Spinorial R operator and Algebraic Bethe Ansatz

Abstract

We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinorvector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor R matrices of low rank orthogonal algebras and the corresponding RT T algebras. Coincidences with fundamental R matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices. 1

Algebraic Bethe Ansatz for spinor R-matrices

“…where ϑ i = n p=1 (p − 1 2 ) i p and the overline means that the order of multiplying tensorands is reversed resulting in an overall sign; for instance, e (2) i j = e The inverse of w will be denoted by w.…”

“…Lemma 2.16. The spinor-spinor R-matrices of U ex q (Lso 6 ) are elements of End(V ±(2) ⊗V ± (2) ) and…”

“…In particular, (L (k+1) ) 0 ∼ = or (V ±(k) ) ⊗ when L (n) = L V or L ±S , respectively. Finally, we define the nested level-1 quantum space to be L (1) := (L (2)…”

“…As a consequence, there are four so 2n -invariant spinor-spinor R-matrices indexed by chirality of the corresponding spinor representations thus adding extra difficulties to the nesting procedure. This diagonalization procedure was recently addressed in a new perspective in [2] by Karakhanyan and Kirschner. An important novelty in their work was that the spinor-spinor R-matrices were written in terms of the Euler Beta function rather than in terms of recurrent relations presented in [1] (see also [3]).…”

“…In the present paper we address the long-standing problem of diagonalizing transfer matrices that obey quadratic relations defined by the q-deformed so 2n+1 -and so 2n -invariant spinorspinor R-matrices. We propose a new construction of spinor-spinor and spinor-vector R-matrices in terms of supermatrices (this replaces gamma matrices used in [1] and [2]) and provide explicit recurrence relations. These R-matrices are then used to construct spinor-type transfer matrices for U q 2 (so 2n+1 )and U q (so 2n )-symmetric spin chains with twisted diagonal periodic boundary conditions.…”