Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains

摘要

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated withD-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 ofso(2r). We use the isomorphism between the first fundamental representation ofD3and the6ofA3, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators forr=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 ofDr. 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 forso(2r)spin chains by verifying certain QQ-relations forD4at low lengths.