arXiv subjects

Inna Mashanova-Golikova

Publications and source records attributed to Inna Mashanova-Golikova.

3 recordsLinked to original sources

arXiv2026

Bethe subalgebras in Yangians and Kirillov-Reshetikhin crystals

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra and $G$ its adjoint group. For each $C\in G$, we consider the Bethe subalgebra $B(C)\subset Y(\mathfrak{g})$, a commutative subalgebra encoding the integrals of the generalized $XXX$ spin chain. Adapting the construction of arXiv:1708.05105 of $\mathfrak{g}$-crystals on spectra of inhomogeneous Gaudin subalgebras in $U(\mathfrak{g})$, we construct a natural $\hat{\mathfrak{g}}$-crystal structure on the spectra of $B(C)$ in Kirillov--Reshetikhin $Y(\mathfrak{g})$-modules in type $A$. We conjecture that such a construction exists for arbitrary $\mathfrak{g}$ and recovers Kirillov--Reshetikhin crystals. The main technical ingredient is a degeneration of Bethe subalgebras $B(C)$ to commutative subalgebras $\mathcal{A}_χ^{\mathrm{u}} \subset U(\mathfrak{g}[t])$, depending on $χ\in\mathfrak{g}$. We call these subalgebras universal inhomogeneous Gaudin subalgebras and show that they arise from the Feigin--Frenkel center at the critical level. This allows us to identify the affine crystals above with Kirillov--Reshetikhin crystals. We then apply these results to prove the monodromy conjecture of Ilin and the second and third authors for the spectra of the algebras $B(C)$ and for the spectra of quantum cohomology rings of type $A$ quiver varieties.

math.RT↗

arXiv2021

Spectra of Bethe subalgebras of $Y(\mathfrak{gl}_n)$ in tame representations

We study the eigenproblem for Bethe subalgebras of the Yangian $Y(\mathfrak{gl}_n)$ in tame representations, i.e. in finite dimensional representations which admit Gelfand-Tsetlin bases. Namely, we prove that for any tensor product of skew modules $V=\otimes_{i=1}^k V_{λ_i \setminus μ_i}(z_i)$ over the Yangian $Y(\mathfrak{gl}_n)$ with generic $z_i$'s, the family of Bethe subalgebras $B(X)$ with $X$ being a regular element of the maximal torus of $GL_n$ (or, more generally, with $X \in \overline{M_{0,n+2}}$) acts with a cyclic vector on $V$. Moreover, for $X$ in the real form of $\overline{M_{0,n+2}}$ which is the closure of regular unitary diagonal matrices we show, that the family of subalgebras $B(X)$ acts with simple spectrum on $\otimes_{i=1}^k V_{λ_i \setminus μ_i}(z_i)$ for generic $z_i$'s where all $V_{λ_i \setminus μ_i}(z_i)$ are Kirillov-Reshetikhin modules. In the subsequent paper we will use this to define a KR-crystal structure on the spectrum of a Bethe subalgebra on $V$.

math.RT↗

arXiv2019

Simplicity of spectra for Bethe subalgebras in $Y(\mathfrak{gl}_2)$

We consider Bethe subalgebras B(C) in the Yangian $Y(\mathfrak{gl}_2)$ with $C$ regular $2\times 2$ matrix. We study the action of Bethe subalgebras of $Y(\mathfrak{gl}_2)$ on finite-dimensional representations of $Y(\mathfrak{gl}_2)$. We prove that $B(C)$ with real diagonal $C$ has simple spectrum on any irreducible $Y(\mathfrak{gl}_2)$-module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of $B(C)$.

math.QA↗