Searcharxiv⌕ Search

arXiv subjects

Andrey Mudrov

Publications and source records attributed to Andrey Mudrov.

At least 19 recordsLinked to original sources
arXiv2026

R-matrix via Hasse diagrams

We calculate the R-matrix for the exceptional Lie superalgebra $\mathfrak{d}(2,1;κ)$ in the smallest representation of its quantum supergroup, using a method of Hasse diagrams.

math.QA↗
arXiv2026

Affine supersymmetric pairs

We classify Satake diagrams for general linear and orthosymplectic non-twisted affine Lie superalgebras and prove that each of them generates a family of proper spherical subalgebras. Furthermore, we demonstrate that every such family contains subalgebras with matrix invariants, which are viewed as classical analogs of K-matrices solving supersymmetric Reflection equation.

math.QA↗
arXiv2025

Mickelsson algebras and inverse Shapovalov form

Let $\mathcal{A}$ be an associative algebra containing the classical or quantum universal enveloping algebra $U$ of a semi-simple complex Lie algebra. Let $\mathcal{J}\subset \mathcal{A}$ designate the left ideal generated by positive root vectors in $U$. We construct the reduction algebra of the pair $(\mathcal{A},\mathcal{J})$ via the inverse Shapovalov form of $U$.

math.QA↗
arXiv2024

Mickelsson algebras via Hasse diagrams

Let $\mathcal{A}$ be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra $\mathfrak{g}$. We present a construction of the Mickelsson algebra $Z(\mathcal{A},\mathfrak{g})$ relative to the left ideal in $\mathcal{A}$ generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum $\mathfrak{g}$-modules. We give an explicit expression for a PBW basis in $Z(\mathcal{A},\mathfrak{g})$ in the case when $\mathcal{A}=U(\mathfrak{a})$ of a finite-dimensional Lie algebra $\mathfrak{a}\supset \mathfrak{g}$. For $\mathcal{A}=U_q(\mathfrak{a})$ and $\mathfrak{g}$ the commutant of a Levi subalgebra in $\mathfrak{a}$, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to $\mathbb{C}[[\hbar]]$.

math.QA↗
arXiv2024

Vector bundles on quantum conjugacy classes

Let $\mathfrak{g}$ be a simple complex Lie algebra of a classical type and $U_q(\mathfrak{g})$ the corresponding Drinfeld-Jimbo quantum group at $q$ not a root of unity. With every point $t$ of the fixed maximal torus $T$ of an algebraic group $G$ with Lie algebra $\mathfrak{g}$ we associate an additive category $\mathcal{O}_q(t)$ of $U_q(\mathfrak{g})$-modules that is stable under tensor product with finite-dimensional quasi-classical $U_q(\mathfrak{g})$-modules. We prove that $\mathcal{O}_q(t)$ is essentially semi-simple and use it to explicitly quantize equivariant vector bundles on the conjugacy class of $t$.

math.QA↗
arXiv2023

Shapovalov elements of classical and quantum groups

Shapovalov elements $θ_{β,m}$ of the classical or quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ are parameterized by a positive root $β$ and a positive integer $m$. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of $θ_{β,m}$ to a product of $θ_{β,1}$ and calculate $θ_{β,1}$ as a residue of a matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm. This way we explicitly express $θ_{β,m}$ of a classical simple Lie algebra through the Cartan-Weyl basis in $\mathfrak{g}$. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum $L$-operator) in fundamental representations.

math.QA↗
arXiv2022

Factorization of Shapovalov elements

Shapovalov elements $θ_{β,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $β$ and a positive integer $m$. They relate the canonical generator of a reducible Verma module with highest vectors of its Verma submodules. For $m=1$, they can be explicitly obtained as matrix elements of the inverse Shapovalov form. We extend this approach to $m>1$ for all $β$ but three roots in $\mathfrak{g}_2$, $\mathfrak{f}_4$, and $\mathfrak{e}_8$, presenting $θ_{β,m}$ as a product of matrix elements of weight $β$.

