No braiding for quantum sl2 reps - A short proof

Let $G$ be a finite non-abelian group, let $Φ\in Z^3(G,\mathbb C^\times)$ be normalized, and let $V,W$ be finite-dimensional simple objects of ${}_G^G\mathcal{YD}^Φ$. We classify, up to interchange, the braided-indecomposable pairs $(V,W)$ whose supports generate $G$ and for which $\mathcal B(V\oplus W)$ is finite-dimensional. The classification consists of eight cases with five possible support quandles. In every case the Cartan graph is standard of type $A_2$, $B_2$, or $G_2$, and the dimension is determined explicitly. A new phenomenon occurs {in the $Γ_2$ case}: the twisted setting admits a family of type $G_2$ absent from the ordinary $Γ_2$ classification and we construct an explicit example over a non-abelian group of order $16$.

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arXiv2026

Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

math.QA↗

arXiv2026

On adjoint double shuffle relations and adjoint reduction relations

In this paper, we give a positive answer to Jarossay's question under the adjoint reduction relations. Namely, we prove that the extended double shuffle relations together with the relation $\!\!\!\mod π^2$ are equivalent to the adjoint double shuffle relations together with the adjoint reduction relations.

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