# Higher dimensional multiparameter unitary and nonunitary braid matrices: Even dimensions

@article{Abdesselam2007HigherDM,
title={Higher dimensional multiparameter unitary and nonunitary braid matrices: Even dimensions},
author={Boucif Abdesselam and Amitabha Chakrabarti and Vladimir Dobrev and S. G. Mihov},
journal={Journal of Mathematical Physics},
year={2007},
volume={48},
pages={103505-103505}
}
• Published 19 June 2007
• Mathematics, Physics
• Journal of Mathematical Physics
A class of (2n)2×(2n)2 multiparameter braid matrices are presented for all n(n⩾1). Apart from the spectral parameter θ, they depend on 2n2 free parameters mij(±), i,j=1,…,n. For real parameters, the matrices R(θ) are nonunitary. For purely imaginary parameters, they became unitary. Thus, a unification is achieved with odd dimensional multiparameter solutions presented before.
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We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of (2n)2×(2n)2 dimensions with 2n2 free parameters (n=1,2,3,…). The simplest,
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Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are
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We construct exotic bialgebras that arise from multiparameter 9 × 9 R-matrices, some of which are new. We also construct the dual bialgebras of two of these exotic bialgebras.
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The exotic bialgebras that arise from a 9 × 9 unitary braid matrix are constructed. The dual bialgebra of one of these exotic bialgebras is presented.
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We study chain Hamiltonians derived from a class of multidimensional, multiparameter braid matrices introduced and explored in a series of previous papers. The N2 × N2 braid matrices (for all N)
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For a class of multiparameter statistical models based on $N^2\times N^2$ braid matrices the eigenvalues of the transfer matrix ${\bf T}^{(r)}$ are obtained explicitly for all $(r,N)$. Our formalism
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Quantum Inf. Process.
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A method to construct “X” form unitary Yang-Baxter matrices, which act on the tensor product space, and can obtain a set of entangled states for ( 2j1 + 1) × (2j2 +-1)-dimensional system with these Yang- B Baxter matrices.
(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p

## References

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• 2007
We construct (2n)2×(2n)2 unitary braid matrices R for n⩾2 generalizing the class known for n=1. A set of (2n)×(2n) matrices (I,J,K,L) is defined. R is expressed in terms of their tensor products
Braid matrices R(θ), corresponding to vector representations, are spectrally decomposed obtaining a ratio fi(θ)/fi(−θ) for the coefficient of each projector Pi appearing in the decomposition. This
A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R(θ), θ being the spectral parameter. Only odd values of N are considered
• Mathematics
• 2006
Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such N2×N2 matrices for odd
• Physics
• 2004
This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang–Baxter equation is a universal
• Physics, Environmental Science
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In this paper, small-scale statistics in stably stratified turbulence is studied theoretically and numerically. Expressions for the spectra of the velocity correlation, density fluctuation and
• Mathematics
Quantum Inf. Process.
• 2007
The Bell matrix is defined to yield all the Greenberger–Horne–Zeilinger (GHZ) states from the product basis, proved to form a unitary braid representation and presented as a new type of solution of the quantum Yang–Baxter equation.