arXiv subjects

Erik Brodsky

Publications and source records attributed to Erik Brodsky.

3 recordsLinked to original sources

arXiv2026

Constructing Casimirs for the quantum reflection equation

We explore the quantum reflection equation in the case of a full-dimensional $n\times n$ matrix $\mathbb A$. Using the technique of $q$-determinants, we find closed expressions for the quantum Pfaffian (in the case of even $n$) and for new quantum Casimirs that are ratios of special quantum (Sklyanin) minors of $\mathbb A$. Special attention is paid to the self-adjointness property of the newly constructed Casimirs. We put forward a conjecture that, adopting a natural ansatz for a quantum deformation of a Casimir of a classical algebra, self-adjointness implies commutativity. We also explore the relationship to networks of quantum cluster variables.

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arXiv2026

Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases

The Type D asymmetric simple exclusion process (ASEP), a particle system involving two classes of particles, was studied from both probabilistic and algebraic perspectives by Kuan et al. (arXiv:2011.13473). From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP introduced by Kuan (arXiv:1908.02359) and analyze the outcome on the generator matrix, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of $\mathcal{U}_q(\mathfrak{so}_6)$, using crystal bases to analyze and manipulate various representations of $\mathcal{U}_q(\mathfrak{so}_6)$. We know that the same generators for the normal ASEP are produced by stochastic fusion and by a suitable ground state transformation on a central element of $\mathcal{U}_q(\mathfrak{sl}_2)$ in a symmetric tensor product representation. However, in the case of the Type D ASEP, we find that the probabilistic and algebraic generator matrices are not the same and thus represent different processes. We conclude that the relationship between stochastic fusion and a ground state transformation (specifically, what we term the Type A ground state transformation) established by Kuan (arXiv:1908.02359) does not generalize to all finite-dimensional simple Lie algebras.

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arXiv2023

Power sum elements in the $G_2$ skein algebra

We study the skein algebras of surfaces associated to the exceptional Lie group $G_2,$ using Kuperberg webs. We identify two 2-variable polynomials, $P_n(x,y)$ and $Q_n(x,y),$ and use threading operations along knots to construct a family of central elements in the $G_2$ skein algebra of a surface, $\mathcal{S}_q^{G_2}(Σ),$ when the quantum parameter $q$ is a $2n\text{-th}$ root of unity. We verify these elements are central using elementary skein-theoretic arguments. We also obtain a result about the uniqueness of the so-called transparent polynomials $P_n$ and $Q_n.$ Our methods involve a detailed study of the skein modules of the annulus and the twice-marked annulus.

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