Reflection matrices from Hadamard-type Temperley-Lieb R-matrices (original) (raw)

3 Reflection matrices from Hadamard-type Temperley-Lieb R-matrices

2016

We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank-n Hadamard type representations of the Temperley-Lieb algebra T L N (√ n). They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined from the "Master matrix" associated to the chosen realization of Temperley-Lieb algebra

Temperley-Lieb R-matrices from generalized Hadamard matrices

Theoretical and Mathematical Physics, 2014

New sets of rank n-representations of Temperley-Lieb algebra T L N (q) are constructed. They are characterized by two matrices obeying a generalization of the complex Hadamard property. Partial classifications for the two matrices are given, in particular when they reduce to Fourier or Butson matrices.

Reflection k-matrices related to Temperley-Lieb R-matrices

Theoretical and Mathematical Physics, 2011

The general solutions of the reflection equation associated with Temperley-Lieb Rmatrices are constructed. Their parametrization is defined and the Hamiltonians of corresponding integrable spin systems are given.

Quantum symmetry algebras of spin systems related to Temperley–Lieb R-matrices

Journal of Mathematical Physics, 2008

A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U q with a representation ring equivalent to the representation ring of the sl 2 Lie algebra. This algebra U q is the symmetry algebra of the corresponding open spin chain.

The representations of Temperley-Lieb algebras and entanglement in a Yang-Baxter system

arXiv (Cornell University), 2009

A method of constructing Temperley-Lieb algebras(TLA) representations has been introduced in [Xue et.al arXiv:0903.3711]. Using this method, we can obtain another series of n 2 × n 2 matrices U which satisfy the TLA with the single loop d = √ n. Specifically, we present a 9 × 9 matrix U with d = √ 3. Via Yang-Baxterization approach, we obtain a unitaryȒ(θ, ϕ1, ϕ2)-matrix, a solution of the Yang-Baxter Equation. This 9 × 9 Yang-Baxter matrix is universal for quantum computing.

Set-theoretic Yang–Baxter & reflection equations and quantum group symmetries

Letters in Mathematical Physics, 2021

Connections between set-theoretic Yang–Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra {\mathcal {H}}_N(q=1)HN(q=1).Weshowinthecaseofthereflectionalgebrathatthereexistsa“boundary”finitesub−algebraforsomespecialchoiceof“boundary”elementsoftheB−typeHeckealgebraH N ( q = 1 ) . We show in the case of the reflection algebra that there exists a “boundary” finite sub-algebra for some special choice of “boundary” elements of the B-type Hecke algebraHN(q=1).Weshowinthecaseofthereflectionalgebrathatthereexistsa“boundary”finitesub−algebraforsomespecialchoiceof“boundary”elementsoftheB−typeHeckealgebra{\mathcal {B}}_N(q=1, Q)$$ B N ( q = 1 , Q ) . We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements...

On quantum matrix algebras satisfying the Cayley - Hamilton - Newton identities

Journal of Physics A: Mathematical and General, 1999

The Cayley-Hamilton-Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras M(R,F ) defined by a pair of compatible solutions of the Yang-Baxter equation. This class includes the RTTalgebras as well as the Reflection equation algebras. * On leave of absence from P. N. Lebedev

The center of the reflection equation algebra via quantum minors

Journal of Algebra, 2019

We give simple formulas for the elements c k appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group Uq(gl N), answering a question of Kolb and Stokman. The c k 's are certain canonical generators of the center of the REA, and hence of Uq(gl N) itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known. As byproducts, we prove a quantum Girard-Newton identity relating the c k 's to the so-called quantum power traces, and we give a new presentation for the quantum group Uq(gl N), as a localization of the REA along certain principal minors.