math.QA↗
arXiv2021

Pseudo-parabolic category over quaternionic projective plane

Quaternionic projective plane $\mathbb{H} P^2$ is the next simplest conjugacy class of the symplectic group $SP(6)$ with pseudo-Levi stabilizer subgroup after the sphere $\mathbb{S}^4\simeq \mathbb{H} P^1$. Its quantization gives rise to a module category $\mathcal{O}_t\bigl(\mathbb{H} P^2\bigr)$ over finite-dimensional representations of $U_q\bigl(\mathfrak{s}\mathfrak{p}(6)\bigr)$, a full subcategory in the category $\mathcal{O}$. We prove that $\mathcal{O}_t\bigl(\mathbb{H} P^2\bigr)$ is semi-simple and equaivalent to the category of quantized equivariant vector bundles on $\mathbb{H} P^2$.

math.QA↗
arXiv2020

Rapid parameter determination of discrete damped sinusoidal oscillations

We present different computational approaches for the rapid extraction of the signal parameters of discretely sampled damped sinusoidal signals. We compare time- and frequency-domain-based computational approaches in terms of their accuracy and precision and computational time required in estimating the frequencies of such signals, and observe a general trade-off between precision and speed. Our motivation is precise and rapid analysis of damped sinusoidal signals as these become relevant in view of the recent experimental developments in cavity-enhanced polarimetry and ellipsometry, where the relevant time scales and frequencies are typically within the $\sim1-10\,μ$s and $\sim1-100$MHz ranges, respectively. In such experimental efforts, single-shot analysis with high accuracy and precision becomes important when developing experiments that study dynamical effects and/or when developing portable instrumentations. Our results suggest that online, running-fashion, microsecond-resolved analysis of polarimetric/ellipsometric measurements with fractional uncertainties at the $10^{-6}$ levels, is possible, and using a proof-of-principle experimental demonstration we show that using a frequency-based analysis approach we can monitor and analyze signals at kHz rates and accurately detect signal changes at microsecond time-scales.

physics.ins-det↗
arXiv2019

Equivariant vector bundles over quantum spheres

We quantize homogeneous vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$ and prove their complete reducibility.

math.QA↗
arXiv2019

Contravariant forms and extremal projectors

Tensor product of irreducible modules of highest weight over a semi-simple quantum group is semi-simple if and only if a natural contravariant form is non-degenerate when restricted to the span of singular vectors. We express this restriction through the extremal projector of the quantum group providing a computationally feasible criterion for complete reducibility of tensor products.

math.QA↗
arXiv2018

Equivariant vector bundles over quantum projective spaces

We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. In this way, we prove complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.

math.QA↗
arXiv2016

Quantum exceptional group $G_2$ and its conjugacy classes

We construct quantization of semisimple conjugacy classes of the exceptional group $G=G_2$ along with and by means of their exact representations in highest weight modules of the quantum group $U_q(\mathfrak{g})$. With every point $t$ of a fixed maximal torus we associate a highest weight module $M_t$ over $U_q(\mathfrak{g})$ and realize the quantized polynomial algebra of the class of $t$ by linear operators on $M_t$. Quantizations corresponding to points of the same orbit of the Weyl group are isomorphic.

math.QA↗
arXiv2016

R-matrix and inverse Shapovalov form

We construct the inverse Shapovalov form of a simple complex quantum group from its universal R-matrix based on a generalized Nagel-Moshinsky approach to lowering operators. We establish a connection between this algorithm and the ABRR equation for dynamical twist.

math.QA↗
arXiv2015

Regularization of Mickelsson generators for non-exceptional quantum groups

Let $\mathfrak{g}'\subset \mathfrak{g}$ be the pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces $\mathbb{C}^{N-2}\subset \mathbb{C}^N$ and $U_q(\mathfrak{g}')\subset U_q(\mathfrak{g})$ the pair of quantum groups with triangular decomposition $U_q(\mathfrak{g})=U_q(\mathfrak{g}_-)U_q(\mathfrak{g}_+)U_q(\mathfrak{h})$. Let $Z_q(\mathfrak{g},\mathfrak{g}')$ be the corresponding step algebra and regard its generators as rational trigonometric functions $\mathfrak{h}^*\to U_q(\mathfrak{g}_\pm)$. We describe their regularization such that the resulting generators do not vanish when specialized at any weight.

math.QA↗