1 figure, Addendum and new references are added. To appear in Journal of Mathematical Physics

References in corpus (10)

• Quantum groups and nonabelian braiding in quantum Hall systems
• Geometrical Tools for Quantum Euclidean Spaces
• On Combined Standard-Nonstandard or Hybrid (q,h)-Deformations
• Spectral Decomposition and Baxterisation of Exotic Bialgebras and Associated Noncommutative Geometries
• A nested sequence of projectors and corresponding braid matrices $\hat{R}(\theta )$: (1) Odd dimensions
• Duality and Representations for New Exotic Bialgebras
• Aspects of a new class of braid matrices: roots of unity and hyperelliptic $q$ for triangularity, L-algebra,link-invariants, noncommutative spaces
• A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices
• Exotic Bialgebra S03: Representations, Baxterisation and Applications
• A new class ${\hat{o}}_N$ of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable $S$-matrix

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