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Exploring Noncommutative Polynomial Equation Methods for Discrete-Time Quaternionic Control
Michael Sebek
https://arxiv.org/abs/2506.08034 https://

Exploring Noncommutative Polynomial Equation Methods for Discrete-Time Quaternionic Control
We present new polynomial-based methods for discrete-time quaternionic systems, highlighting how noncommutative multiplication modifies classical control approaches. Defining quaternionic polynomials via a backward-shift operator, we examine left and right fraction representations of transfer functions, showing that right zeros correspond to similarity classes of quaternionic matrix right eigenvalues. We then propose a feedback design procedure that generalizes pole placement to quaternions - a…

The Dixmier problem for skew PBW extensions and rings
William Fajardo, Oswaldo Lezama
https://arxiv.org/abs/2506.09285 https://arxiv.…

The Dixmier problem for skew PBW extensions and rings
In this paper we discuss for skew $PBW$ extensions the famous Dixmier problem formulated by Jacques Dixmier in 1968. The skew $PBW$ extensions are noncommutative rings of polynomial type and covers several algebras and rings arising in mathematical physics and noncommutative algebraic geometry. For this purpose, we introduce the Dixmier algebras and we will study the Dixmier problem for algebras over commutative rings, in particular, for $\mathbb{Z}$-algebras, i.e., for arbitrary rings. The res…

Quantum noncommutative ABJM field theory: four- and six-point functions
Carmelo P. Martin, Josip Trampetic, Jiangyang You
https://arxiv.org/abs/2507.04823 …

Quantum noncommutative ABJM field theory: four- and six-point functions
Following our previous paper Quantum noncommutative ABJM theory: first steps, JHEP {\bf 1804} (2018) 070), in this article we investigate one-loop 1PI four-, and six-point functions by using the component formalism in the Landau gauge and show that they are UV finite and have well-defined $(θ^{μν}\rightarrow 0)$ limit. Those results also hold for all one-loop functions which are UV finite by power counting. In summary, taking into account results from previous paper, JHEP {\bf 1804} (2018) 0…

Noncommutative discrete equations, symmetries and reductions
Pavlos Xenitidis
https://arxiv.org/abs/2507.04472 https://arxiv.org/pdf/…

Noncommutative discrete equations, symmetries and reductions
Employing the Lax pairs of the noncommutative discrete potential Korteweg--de Vries (KdV) and Hirota's KdV equations, we derive differential--difference equations that are consistent with these systems and serve as their generalised symmetries. Miura transformations mapping these equations to a noncommutative modified Volterra equation and its master symmetry are constructed. We demonstrate the use of these symmetries to reduce the potential KdV equation, leading to a noncommutative discrete Pa…

The connection between the chromatic function and the Redei-Berge function
Stefan Mitrovi\'c, Tanja Stojadinovi\'c
https://arxiv.org/abs/2506.08841

The connection between the chromatic function and the Redei-Berge function
There is a natural way to assign both graph and digraph to every poset. Furthermore, any graph has its chromatic function, while any digraph has its Redei-Berge function. On the level of posets, these two functions are almost identical. Here, we prove that this connection is actually a reflection of the connection between the noncommutative generalizations of these two functions. The simplicity of this relationship enables us to easily translate the properties proved for one of them to the case…

Replaced article(s) found for math.OA. https://arxiv.org/list/math.OA/new/
[1/1]:
Noncommutative Choquet theory
https://arxi…

The noncommutative residue, divergence theorems and the spectral geometry functional
Jian Wang, Yong Wang
https://arxiv.org/abs/2506.18038 https://

The noncommutative residue, divergence theorems and the spectral geometry functional
In this paper, a simple proof of the divergence theorem is given by using the Dirac operator and noncommutative residues. Then we extend the divergence theorem to compact manifolds with boundary by the noncommutative residue of the B-algebra. Furthermore, we present some spectral geometric functionals which we call spectral divergence functionals, and we calculate the spectral divergence functionals of manifolds with (or without) boundary.

