2025/05/26 by Cheong, Wan Keng, Lam, Ngau · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2505.19661
Let \mathfrakg denote the classical Lie algebra \mathfrakgld, \mathfraksp2d, or \mathfrakso2d with a fixed *-structure σ. Let M1, …, M_ℓ be unitarizable \mathfrakg-modules (with respect to σ), and let \bf z=(z1, …, z_ℓ) ∈ ℂ^ℓ. We investigate the action of the Bethe algebra B_\mathfrakgμ for \mathfrakg with respect to μ∈ \mathfrakg^* on the tensor product \underlineM(\bf z):=M1(z1) ⊗ ⋯ ⊗ M_ℓ(z_ℓ) of evaluation \mathfrakg[t]-modules. We show that if μ∘ σ equals the complex conjugation of μ, then B_\mathfrakgμ is diagonalizable on any finite-dimensional B_\mathfrakgμ-submodule of \underlineM(\bf z) for \bf z ∈ ℝ^ℓ. This, together with the result derived from the duality of Bethe algebras (see below), suggests that a simple spectrum conjecture for B_\mathfrakgμ should hold. We establish a duality of Bethe algebras for the general linear Lie (super)algebras \mathfrakgld and \mathfrakglp+m|q+n. As an application, we show that under a generic condition, the Bethe algebra for \mathfrakglp+m|q+n with respect to \bf z ∈ ℂp+q+m+n is diagonalizable with a simple spectrum on any weight space of L1(w1) ⊗ ⋯ ⊗ Ld(wd), where the Li are (infinite-dimensional) unitarizable highest weight \mathfrakglp+m|q+n-modules corresponding to generalized partitions of depth 1, and w1, …, wd ∈ ℂ. We also obtain the corresponding result for \mathfrakglp+m by setting q=n=0.