MD Nahidul Hasan Sabit
Apr 23, 2026·quant-ph·PDF We study spectral properties of quantum many-body Hamiltonians through a subsystem-based framework. Given a Hamiltonian of the form $H = \sum_{X \subseteq Λ} Φ(X)$ acting on a tensor product Hilbert space, we associate to each subset $S \subseteq Λ$ a subsystem Hamiltonian $H_S$ and its spectrum $\mathcal{S}(S) = σ(H_S)$. This produces a family of spectra indexed by subsystems, allowing spectral data to be organized according to interaction structure. We show that subsystem Hamiltonians admit local approximations: $H_S$ can be approximated by operators supported on finite neighborhoods with an error bounded by $\|H_S - H_{S,r}\| \le |S| e^{-μr} \|Φ\|_μ$. As a consequence, subsystem spectra are stable under truncation in the sense that $d_H(\mathcal{S}(S), σ(H_{S,r})) \le |S| e^{-μr} \|Φ\|_μ.$ We then prove that for disjoint subsets $S_1, S_2 \subseteq Λ$, the subsystem spectrum is approximately additive: $d_H\big(\mathcal{S}(S_1 \cup S_2), \mathcal{S}(S_1) + \mathcal{S}(S_2)\big) \le (|S_1| + |S_2|) e^{-μD} \|Φ\|_μ,$ where $D = d(S_1, S_2)$. In the finite-range case, this relation becomes exact. The results show that spectral properties reflect the locality of interactions not only at the level of operators, but also at the level of spectra. The framework provides a way to study many-body systems in which interaction geometry directly shapes spectral behavior.
Quan Chen
We prove that if a finite group $G$ acts outerly on a McDuff $\rm II_1$ factor $M$, then $\mathsf{Rep}(G/KL)$ is a braided monoidal full subcategory of the categorical Connes' $\tildeχ(M\rtimes G)$ defined in arXiv:2111.06378, where $K$ and $L$ are the centrally trivial and approximately inner parts in $G$ respectively. When $L$ is trivial, we give an explicit formula for the $G/K$-gauging procedure on $\tildeχ(M\rtimes G)$. This is the categorical generalization of Connes' short exact sequence on $χ(M\rtimes G)$. Using this machinery, for any finite group $G$, we construct a McDuff $\rm II_1$ factor $M$, whose $\tildeχ(M)$ is braided equivalent to $\mathsf{Rep}(G)$. This is the first example of a braided fusion category which is not modular as $\tildeχ$.
Kyungtak Hong, Alexander Tsymbaliuk
We present the RLL-realization of extended orthosymplectic quantum supergroups for any parity sequence, with R-matrices evaluated in the earlier work arxiv:2408.16720. Our isomorphism is compatible with the internal structure of generalized doubles. We also relate different sign conventions through 2-cocycle twists. Furthermore, we establish a factorization of the reduced R-matrix within the RLL-realization.
Thomas Creutzig, Volodymyr Kovalchuk, Andrew R. Linshaw, Arim Song, Uhi Rinn Suh
The universal $2$-parameter vertex algebra $\mathcal{W}_{\infty}$ of type $\mathcal{W}(2,3,\dots)$ is a classifying object for vertex algebras of type $\mathcal{W}(2,3,\dots,N)$ for some $N$; under mild hypotheses, all such vertex algebras arise as quotients of $\mathcal{W}_{\infty}$. In 2017, Gaiotto and Rapčák introduced a family of such vertex algebras called $Y$-algebras, and conjectured that they fall into groups of three that are mutually isomorphic. This is a common generalization of both Feigin-Frenkel duality and the coset realization of principal $\mathcal{W}$-algebras in type $A$, and was proven in 2021 for the simple $Y$-algebras (i.e., one label is zero) by the first and third authors. In this paper, we extend this entire story to the $\mathcal{N}=2$ superconformal setting. First, we prove the 2013 conjecture of Gaberdiel and Candu that there exists a universal $2$-parameter vertex algebra $\mathcal{W}^{\mathcal{N}=2}_{\infty}$ which is an extension of the $\mathcal{N}=2$ superconformal algebra, and has four additional generators in weights $i, i + \frac{1}{2}, i + \frac{1}{2}, i+1$, for each integer $i > 1$. This admits many $1$-parameter quotients which we call $\mathcal{N}=2$ supersymmetric $Y$-algebras, and we prove the dualities among these algebras which were conjectured in 2018 by Prochazka and Rapčák. A special case is the coset realization of the principal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{sl}_{n+1|n})$ which was conjectured in 1992 by Ito. As a corollary, we obtain the strong rationality of $\mathcal{W}_k(\mathfrak{sl}_{n+1|n})$ for $k = -1 + \frac{1}{n+a+1}$ for all positive integers $n,a$, and we describe its module category. This generalizes Adamović's 1999 result on $\mathcal{N}=2$ minimal models, which is the case $n=1$.