Almost uniform convergence for noncommutative Vilenkin-Fourier series
Yong Jiao, Sijie Luo, Tiantian Zhao, Dejian Zhou
https://arxiv.org/abs/2506.14431 htt…

Almost uniform convergence for noncommutative Vilenkin-Fourier series
In the present paper, we study almost uniform convergence for noncommutative Vilenkin-Fourier series. Precisely, we establish several noncommutative (asymmetric) maximal inequalities for the Cesàro means of the noncommutative Vilenkin-Fourier series, which in turn give the corresponding almost uniform convergence. The primary strategy in our proof is to explore a noncommutative generalization of Sunouchi square function operator, and the very recent advance of the noncommutative Calderón-Zygm…

On constructing deformations of noncommutative algebras
Agata Smoktunowicz
https://arxiv.org/abs/2507.04436 https://arxiv.org/pdf/250…

On constructing deformations of noncommutative algebras
This paper presents a method for constructing flat deformations of associative algebras. We will refer to this method as method two because it is a generalisation of the method obtained in [1]. The deformations obtained using the first two methods are of polynomial type. We also present a third method, for the construction of deformations from N to A, for a semisimple algebra A. As an applied example, we answer a question posed by Michael Wemyss on deformations of Acons.

The Free Functional Calculus in General
Julian Bushelli
https://arxiv.org/abs/2506.00170 https://arxiv.org/pdf/2506.00170

The Free Functional Calculus in General
The classical theory of free analysis generalizes the noncommutative (nc) polynomials and rational functions, easily providing such results as an nc analogue of the Jacobian conjecture. However, the classical theory misses out on important functions, such as the Schur complement. This paper presents a generalization of free functions, viewing them as a natural categorial structure: functors between functor categories that commute with natural transformation. We study this construction on genera…

This https://arxiv.org/abs/2506.00699 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

What is a double star-product?
Double Poisson brackets, introduced by M. Van den Bergh in 2004, are noncommutative analogs of the usual Poisson brackets in the sense of the Kontsevich-Rosenberg principle: they induce Poisson structures on the space of $N$-dimensional representations $\operatorname{Rep}_N(A)$ of an associative algebra $A$ for any $N$. The problem of deformation quantization of double Poisson brackets was raised by D. Calaque in 2010, and had remained open since then. In this paper, we address this problem b…

Trace Formulas in Noncommutative Geometry
Eva-Maria Hekkelman
https://arxiv.org/abs/2506.21950 https://arxiv.org/pdf/2506.21950

Trace Formulas in Noncommutative Geometry
Trace formulas appear in many forms in noncommutative geometry (NCG). In the first part of this thesis, we obtain results for asymptotic expansions of trace formulas like heat trace expansions by adapting the theory of Multiple Operator Integration to NCG. More broadly, this construction provides a natural language for operator integrals in NCG, which systematises and simplifies operator integral arguments throughout the literature. Towards this end, we construct a functional calculus for abstr…

f4ncgb: High Performance Gr\"obner Basis Computations in Free Algebras
Maximilian Heisinger, Clemens Hofstadler
https://arxiv.org/abs/2505.19304 https…

f4ncgb: High Performance Gröbner Basis Computations in Free Algebras
We present f4ncgb, a new open-source C++ library for Gröbner basis computations in free algebras, which transfers recent advancements in commutative Gröbner basis software to the noncommutative setting. As our experiments show, f4ncgb establishes a new state-of-the-art for noncommutative Gröbner basis computations. We also discuss implementation details and design choices.

This https://arxiv.org/abs/2405.17286 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotics of extensions of simple $\mathbb Q$-algebras
We answer various questions concerning the distribution of extensions of a given central simple algebra $K$ over a number field. Specifically, we give asymptotics for the count of inner Galois extensions $L/K$ of fixed degree and center with bounded discriminant. We also relate the distribution of outer extensions of $K$ to the distribution of field extensions of its center $Z(K)$. This paper generalizes the study of asymptotics of field extensions to the noncommutative case in an analogous man…

$L^p$-Theory and Noncommutative Geometry in Quantum Harmonic Analysis
Saeed Hashemi Sababe, Ismail Nikoufar
https://arxiv.org/abs/2506.18907 https://

$L^p$-Theory and Noncommutative Geometry in Quantum Harmonic Analysis
Quantum harmonic analysis extends classical harmonic analysis by integrating quantum mechanical observables, replacing functions with operators and classical convolution structures with their noncommutative counterparts. This paper explores four interrelated developments in this field: (i) a noncommutative $L^p$-theory tailored for quantum harmonic analysis, (ii) the extension of quantum harmonic analysis beyond Euclidean spaces to include Lie groups and homogeneous spaces, (iii) its deep conne…

The strong convergence phenomenon
Ramon van Handel
https://arxiv.org/abs/2507.00346 https://arxiv.org/pdf/2507.00346

