2024/07/08 by Aleksei Ilin, Joel Kamnitzer, Ilin, Aleksei +3 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2407.06424
openalex publication_date 2024/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
We introduce and study the family of trigonometric Gaudin subalgebras in U g⊗ n for arbitrary simple Lie algebra g. This is the family of commutative subalgebras of maximal possible transcendence degree that serve as a universal source for higher integrals of the trigonometric Gaudin quantum spin chain attached to g. We study the parameter space that indexes all possible degenerations of subalgebras from this family. In particular, we show that (rational) inhomogeneous Gaudin subalgebras of U g⊗ n previously studied in \citeffry arise as certain limits of trigonometric Gaudin subalgebras. Moreover, we show that both families of commutative subalgebras glue together into the one parameterized by the space \mathcal Fn, which is the total space of degeneration of the Deligne-Mumford space of stable rational curves Mn+2 to the moduli space of cactus flower curves Fn recently introduced in \citeiklpr. As an application, we show that trigonometric Gaudin subalgebras act on tensor products of irreducible finite-dimensional g-modules without multiplicities, under some explicit assumptions on the parameters in terms of two different real forms of Mn+2. This gives rise to a monodromy action of the affine cactus group on the set of eigenstates for the trigonometric Gaudin model. We also explain the relation between the trigonometric Gaudin model and the quantum cohomology of affine Grassmannians slices.