Mapping Class Groups do not have Kazhdan's Property (T)

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$ do have free analogues $S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$. There are many examples of real algebraic submanifolds $X\subset S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $\int:C(X)\to\mathbb C$, that we will study here. We will mostly focus on free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely real/complex, classical/free.

math.QA↗

arXiv2026

Complex Hadamard Matrices - Quantum Symmetries, Equivalence and Non-Local Games

We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local ``Hadamard equivalence'' game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak--Sołtan--Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.

math.QA↗

arXiv2026

Ordinary modules for affine vertex operator superalgebras

Let $\mathfrak{g}$ be a basic classical Lie superalgebra and let $\widehat{\mathfrak{g}}$ be the corresponding affine Lie superalgebra. In this paper, we first prove that a Cartan subalgebra acts semisimply on ordinary modules for the simple affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(k,0)$ at boundary admissible level $k$. Then we prove that the category $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ of ordinary $L_{\widehat{\mathfrak{g}}}(k,0)$-modules is finite, semisimple and $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is exactly the category $KL_k(\mathfrak{g})$ of finite-length generalized modules for the affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(k,0)$.Thus $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is a braided tensor supercategory. Furthermore, we obtain the rigidity of the supercategory $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ and thus it is a ribbon supercategory. Finally, we conclude that $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is a ribbon fusion supercategory.

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