The strong convergence phenomenon
In a seminal 2005 paper, Haagerup and Thorbjørnsen discovered that the norm of any noncommutative polynomial of independent complex Gaussian random matrices converges to that of a limiting family of operators that arises from Voiculescu's free probability theory. In recent years, new methods have made it possible to establish such strong convergence properties in much more general situations, and to obtain even more powerful quantitative forms of the strong convergence phenomenon. These, in tu…

The spectral torsion for the one form rescaled Dirac operator
Jian Wang, Yong Wang
https://arxiv.org/abs/2505.23319 https://arxiv.org…

The spectral torsion for the one form rescaled Dirac operator
The spectral torsion is defined by three vector fields and Dirac operators and the noncommutative residue. Motivated by the spectral torsion and the one form rescaled Dirac operator, we give some new spectral torsion which is the extension of spectral torsion for Dirac operators, and compute the spectral torsion for the one form rescaled Dirac operator on even-dimensional spin manifolds without boundary.

Non-Hermitian Nested Hopf-Links and Conjoint Open-Arcs in Synthetic Non-Abelian Gauge Photonic Lattices
Samit Kumar Gupta
https://arxiv.org/abs/2506.21986 …

Non-Hermitian Nested Hopf-Links and Conjoint Open-Arcs in Synthetic Non-Abelian Gauge Photonic Lattices
Non-Hermitian physics enriches the topological attributes of non-Abelian systems. Non-Abelian systems characterized by noncommutative braid patterns are associated with intriguing physical features and applications. Non-Abelian braiding of the non-Hermitian bands and anomalous skin mode localization may emerge due to a host of competing physical effects. The quest for the generality of their physical origin and the associated new phenomena, therefore, constitutes a pertinent question to conside…

Replaced article(s) found for quant-ph. https://arxiv.org/list/quant-ph/new
[3/3]:
Quantum Mechanics in Curved Space(time) with a Noncommutative Geometric Perspective

What is a double star-product?
Nikita Safonkin
https://arxiv.org/abs/2506.00699 https://arxiv.org/pdf/2506.00699

What is a double star-product?
Double Poisson brackets, introduced by M. Van den Bergh in 2004, are noncommutative analogs of the usual Poisson brackets in the sense of the Kontsevich-Rosenberg principle: they induce Poisson structures on the space of $N$-dimensional representations $\operatorname{Rep}_N(A)$ of an associative algebra $A$ for any $N$. The problem of deformation quantization of double Poisson brackets was raised by D. Calaque in 2010, and had remained open since then. In this paper, we address this problem b…

A ringed-space-like structure on coalgebras for noncommutative algebraic geometry
So Nakamura
https://arxiv.org/abs/2506.14689 https://

A ringed-space-like structure on coalgebras for noncommutative algebraic geometry
Inspired by the perspective of Reyes' noncomutative spectral theory, we attempt to develop noncommutative algebraic geometry by introducing ringed coalgebras, which can be thought of as a noncommutative generalization of schemes over a field $k$. These objects arise from fully residually finite-dimensional(RFD) algebras introduced by Reyes and from schemes locally of finite type over $k$. The construction uses the Heyneman-Sweedler finite dual coalgebra and the Takeuchi underlying coalgebra. Wh…

A direct method in noncommutative integrable systems
Shi-Hao Li, Shou-Feng Shen, Guo-Fu Yu, Jun-Yang Zhang
https://arxiv.org/abs/2506.18661 https://…

A direct method in noncommutative integrable systems
We present a constructive framework for deriving noncommutative (NC) integrable equations directly from quasi-determinant solutions. Building upon the quasi-Wronskian structure, we extend the classical direct method to the NC setting, where standard determinant identities are replaced by algebraic relations intrinsic to quasi-determinants. By analyzing derivative identities satisfied by quasi-determinants, we recover the NC Kadomtsev-Petviashvili (ncKP) and Date-Jimbo-Kashiwara-Miwa (ncDJKM) eq…

The homological spectrum for monoidal triangulated categories
Daniel K. Nakano, Kent B. Vashaw, Milen T. Yakimov
https://arxiv.org/abs/2506.19946 https://

The homological spectrum for monoidal triangulated categories
The authors develop a notion of homological prime spectrum for an arbitrary monoidal triangulated category, ${\mathbf C}$. Unlike the symmetric case due to Balmer, the homological primes of ${\mathbf C}$ are not defined as the maximal Serre ideals of the small module category ${\sf mod}\text{-}{\mathbf C}$, or via a noncommutative ring theory inspired version of this construction. Instead, the authors work with an extended comparison map from the Serre prime spectrum $\operatorname{Spc} ({\sf m…

