The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters

Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{\otimes 2} in two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang-Baxter equatio…

Solutions to the Yang-Baxter equations with $osp_q(1|2)$ symmetry: Lax operators

We find a new $4\times4$ solution to the $osp_q(1|2)$-invariant Yang-Baxter equation with simple dependence on the spectral parameter and propose $2\times 2$ matrix expressions for the corresponding Lax operator. The general inhomogeneous universal spectral-parameter dependent $R$-matrix is derived. It is proven, that there are two independent solutions to the homogeneous $osp_q(1|2)$-invariant YBE, defined on the fundamental three dimensional representations. One of them is the particular case of the universal matrix, while the second one does not admit generalization to the higher dimensiona…

Yang-Baxter R-operators for osp superalgebras

We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic $\hat{\cal R}$-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic $\hat{\cal R}$-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such $\hat{\cal R}$ operators in explicit form for special cases of the $osp(n|2m)$ algebra, in particular for a few low-rank cases. We also propose a novel, simpl…

Yang-Baxter 能否蕴含 Lie 代数？

Can Yang-Baxter imply Lie algebra?

本文回顾 Vogel 对 Vassiliev 展开系数的三参数多项式参数化，指出其构造暗示无限维 Lie 代数或可扩展 Chern-Simons 理论，并提供从纽结不变量反推单 Lie 代数分类的逆向公理化路径。

New solutions to the $s\ell_q(2)$-invariant Yang-Baxter equations at roots of unity

We find new solutions to the Yang-Baxter equations with the $R$-matrices possessing $sl_q(2)$ symmetry at roots of unity, using indecomposable representations. The corresponding quantum one-dimensional chain models, which can be treated as extensions of the XXZ model at roots of unity, are investigated. We consider the case $q^4=1$. The Hamiltonian operators of these models as a rule appear to be non-Hermitian. Taking into account the correspondence between the representations of the quantum algebra $sl_q(2)$ and the quantum super-algebra $osp_t(1|2)$, the presented analysis can be extended to…

Reflection $K$-matrices for a nineteen vertex model with $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]$ symmetry

We derive the solutions of the boundary Yang-Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]\simeq U_{q}[C\left(2\right)^{\left(2\right)}]$. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection $K$-matrix are different from zero. In the type II solution, the reflection $K$-matrix is even (every element of the $K$-mat…