Transfer matrices of rational spin chains via novel BGG-type resolutions

Abstract

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras g, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian Y (g). These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study [FPT, FT] of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional g-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by B−-orbits. ROUVEN FRASSEK, IVAN KARPOV, AND ALEXANDER TSYMBALIUK 7.2. Two-term factorisation in A-type 40 7.3. Two-term factorisation in C-type 45 7.4. Two-term factorisation in D-type 48 8. Resolutions for transfer matrices of BD-types: first fundamental representations 50 8.1. Oscillator realization in types BD (parabolic Verma) 50 8.2. More oscillator realizations in types BD via underlying symmetries 51 8.3. Type BD transfer matrices 53 9. Factorisation for quadratic BD-types 56 10. Further generalizations 59 References 60Let us note that p {1,...,r} = g, while p ∅ coincides with the Borel subalgebra b = h ⊕ α∈∆ + Ce α .It is well-known that all parabolic subalgebras p satisfying b ⊆ p ⊆ g are necessarily of that form. Such p S further decomposes into the semidirect product:of the semisimple part l (the Levi subalgebra) and the unipotent part u (the unipotent radical).The subgroup W l of W generated by the simple reflections {s α j } j∈S is the Weyl group of l. Then ∆ l = ∆ + l ⊔ (−∆ + l ) with ∆ + l = ∆ + S

Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin

“…Baxter initially introduced the Q-operator to solve the eight-vertex model without specifying a reference state [7], and it represents an important member of the family of commuting transfer matrices of integrable systems. Besides the developments arising from the study of the representation theory of quantum groups and solutions of the Yang-Baxter equation, see in particular [8][9][10][11][12][13][14][15][16][17][18][19][20], important contributions came from the ODE/IM correspondence [21] and more recently from 4d Chern-Simons theory [22,23]. The majority of the results however concerns spin chains with periodic boundary conditions.…”