A noncommutative construction of families of biunitary matrices and application to subfactors
Keshab Chandra Bakshi, Satyajit Guin, Guruprasad
https://arxiv.org/abs/2506.14351

A noncommutative construction of families of biunitary matrices and application to subfactors
We introduce a construction that, given a pair (u,v) of complex Hadamard matrices of the same order, generates infinitely many biunitary matrices of varying (and distinct) orders. As a key application, this framework yields nested sequences of vertex model subfactors that are not a tower of downward basic construction. Notably, the construction is noncommutative: interchanging the matrices (i.e., considering (v,u) instead of (u,v)) can lead to non-isomorphic subfactors. Focusing on the Hadamard…

Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields
Shane Farnsworth
https://arxiv.org/abs/2506.21496 https://arxiv.…

Spectral Geometry with Exceptional Symmetry and Charged Higgs Fields
We lay the foundations for a general approach to nonassociative spectral geometry as an extension of Connes' noncommutative geometry by explaining how to construct finite-dimensional, discrete spectral geometries with exceptional symmetry, and gauge covariant Dirac operators. We showcase an explicit construction of a geometry corresponding to the internal space of a $G_2\times G_2$ gauge theory with charged scalar content and scalar representations restricted by novel conditions arising from th…

Dimensions of compositions modulo a prime
Jia Huang
https://arxiv.org/abs/2506.16953 https://arxiv.org/pdf/2506.16953

Dimensions of compositions modulo a prime
The (ordinary) representation theory of the symmetric group is fascinating and has rich connections to combinatorics, including the Frobenius correspondence to the self-dual graded Hopf algebra of symmetric functions. The $0$-Hecke algebra (of type $A$) is a deformation of the group algebra of the symmetric group, and its representation theory has an analogous correspondence to the dual graded Hopf algebras of quasisymmetric functions and noncommutative symmetric functions. Macdonald used the h…

This https://arxiv.org/abs/2501.15859 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Free Hamilton Algebra
Over an arbitrary field $\mathbb{F}$, let $p$ and $q$ be monic polynomials with degree $2$ in $\mathbb{F}[t]$. The free Hamilton algebra of the pair $(p,q)$ is the free noncommutative algebra in two generators $a$ and $b$ subject only to the relations $p(a)=0=q(b)$. Free Hamilton algebras are models of free products of two $2$-dimensional algebras over $\mathbb{F}$. They can be viewed as the most elementary nontrivial noncommutative algebras over fields. It has been recently observed that the…

Transportation cost and contraction coefficient for channels on von Neumann algebras
Roy Araiza, Marius Junge, Peixue Wu
https://arxiv.org/abs/2506.04197 h…

Transportation cost and contraction coefficient for channels on von Neumann algebras
We present a noncommutative optimal transport framework for quantum channels acting on von Neumann algebras. Our central object is the Lipschitz cost measure, a transportation-inspired quantity that evaluates the minimal cost required to move between quantum states via a given channel. Accompanying this is the Lipschitz contraction coefficient, which captures how much the channel contracts the Wasserstein-type distance between states. We establish foundational properties of these quantities, in…

4-type subvarieties of the variety of associative algebras
B. Sartayev
https://arxiv.org/abs/2506.02471 https://arxiv.org/pdf/2506.02…

4-type subvarieties of the variety of associative algebras
In this paper, we consider four subvarieties of the variety of associative algebras. We study these subvarieties from the point of view of operads and show connections with well-known algebras such as dendriform algebras and noncommutative Novikov algebras. Finally, we define operations, commutator and anti-commutator on these algebras and obtain some identities under these operations.

This https://arxiv.org/abs/2405.14157 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Contractive representations of odometer semigroup
Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag-Solitar monoid with two generators, is a well-known object in group theory. This paper examines the odometer semigroup in relation to representations of bounded linear operators. We focus on noncommutative operators and prove that contractive representations of $O_n$ always admit to nicer representations of $O_n$. We give a complete description of representations of $O_n$ on the Foc…

Detecting local topology with the spectral localizer
Alexander Cerjan, Hermann Schulz-Baldes
https://arxiv.org/abs/2506.14174 https://

Detecting local topology with the spectral localizer
The spectral localizer is a predictive framework for the computation of topological invariants of natural and artificial materials. Here, three crucial improvements on the criterion for the validity of the framework are reported: first, merely a properly defined local spectral gap of the Hamiltonian is required, second, only relative bounds on the Hamiltonian and its noncommutative derivative are relevant, and, third, the numerical constant in a tapering estimate is improved. These developments…

