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Series of the solutions to Yang-Baxter equations: Hecke type matrices and descendant R-, L-operators
Mathematical Physics · Submitted 12 May 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra.
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We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. These series are produced as descendant solutions from the $sl_{q}(2)$-invariant Hecke type $R^{r\;r}(u)$-matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra $osp_q(1|2)$ are obtained precisely. For the homogeneous solutions $R^{r^2-1\;r^2-1}$ there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators $R^{r\mathcal{R}}(u)$, extended Lax operators of Hecke type, also are suggested.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. | contribution 0.97 | 0.96 | 0.58 | asserted 1.00 |
| s1 | These series are produced as descendant solutions from the $sl_{q}(2)$-invariant Hecke type $R^{r\;r}(u)$-matrices. | method 0.72 | 0.89 | 0.23 | asserted 1.00 |
| s2 | The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra $osp_q(1|2)$ are obtained precisely. | result 0.60 | 0.91 | 0.15 | asserted 1.00 |
| s3 | For the homogeneous solutions $R^{r^2-1\;r^2-1}$ there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. | result 0.55 | 0.93 | 0.17 | asserted 1.00 |
| s4 | Centralizer operators defined on the products of the composite states are discussed. | contribution 0.91 | 0.69 | 0.06 | asserted 0.88 |
| s5 | The inhomogeneous series of the operators $R^{r\mathcal{R}}(u)$, extended Lax operators of Hecke type, also are suggested. | contribution 0.90 | 0.87 | 0.12 | hedged possibility 0.72 |
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Original abstract
We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. These series are produced as descendant solutions from the $sl_{q}(2)$-invariant Hecke type $R^{r\;r}(u)$-matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra $osp_q(1|2)$ are obtained precisely. For the homogeneous solutions $R^{r^2-1\;r^2-1}$ there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators $R^{r\mathcal{R}}(u)$, extended Lax operators of Hecke type, also are suggested.
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It slightly differs from the notations we have used in [ref] , [ref] .
Fusion Rules of the Lowest Weight Representations of osp_q(1|2) at Roots of Unity: Polynomial Realization and Degeneration at Roots of Unity · Introduction · 0.83
Cites as background 1 paper
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The Yang-Baxter equations (YBE), appeared in the early investigations on the problem of the exactly solvability in 2d statistical physics, as well as in $(1+1)$-dimensional scattering theory, and being one of the key relations of the QISM (Quantum Inverse Scattering Method), still remain an actual and attractive subject of the statistical and mathematical physics, with expanding area of applications [ref] - [ref] .
Braiding Operators are Universal Quantum Gates · Introduction · 1.00
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