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

Rouven Frassek

doi:10.1016/j.nuclphysb.2020.115063

Abstract

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.

Highlights

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with D-type Lie algebras
Of determining explicit R-matrices did not attract much attention in the following years, for which the reason may well be that closed expressions for representations corresponding to nonextremal nodes of the Dynkin diagram are difficult to obtain
The multiplicity of linear independent solutions that we find is given by the dimension of the corresponding fundamental representation

Summary

Fundamental R-matrix

The fundamental R-matrix for so(2r) corresponding to the first fundamental representation was obtained in [1]. We introduced the identity I, the permutation P and the K matrix. The prime denotes the transposition Ea′ b = E−b,−a This symmetry stems from the so(2r) invariance of the R-matrix R in (2.1). We introduce two sets of permutation matrices that satisfy the invariance condition (2.11). The second permutation matrix is labelled by two indices i, j = 1, . When applied to L it permutes the ith and jth rows and columns and the −ith and −jth rows and columns

Lax matrices for D3

Minimal solutions of oscillator type for Dr

Factorisation formulas

Conclusion