Quantum Lie algebras via modified Reflection Equation Algebra

The Levin-Wen string-nets of a spherical fusion category $\mathcal{C}$ describe, by results of Kirillov and Bartlett, the representations of mapping class groups of closed surfaces obtained from the Turaev-Viro construction applied to $\mathcal{C}$. We provide a far-reaching generalization of this statement to arbitrary pivotal finite tensor categories, including non-semisimple or non-spherical ones: We show that the finitely cocompleted string-net modular functor built from the projective objects of a pivotal finite tensor category is equivalent to Lyubashenko's modular functor built from the Drinfeld center $Z(\mathcal{C})$.

math.QA↗

arXiv2026

The Quantum Double of Hopf Algebras Realized via Partial Dualization and the Tensor Category of Its Representations

In this paper, we aim to study the (generalized) quantum double $K^{\ast\mathrm{cop}}\bowtie_σH$ determined by a (skew) pairing between finite-dimensional Hopf algebras $K^{\ast\mathrm{cop}}$ and $H$, especially the tensor category $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_σH)$ of its finite-dimensional representations. Specifically, we show that $K^{\ast\mathrm{cop}}\bowtie_σH$ is a left partially dualized (quasi-)Hopf algebra of $K^\mathrm{op}\otimes H$, and use this formulation to establish tensor equivalences from $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_σH)$ to the categories ${}^K_K\mathcal{M}^K_H$ and ${}^{K^\ast}_{K^\ast}\mathcal{M}^{H^\ast}_{K^\ast}$ of two-sided two-cosided relative Hopf modules, as well as the category ${}_H\mathfrak{YD}^K$ of relative Yetter-Drinfeld modules.

math.QA↗

arXiv2026

Shifted symplectic structures and Poisson vertex algebra

We identify Poisson vertex algebra (PVA) Hamiltonian structures on arc spaces with Maurer--Cartan data arising from degree-$1$ graded symplectic geometry. Let $X$ be a smooth scheme of finite type and set $Y=T^*[1]X$. We prove that a degree-$2$ Hamiltonian on $Y_\infty$ defines a sheaf of PVAs on $X_\infty$ precisely when its class in the Lie algebra of local functionals satisfies the classical master equation; conversely, every such PVA bracket determines a unique degree-$2$ local functional. We further show that classical $R$-matrices are exactly Maurer--Cartan deformations of the corresponding PVA Hamiltonian structure. As applications, we classify a class of global scalar PVA structures on $\mathbb P^1$ and apply the $R$-matrix formalism to finite and affine classical $\mathcal W$-algebras. For the subregular $\mathfrak{sl}_3$ example, we construct a compatible Poisson pencil and the centrally reduced RDW system, whose generic reduction is a genus-two hyperelliptic system admitting an explicit Mumford-type Lax representation. Its affine counterpart yields constant and first-order differential $R$-deformations, together with commuting Hamiltonian flows.

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