Real Non-Commutative Convexity I
David P. Blecher, Caleb McClure
https://arxiv.org/abs/2506.13512 https://arxiv.org/pdf/2506.13512

Real Non-Commutative Convexity I
We initiate the theory of real noncommutative convex sets, following the recent and profound complex case developed by Davidson and Kennedy. The present paper focuses on the real case of the topics from the first several sections of their profound memoir.. Later results will be discussed in a future paper. We give many foundational structural results for real operator systems and their associated noncommutative (nc) convex sets, and elucidate how the complexification interacts with the basic co…

This https://arxiv.org/abs/2410.03475 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Noncommutative metric geometry of quantum circle bundles
In this paper we investigate quantum circle bundles from the point of view of compact quantum metric spaces. The raw input data is a circle action on a unital $C^*$-algebra together with a quantum metric of spectral geometric origin on the fixed point algebra. Under a few extra conditions on the spectral subspaces we show that the spectral geometric data on the base algebra can be lifted to the total algebra. Notably, the lifted spectral geometry is independent of the choice of frames and is pe…

Quantisation ideals, canonical parametrisations of the unipotent group and consistent integrable systems
M. A. Chirkov, A. V. Mikhailov, D. V. Talalaev
https://arxiv.org/abs/2505.20253

Quantisation ideals, canonical parametrisations of the unipotent group and consistent integrable systems
Using the methods of quantisation ideals, we construct a family of quantisations corresponding to Case alpha in Sergeev's classification of solutions to the tetrahedron equation. This solution describes transformations between special parametrisations of the space of unipotent matrices with noncommutative coefficients. We analyse the classical limit of this family and construct a pencil of compatible Poisson brackets that remain invariant under the re-parametrisation maps (mutations). This deco…

Proper cocycles, measure equivalence and $L_p$-Fourier multipliers
Simeng Wang, Runlian Xia, Gan Yao
https://arxiv.org/abs/2506.18320 https://

Proper cocycles, measure equivalence and $L_p$-Fourier multipliers
We develop a new transference method for completely bounded $L_p$-Fourier multipliers via proper cocycles arising from probability measure-preserving group actions. This method extends earlier results by Haagerup and Jolissaint, which were limited to the case $p = \infty$. Based on this approach, we present a new and simple proof of the main result in [Hong-Wang-Wang, Mem. Amer. Math. Soc. 2024] regarding the pointwise convergence of noncommutative Fourier series on amenable groups, refining th…

This https://arxiv.org/abs/2209.01408 has been replaced.
link: https://scholar.google.com/scholar?q=a

Adequacy of nonsingular matrices over commutative principal ideal domains
The notion of the adequacy of commutative domains was introduced by Helmer in Bull. Amer.Math. Soc., 49 (1943), 225--236. In the present paper we extend the concept of adequacy to noncommutative Bézout rings. We show that the set of nonsingular second-order matrices over a commutative principal ideal domain is adequate.

This https://arxiv.org/abs/2310.04052 has been replaced.
link: https://scholar.google.com/scholar?q=a

Spectral metrics on quantum projective spaces
We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dabrowski. In particular, we establish that the Connes metric metrizes the weak-* topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spect…

This https://arxiv.org/abs/2403.06767 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Continuity and equivariant dimension
We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fiber…

This https://arxiv.org/abs/2304.07852 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Simplicity of crossed products by FC-hypercentral groups
In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A \rtimes_λG$ is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that contains all virtually nilpotent groups, and in the finitely-generated setting, coincides with the set of groups which have polynomial growth. We further completely characterize the ideal intersection property under the assumption that the group is FC, meaning that every e…

Brown-Voiculescu entropy revisited
Bhishan Jacelon, Robert Neagu
https://arxiv.org/abs/2506.18352 https://arxiv.org/pdf/2506.18352

Brown-Voiculescu entropy revisited
Aided by the tools and outlook provided by modern classification theory, we take a new look at the Brown-Voiculescu entropy of endomorphisms of nuclear C*-algebras. In particular, we introduce `coloured' versions of noncommutative topological entropy suitable for C*-algebras A of finite nuclear dimension or finite decomposition rank. In the latter case, assuming further that A is simple, separable, unital, satisfies the UCT and has finitely many extremal traces, we prove a variational type prin…