arXiv · math/9808016
A survey of results on quantum Poincare groups and quantum Minkowski spaces is presented.
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P. Podles. 1999-01-08. Quantum Minkowski spaces. https://doi.org/10.1063/1.57123
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Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka--Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category $\mathcal{NC}$ of non-crossing partitions and the category $\mathcal{P}$ of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by $\mathcal{NC}$ together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, using an argument of ChatGPT to account for a five-dimensional gap, there is no exotic category distinguished from $\mathcal{NC}$ or $\mathcal{P}$ at moments of order six.
Let $p>5$ be a prime. In this paper, we show that a pre-modular fusion category of global dimension $2p$ is either pointed or tensor equivalent to $\text{Rep}(D_{2p})$.
We determine the graded automorphism group of the multiparameter quantum matrix algebra $\mathcal O_{λ,\mathbf p}(M_n(\Bbbk))$ over an arbitrary field $\Bbbk$ for all $n\ge2$ and $λ\ne1$. No other restrictions are assumed on $\mathbf p$ or $λ$. Every graded automorphism is a torus automorphism, possibly followed by transposition or by simultaneous reversal of the row and column indices. We give the precise parameter conditions for these two involutions. As an application, we also determine the full automorphism group when the multiplicative group generated by the parameters is torsion-free.