Marco Mackaay, Vanessa Miemietz, Pedro Vaz
In this article, we develop a generalization of finitary birepresentation theory applicable to Soergel bimodules for infinite Coxeter groups. We establish a reduction process for the classification of simple birepresentations of almost finitary bicategories, and consider in detail the case of Soergel bimodules in extended affine type A.
Petter Andreas Bergh
It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.
Naihuan Jing, Yinlong Liu, Jian Zhang
In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality between the quantum superalgebra $U_q(\mathfrak{gl}_{m|n})$ and the Iwahori-Hecke algebra $\mathcal{H}_r$ of type A, we explicitly construct basis vectors of the $(U_q(\mathfrak{gl}_{m|n}), \mathcal{H}_r)$-bimodule on the tensor product space $(\mathbb{C}^{m|n})^{\otimes r}$. Using this construction, we interpret the quantum super immanants via weight spaces of covariant tensor representations of $U_q(\mathfrak{gl}_{m|n})$.
Bintao Cao, Ye Huang
We define the Verma vector system for each finite dimensional irreducible representation of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ with the highest weight $λ,$ via the conditions that making a tableau with shape $λ$ to be a Kashiwara-Nakashima tableau. We then show the linearly independence of this vector system. It turns out to be a basis of the finite dimensional irreducible representation $L(λ)$ of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ with the highest weight $λ,$ which analogs to the Verma basis of representations of $\mathfrak{sp}_4,$ called the Verma basis of the finite dimensional irreducible representation of $\mathfrak{spo}(4|1)$.
Bintao Cao, Ye Huang
We construct a one-to-one correspondence between the Verma basis vectors of a finite dimensional irreducible representation $L(λ)$ of the symplectic Lie algebra $\mathfrak{sp}_4$ and the Kashiwara-Nakashima tableaux of $\mathfrak{sp}_4$ with shape $λ$ naturally. We also give a proof of the linear independence of the Verma vector system directly.
Charles Igel, Jeremy Mancinas, Juan Villarreal
This work is motivated by recent developments in celestial holography. In \cite{CP}, the authors interpreted QCD collinear singularities in terms of operator product expansions in a two-dimensional CFT. We reformulate the algebraic structures arising in their work using the formalism of non-linear Lie conformal algebras developed in \cite{SK}.
Gautier-Edouard Edouard Filardo
We introduce a traceable operator-algebraic framework for incompressible transport on M= T3 (and, more generally, compact Riemannian manifolds endowed with a smooth invariant probability measure). Given an autonomous divergence-free velocity field u, the time-1 map $Φ$ induces the Koopman unitary U on L2(M) and the crossed-product finite von Neumann algebra Mu\,:= L$\infty$(M) ___$α$ Z= W$\star$(L$\infty$(M),U), equipped with its canonical faithful normal trace $τ$u. Within Mu we define tracial complexity functionals from commutators [U,Mf] (with Mf the multiplication operators) and associated positive elements, and we connect these quantities to Fuglede--Kadison determinants and entropy-like tracial functionals. In parallel, we introduce bounded regularized advection operators T(s) u\,:= KsTuKs as differential-level probes of transport oncommutativity, and we recall the Lie-bracket commutator identity at the formal generator level. This provides a natural algebraic setting in which tracial invariants are well posed and, in principle, computable on discretizations (e.g. cavity flow and vortex benchmarks).
Shoki Sato
The classification of finite groups under monoidal equivalence is a fundamental topic in the study of finite quantum groups. While a complete classification has been established for all groups of order strictly less than 64, the case for order 64 has remained limited to the construction of specific examples. In this study, we achieve the complete classification for groups of order 64 by developing an original computational approach using GAP. We describe our methodology and demonstrate that there exist exactly two pairs of non-isomorphic isocategorical groups of this order.
Ryo Hayami
In this paper, we investigate a quandle structure induced by an augmented rack arising from a gauge transformation group. We construct a quandle from a principal bundle and its discrete generalization. When we see a group as a (discrete) principal bundle over a point, this quandle becomes equivalent to the generalized Alexander quandle for its inner automorphism. Moreover, we construct a Lie and Noether quandle structure from a smooth gauge transformation.
Mohamed Elhamdadi, Bryce Virgin
In [Self-distributive structures in physics. Internat. J. Theoret. Phys. 64 (2025), no. 3, Paper No. 73], Fritz was motivated by the structure of Hamiltonian/Heisenberg mechanics to define the notion of "Lie Quandle", which he argued are nonlinear generalizations of finite dimensional real Lie algebras. In this article, we will investigate a linear/nonlinear correspondence to which Fritz' is a special case, classify a class of generalizations of these objects, as well as describe some results in the direction of a nonlinear analogue of Noether's first theorem first described by Fritz.
Niels Lauritzen, Jesper Funch Thomsen
We study when algebra endomorphisms can be lifted to first-order flat lifts. To a first-order flat lift of an algebra and an endomorphism, we associate a canonical class in Hochschild cohomology with coefficients in a naturally twisted bimodule. The cohomology class vanishes exactly when the endomorphism admits a multiplicative lift. For an Azumaya algebra of constant rank over a formally smooth center, we prove that an endomorphism lifts if and only if the induced endomorphism of the center preserves the Poisson structure given by the lift of the algebra.
