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Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains
Nuclear Physics B ( IF 2.8 ) Pub Date : 2020-05-29 , DOI: 10.1016/j.nuclphysb.2020.115063
Rouven Frassek

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with D-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of so(2r). We use the isomorphism between the first fundamental representation of D3 and the 6 of A3, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for r=3. The results are used to generalise the Lax matrices to arbitrary rank for representations corresponding to the extremal nodes of the simply laced Dynkin diagram of Dr. The multiplicity of independent solutions at each extremal node is given by the dimension of the fundamental representation. We further derive certain factorisation formulas among these solutions and build transfer matrices with oscillators in the auxiliary space from the introduced degenerate Lax matrices. Finally, we provide some evidence that the constructed transfer matrices are Baxter Q-operators for so(2r) spin chains by verifying certain QQ-relations for D4 at low lengths.



中文翻译:

与D型Yangian相关的振荡器实现:朝向正交自旋链的算子Q系统

我们提出了一系列新颖的Lax算子,它们对应于与D型李代数相关的Yangian的RTT实现。这些Lax算子是振荡器类型的,即,算子的一个空间是无穷大的,而另一个空间在的第一个基本表示中所以2[R。我们使用同构之间的第一个基本表示d36一种3已知简并振荡器类型Lax矩阵的,以导出用于 [R=3。结果用于将Lax矩阵泛化为任意等级,以表示与简单带状Dynkin图的极值节点相对应的表示形式。d[R。每个极值节点的独立解的多样性由基本表示的维数给出。我们进一步在这些解决方案中得出某些分解公式,并根据引入的简并Lax矩阵在辅助空间中用振荡器建立传递矩阵。最后,我们提供了一些证据,证明所构造的传递矩阵是Baxter Q-operator for所以2[R 通过验证某些QQ关系来旋转链 d4 低长度。

更新日期:2020-05-29
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