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1. Universal Tensor Categories Generated by Dual Pairs

   https://doi.org/10.1007/s10485-021-09640-2  

   Chirvasitu, Alexandru; Penkov, Ivan (October 2021, Applied Categorical Structures)

   Abstract Let $$V_*\otimes V\rightarrow {\mathbb {C}}$$ V ∗ ⊗ V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and $$V_*$$ V ∗ . The Mackey Lie algebra $${\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)$$ g = gl M ( V , V ∗ ) corresponding to this pairing consists of all endomorphisms $$\varphi $$ φ of V for which the space $$V_*$$ V ∗ is stable under the dual endomorphism $$\varphi ^*: V^*\rightarrow V^*$$ φ ∗ : V ∗ → V ∗ . We study the tensor Grothendieck category $${\mathbb {T}}$$ T generated by the $${\mathfrak {g}}$$ g -modules V , $$V_*$$ V ∗ and their algebraic duals $$V^*$$ V ∗ and $$V^*_*$$ V ∗ ∗ . The category $${{\mathbb {T}}}$$ T is an analogue of categories considered in prior literature, the main difference being that the trivial module $${\mathbb {C}}$$ C is no longer injective in $${\mathbb {T}}$$ T . We describe the injective hull I of $${\mathbb {C}}$$ C in $${\mathbb {T}}$$ T , and show that the category $${\mathbb {T}}$$ T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category $$_I{\mathbb {T}}$$ I T of objects in $${\mathbb {T}}$$ T which are free as I -modules. Our main result is that the category $${}_I{\mathbb {T}}$$ I T is also Koszul, and moreover that $${}_I{\mathbb {T}}$$ I T is universal among abelian $${\mathbb {C}}$$ C -linear tensor categories generated by two objects X , Y with fixed subobjects $$X'\hookrightarrow X$$ X ′ ↪ X , $$Y'\hookrightarrow Y$$ Y ′ ↪ Y and a pairing $$X\otimes Y\rightarrow {\mathbf{1 }}$$ X ⊗ Y → 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $${\mathbb {T}}$$ T and $${}_I{\mathbb {T}}$$ I T . 

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2. Hamiltonian quantization of complex Chern-Simons theory at level-k

   https://doi.org/10.1007/JHEP12(2025)158  

   Han, Muxin (December 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group$$\text{SL}(2,{\mathbb{C}})$$at an even level$$k\in {\mathbb{Z}}_{+}$$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The *-representation of the operator algebra is carried by the infinite dimensional Hilbert space$${\mathcal{H}}_{\overrightarrow{\lambda }}$$and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$. The quantum group$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on am-holed sphere Σ_0,m, the physical Hilbert space$${\mathcal{H}}_{\text{phys}}$$is identified by imposing the gauge invariance and the flatness constraint. The states in$${\mathcal{H}}_{\text{phys}}$$are the$${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$-invariant linear functionals on a dense domain in$${\mathcal{H}}_{\overrightarrow{\lambda }}$$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of Σ_0,mare diagonalized. 

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3. Inequalities for the Radon Transform on Convex Sets

   https://doi.org/10.1093/imrn/rnab122  

   Giannopoulos, Apostolos; Koldobsky, Alexander; Zvavitch, Artem (May 2021, International Mathematics Research Notices)

   Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem. 

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4. Scheme-theoretic coisotropic reduction

   https://doi.org/10.1093/qmath/haag005  

   Crooks, Peter; Mayrand, Maxence (February 2026, The Quarterly Journal of Mathematics)

   ABSTRACT We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $$\mathbb {k}=\mathbb {R}$$ or $$\mathbb {k}=\mathbb {C}$$, and is formulated for an affine symplectic groupoid $$\mathcal {G}\,\,\substack{\longrightarrow \\ \longrightarrow }\,\,X$$, an affine Hamiltonian $$\mathcal {G}$$-scheme $$\mu :M\longrightarrow X$$, a coisotropic subvariety $$S\subseteq X$$, and a stabilizer subgroupoid $$\mathcal {H}\,\,\substack{\longrightarrow \\\longrightarrow }\,\,S$$. Our first main result is that the Poisson bracket on $$\mathbb {k}[M]$$ induces a Poisson bracket on the subquotient $$\mathbb {k}[\mu ^{-1}(S)]^{\mathcal {H}}$$. The Poisson scheme $$M\mathbin {//}_{{S,\mathcal {H}}}\mathcal {G}:= \mathrm{Spec}(\mathbb {k}[\mu ^{-1}(S)]^{\mathcal {H}})$$ is then declared to be a Hamiltonian reduction of M. Other main results include sufficient conditions for $$M\mathbin {//}_{{S,\mathcal {H}}}\mathcal {G}$$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to [6], where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden–Ratiu reduction [19], Mikami–Weinstein reduction [20], Śniatycki–Weinstein reduction [23], and symplectic reduction along general coisotropic submanifolds [6]. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore–Tachikawa conjecture. 

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5. Banach spaces for which the space of operators has 2 ^𝔠 closed ideals

   https://doi.org/10.1017/fms.2021.23  

   Freeman, Daniel; Schlumprecht, Thomas; Zsák, András (January 2021, Forum of Mathematics, Sigma)

   Abstract We formulate general conditions which imply that $${\mathcal L}(X,Y)$$ , the space of operators from a Banach space X to a Banach space Y , has $$2^{{\mathfrak {c}}}$$ closed ideals, where $${\mathfrak {c}}$$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in $${\mathcal L}\left (\ell _p\oplus \ell _q\right )$$ is exactly $$2^{{\mathfrak {c}}}$$ for all $$1<\infty $$ . 

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