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Abstract: We propose that Baxter's Z-invariant six-vertex model at the rational point on a planar but in general not rectangular lattice provides a way to study Yangian invariants. These are identified with eigenfunctions of certain monodromies of an auxiliary inhomogeneous spin chain. As a consequence they are special solutions to the eigenvalue problem of the associated transfer matrix. Excitingly, this allows to construct them using Bethe ansatz techniques. Conceptually, our construction generalizes to general (super) Lie algebras and general representations. Here we present the explicit form of sample invariants for totally symmetric, finite-dimensional representations of in terms of oscillator algebras. In particular, we discuss invariants of three- and four-site monodromies that can be understood respectively as intertwiners of the bootstrap and Yang–Baxter equation. We state a set of functional relations significant for these representations of the Yangian and discuss their solutions in terms of Bethe roots. They arrange themselves into exact strings in the complex plane. In addition, it is shown that the sample invariants can be expressed analogously to Graßmannian integrals. This aspect is closely related to a recent on-shell formulation of scattering amplitudes in planar super Yang–Mills theory. [Copyright &y& Elsevier]

Abstract: We continue our systematic construction of Baxter Q-operators for spin chains, which is based on certain degenerate solutions of the Yang–Baxter equation. Here we generalize our approach from the fundamental representation of to generic finite-dimensional representations in quantum space. The results equally apply to non-compact representations of highest or lowest weight type. We furthermore fill an apparent gap in the literature, and provide the nearest-neighbor Hamiltonians of the spin chains in question for all cases where the representations are described by rectangular Young diagrams, as well as for their infinite-dimensional generalizations. They take the form of digamma functions depending on operator-valued shifted weights. [Copyright &y& Elsevier]

Abstract: We generalize our recent explicit construction of the full hierarchy of Baxter Q-operators of compact spin chains with symmetry to the supersymmetric case . The method is based on novel degenerate solutions of the graded Yang–Baxter equation, leading to an amalgam of bosonic and fermionic oscillator algebras. Our approach is fully algebraic, and leads to the exact solution of the associated compact spin chains while avoiding Bethe ansatz techniques. It furthermore elucidates the algebraic and combinatorial structures underlying the system of nested Bethe equations. Finally, our construction naturally reproduces the representation, due to Z. Tsuboi, of the hierarchy of Baxter Q-operators in terms of hypercubic Hasse diagrams. [Copyright &y& Elsevier]

Abstract: We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang–Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang–Baxter equation serve as elementary, “partonic” building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe Ansatz techniques. [Copyright &y& Elsevier]