arXiv subjects

Aleksei Ilin

Publications and source records attributed to Aleksei Ilin.

9 recordsLinked to original sources

arXiv2025

Bethe subspaces and wonderful models for toric arrangements

We study the family of commutative subspaces in the trigonometric holonomy Lie algebra $t^{\mathrm{trig}}_{\Phi}$, introduced by Toledano Laredo, for an arbitrary root system $\Phi$. We call these subspaces \emph{Bethe subspaces} because they can be regarded as quadratic components of \emph{Bethe subalgebras} in the Yangian corresponding to the root system $\Phi$, that are responsible for integrals of the generalized XXX Heisenberg spin chain. Bethe subspaces are naturally parametrized by the complement of the corresponding toric arrangement . We prove that this family extends regularly to the minimal wonderful model $X_{\Phi}$ of the toric arrangement described by De Concini and Gaiffi, thus giving a compactification of the parameter space for Bethe subspaces. For classical types $A_n, B_n, C_n, D_n$, we show that this extension is faithful. As a special case, when $\Phi$ is of type $A_n$, our construction agrees with the main result of Aguirre--Felder--Veselov on the closure of the set of quadratic Gaudin subalgebras. Our work is also closely related to, and refines in this root system setting, a parallel compactification result of J. Peters obtained for more general toric arrangements arising from quantum multiplication on hypertoric varieties. Next, we show that the Bethe subspaces assemble into a vector bundle over $X_{\Phi}$, which we identify with the logarithmic tangent bundle of $X_{\Phi}$. Finally, we formulate conjectures extending these results to the setting of Bethe subalgebras in Yangians and to the quantum cohomology rings of Springer resolutions. We plan to address this in our next papers.

math.AG↗

arXiv2025

Lightweight Safety Guardrails via Synthetic Data and RL-guided Adversarial Training

We introduce a lightweight yet highly effective safety guardrail framework for language models, demonstrating that small-scale language models can achieve, and even surpass, the performance of larger counterparts in content moderation tasks. This is accomplished through high-fidelity synthetic data generation and adversarial training. The synthetic data generation process begins with human-curated seed data, which undergoes query augmentation and paraphrasing to create diverse and contextually rich examples. This augmented data is then subjected to multiple rounds of curation, ensuring high fidelity and relevance. Inspired by recent advances in the Generative Adversarial Network (GAN) architecture, our adversarial training employs reinforcement learning to guide a generator that produces challenging synthetic examples. These examples are used to fine-tune the safety classifier, enhancing its ability to detect and mitigate harmful content. Additionally, we incorporate strategies from recent research on efficient LLM training, leveraging the capabilities of smaller models to improve the performance of larger generative models. With iterative adversarial training and the generation of diverse, high-quality synthetic data, our framework enables small language models (SLMs) to serve as robust safety guardrails. This approach not only reduces computational overhead but also enhances resilience against adversarial attacks, offering a scalable and efficient solution for content moderation in AI systems.

cs.LG↗

arXiv2024

Gaudin models and moduli space of flower curves

We introduce and study the family of trigonometric Gaudin subalgebras in $U g^{\otimes n}$ 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^{\otimes n}$ previously studied in \cite{ffry} 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 $\overline{\mathcal F}_n$, which is the total space of degeneration of the Deligne-Mumford space of stable rational curves $ \overline M_{n+2} $ to the moduli space of cactus flower curves $ \overline F_n $ recently introduced in \cite{iklpr}. 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 $\overline M_{n+2}$. 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.

math.RT↗

arXiv2024

Bethe subalgebras in Yangians and the wonderful compactification

We study the family of Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ parameterized by the corresponding adjoint Lie group $G$. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group $G_1[[t^{-1}]]$. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincare series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini--Procesi wonderful compactification $\overline{G}\supset G$ and describe the subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in $\mathfrak{g}$.

math.QA↗

arXiv2023

The moduli space of cactus flower curves and the virtual cactus group

The space $ \ft_n = \C^n/\C $ of $n$ points on the line modulo translation has a natural compactification $ \overline \ft_n $ as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; it is natural to imagine points at infinite distance from each other as living on different projective lines. We call such a configuration of points a ``flower curve'', since we picture the projective lines joined into a flower. Within $ \ft_n $, we have the space $ F_n = \C^n \setminus \Delta / \C $ of $ n$ distinct points. We introduce a natural compatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow \overline \ft_n $, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline \ft_n$ and $\overline F_n$, are special fibers of $1$-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus $0$ curves with $n+2$ marked points. We find combinatorial models for the real loci $ \overline \ft_n(\BR) $ and $ \overline F_n(\BR) $. Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The degeneration of a twisted real form of the Deligne-Mumford space to $\overline F_n(\mathbb{R})$ gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.

math.AG↗

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↗

arXiv2021

LT-LM: a novel non-autoregressive language model for single-shot lattice rescoring

Neural network-based language models are commonly used in rescoring approaches to improve the quality of modern automatic speech recognition (ASR) systems. Most of the existing methods are computationally expensive since they use autoregressive language models. We propose a novel rescoring approach, which processes the entire lattice in a single call to the model. The key feature of our rescoring policy is a novel non-autoregressive Lattice Transformer Language Model (LT-LM). This model takes the whole lattice as an input and predicts a new language score for each arc. Additionally, we propose the artificial lattices generation approach to incorporate a large amount of text data in the LT-LM training process. Our single-shot rescoring performs orders of magnitude faster than other rescoring methods in our experiments. It is more than 300 times faster than pruned RNNLM lattice rescoring and N-best rescoring while slightly inferior in terms of WER.

eess.AS↗

arXiv2021

On classical limits of Bethe subalgebras in Yangians

The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain. We describe the associated graded of Bethe subalgebras as subalgebras in $U(\mathfrak{g}[t])$ and in $\mathcal{O}(G_1[[t^{-1}]])$ for all semisimple $C\in G$. We show that associated graded in $U(\mathfrak{g}[t])$ of the Bethe subalgebra assigned to the identity of $G$ is the universal Gaudin subalgebra of $U(\mathfrak{g}[t])$ obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin subalgebra to $\mathfrak{g}$ of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin-Frenkel center at the critical level. Using our general result on associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple $C\in G$ as $C$ goes to an irregular semisimple group element $C_0$. We show that this limit is the product of the smaller Bethe subalgebra and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of $C_0$ in $\mathfrak{g}$. This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument subalgebras.

math.QA↗

arXiv2017

Degeneration of Bethe subalgebras in the Yangian of $\mathfrak{gl}_n$

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y(\mathfrak{gl}_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parametrizes all possible degenerations, is the Deligne-Mumford moduli space of stable rational curves $\overline{M_{0,n+2}}$. All subalgebras corresponding to the points of $\overline{M_{0,n+2}}$ are free and maximal commutative. We describe explicitly the "simplest" degenerations and show that every degeneration is the composition of the simplest ones. The Deligne-Mumford space $\overline{M_{0,n+2}}$ generalizes to other root systems as some De Concini-Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini-Procesi resolution.

math.QA↗