Motivated by a question of A. Skalski and P. M. Sołtan (2016) about inner faithfulness of S. Curran’s map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on points. This enables us not only to answer the aforementioned question in the positive for the case where $n\,=\,4,\,k\,=\,2$, but also to classify the automorphisms of $S_{4}^{+}$, describe all the embeddings ${{O}_{-1}}(2)\,\subset \,S_{4}^{+}$ and show that all the copies of ${{O}_{-1}}(2)$ inside $S_{4}^{+}$are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

Nichols algebras naturally appear in the theory of quantized enveloping algebras of Kac–Moody algebras and in the classification of finite-dimensional pointed Hopf algebras. Assuming that the base field has characteristic zero a list of finite-dimensional rank 2 Nichols algebras of diagonal type is given. Each of them is described in terms of generators and relations. Kharchenko's restricted Poincaré–Birkhoff–Witt basis is visualized using full binary trees.

We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl–Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups ${\mathcal O}_q(\text{GL}_n)$ and ${\mathcal O}_q(\text{SL}_n)$.

Let ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ be the positive part of the quantized enveloping algebra ${{U}_{q}}\left( \mathfrak{s}{{\Iota }_{n+1}} \right)$. Using results of Alev–Dumas and Caldero related to the center of ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra $U\left( \mathfrak{g} \right)$ of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra ${{\overset{\scriptscriptstyle\smile}{U}}_{q}}\left( \mathfrak{g} \right)$. Of greater importance to its representation theory is the fact that ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ is free over a larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$ to ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.