17 problems

Integral and unimodularity conjecture. The element $(\beta\!\rightharpoonup\!\lambda\bowtie r)$ should be a left integral of $D(H)$, at least when $H$ is unimodular; moreover,…

Let $H$ be a finite-dimensional quasi-Hopf algebra with structure elements $\alpha$ and $\beta$, antipode $S$, and Drinfeld double $D(H)$. Let…

Universal-twist conjecture. These twist-like operators should be evaluation representations of universal twists obeying a shifted cocycle condition, so that the relation above is p…

Let $\mathfrak{g}$ be a Lie algebra, let $U(\mathfrak{g})[[{\mathchar'26\mkern-9muh}]]$ be its formal power-series enveloping algebra with standard coproduct $\Delta_0$, let $\Phi$…

Mason–Ng dome algebra conjecture. This quotient coincides with the quasi-Hopf algebra $\mathscr{D}_{\tilde\omega}(\Gamma,D)$ defined by Mason–Ng:

Let $A$ be a finite abelian group and let $\mathbf{Vec}_A^{(\omega,c)}$ be a modular tensor category. Let $\mathcal{V}$ be a pointed vertex operator algebra with … Let $G$ be a fin…

For each $d\in\mathbb{Z}_{\geq 0}$, let $Q(d)$ be the factorizable quasi-Hopf algebra introduced in the cited work, and let $SF^+_d$ denote the even part of the symplectic fermion…

Let $H$ be a Hopf algebra, let $B$ be a left coideal subalgebra, and let $(H/B^+H)^\ast\#B$ be the associated left partially dualized quasi-Hopf algebra. In Remark on invertibility…

Let $L_2$ be the $A_1$ root lattice and let $V:=V_{L_2}$ be the corresponding lattice vertex operator algebra. Let $SU(2)$ be the universal central extension of $SO_3(\mathbb{R})$,…

Let $H$ be a quasi-Hopf algebra of prime dimension $p$. Commutativity and cocommutativity conjecture. The quasi-Hopf algebra $H$ is commutative and cocommutative. This is open even…

Let $\mathcal{V}_{\mathrm{ev}}(N)$ be the even part of the super-vertex operator algebra of $N$ pairs of symplectic fermions, with central charge $c=-2N$. Let $\mathsf{Q}(N,\beta)$…

Let $p\geq 2$, let $q=e^{\pi i/p}$, let $\overline{U}^{(\Phi)}_{q} s\ell(2)$ be the factorisable ribbon quasi-Hopf algebra described above, and let $\mathcal{W}(p)$ be the triplet…

Let $Qu(\mathfrak{D},\lambda,\Phi_{\underline{c}})=u(\mathfrak{D},\lambda,0,\Phi_{\underline{c}})$ be a small quasi-quantum group given in Proposition 4.5, with…

Let $\mathrm N$ be the parameter specifying the number of symplectic fermion pairs, let $\beta$ be the scalar used in the ribbon structure, and let…

A quasi-bialgebra is an algebraic structure with coproduct, counit, and associator satisfying the quasi-bialgebra axioms; a preantipode is a map satisfying the three preantipode ax…

Let $\mathfrak{g}$ be the Lie algebra occurring above, let $\mathfrak{g}[u]$ be its polynomial current algebra, and for non-zero constants $c_1\neq c_2$ let … These $r$-matrices de…

Let $A$ be an ordinary semisimple factorizable Hopf algebra over a field of characteristic zero, with Gaussian sums ${\mathfrak g}$ and ${\mathfrak g}'$. Ordinary Hopf algebra obst…