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We initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time.

This is the first of our papers on quasi-split affine quantum symmetric pairs$\big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big )$, focusing on the real rank one case, i.e.,$\mathfrak g = \mathfrak {sl}_3$equipped with a diagram involution. We construct explicitly a relative braid group action of type$A_2^{(2)}$on the affine$\imath$quantum group$\widetilde {{\mathbf U}}^\imath$. Real and imaginary root vectors for$\widetilde {{\mathbf U}}^\imath$are constructed, and a Drinfeld type presentation of$\widetilde {{\mathbf U}}^\imath$is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine$\imath$quantum groups in the sequels.

Predicting protein localization and understanding its mechanisms are critical in biology and pathology. In this context, we propose a new web application of MULocDeep with improved performance, result interpretation, and visualization. By transferring the original model into species-specific models, MULocDeep achieved competitive prediction performance at the subcellular level against other state-of-the-art methods. It uniquely provides a comprehensive localization prediction at the suborganellar level. Besides prediction, our web service quantifies the contribution of single amino acids to localization for individual proteins; for a group of proteins, common motifs or potential targeting-related regions can be derived. Furthermore, the visualizations of targeting mechanism analyses can be downloaded for publication-ready figures. The MULocDeep web service is available at https://www.mu-loc.org/.