Djordje Bogdanović, Marija Dimitrijević Ćirić, Richard J. Szabo
We construct cubic scalar field theory on $λ$-Minkowski space by combining the Batalin-Vilkovisky formalism with harmonic analysis, and produce two inequivalent noncommutative quantum field theories. The braided theory is based on a braided $L_\infty$-algebra whereby covariance dictates a spectral decomposition into cylindrical Bessel functions that diagonalise the angular Drinfel'd twist; in this theory we find the usual logarithmic ultraviolet divergences and confirm the absence of UV/IR mixing. The standard noncommutative theory is based on a classical $L_\infty$-algebra; in this theory we relate the spectral decompositions into plane wave and cylindrical harmonic eigenmodes of the Klein-Gordan operator, we verify the planar equivalence theorem, and we demonstrate a periodic form of UV/IR mixing in which non-planar correlators are generically ultraviolet finite but become non-analytic on an infinite lattice of exceptional momenta.
Yimin Huang, Tiancheng Qi, Quanshui Wu, Ruipeng Zhu
In this paper, we study the Chevalley property of Cayley-Hamilton Hopf algebras in the sense of De Concini-Procesi-Reshetikhin-Rosso using discriminant ideals. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ whose identity fiber algebra has the Chevalley property, we prove that an irreducible $H$-module $V$ has the property that $V\otimes W$ is a completely reducible $H$-module for every irreducible $H$-module $W$ if and only if $V$ is annihilated by the lowest discriminant ideal of $(H,C,\text{tr})$, which establishes a bridge between the tensor-nondegenerate behaviour of the irreducible representations of $H$ and the lowest discriminant ideal of $(H,C,\text{tr})$. Using discriminant ideals, we prove that an affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ has the Chevalley property if and only if its identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property and all the discriminant ideals of $(H,C,\text{tr})$ are trivial, thereby resolving a question posed by Huang-Mi-Qi-Wu. Moreover, it is shown that the lowest discriminant subvariety $\mathcal{V}_{\ell}$ of the algebraic group $\operatorname{maxSpec}C$ is a closed subgroup, which reflects the rigid nature of $\mathcal{V}_{\ell}$ and is effective in determining the lowest discriminant subvarieties in certain examples of low GK dimension. This rigidity property provides a method, via the lowest discriminant ideals, for constructing a large family of Hopf algebras with the Chevalley property and finite GK dimension. The results are illustrated through applications to the big quantized Borel subalgebras at roots of unity and to certain Artin-Schelter Gorenstein Hopf algebras of low GK dimension. In particular, the framework yields (non-finite) tensor categories with the Chevalley property arising from some big quantum groups at roots of unity.
Jacek Krajczok, Piotr. M. Sołtan
We continue the study of the braided compact quantum group $\mathrm{SU}_q(2)$ for complex $q$ satisfying $0<|q|<1$ introduced by Kasprzak, Meyer, Roy and Woronowicz (J. Noncommut. Geom. 10(4):1611-1625, 2016). We address such aspects as existence of the Haar measure, construct the scaling group, the antipode and its polar decomposition and describe the related braided Hopf algebra. We also study when the braided flip extends to a completely bounded map and establish equivalence between the two approaches to bosonization and braided tensor product taken in the literature (Kasprzak, Meyer, Roy, Woronowicz J. Noncommut. Geom. 10(4):1611-1625, 2016 vs. Meyer, Roy Woronowicz Internat. J. Math. 25(2):1450019, 37, 2014, Roy Int. Math. Res. Not. (14):11791--11828, 2023 and De Commer, Krajczok arXiv:2412.17444, to appear in J. Operator Th.).
Chenglang Yang
In this paper, we study the $n$-point function of $t$-core partitions. The main tool is the topological vertex, originally developed to study the topological string theory for toric Calabi--Yau 3-folds. By virtue of the topological vertex, we introduce the $q$-deformed $n$-point function that generalizes both the ordinary $n$-point function of all integer partitions studied by Bloch--Okounkov and $t$-core partition case treated here. As a consequence, we provide a closed formula for the $n$-point function of $t$-core partitions in terms of theta functions, and prove that the corresponding correlation functions are quasimodular forms.
Darlayne Addabbo
Let $V$ be a grading-restricted vertex algebra and let $A^\infty(V)=U^\infty(V)/Q^\infty(V)$ be the associative algebra constructed by Huang, where $U^\infty(V)$ is the space of column-finite infinite matrices with entries in V and $Q^\infty(V)$ is an ideal of a (nonassociative) algebra structure on $U^\infty(V)$ defined by Huang. Huang introduced families of elements in $Q^\infty(V)$ and conjectured that these elements generate $Q^\infty(V)$. We discover and prove that Huang's elements all satisfy what we call ``the diagonal shift property". On the other hand, in the case that $V$ is the rank one Heisenberg vertex operator algebra, we construct infinitely many linearly independent elements in $Q^\infty(V)$ that do not satisfy the diagonal shift property. As a corollary, we disprove Huang's conjecture.