A quantum symmetric pair consists of a quantum group U and its coideal subalgebra U ı ς with parameters ς (called an ıquantum group). We initiate a Hall algebra approach for the categorification of ıquantum groups. A universal ıquantum group U ı is introduced and U ı ς is recovered by a central reduction of U ı . The modified Ringel-Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algebras of Gorsky and motivated by Bridgeland's work, are extended to the setting of 1-Gorenstein algebras, as shown in Appendix A by the first author. A new class of 1-Gorenstein algebras (called ıquiver algebras) arising from acyclic quivers with involutions is introduced. The modified Ringel-Hall algebras for the Dynkin ıquiver algebras are shown to be isomorphic to the universal quasi-split ıquantum groups of finite type, and a reduced version provides a categorification of U ı ς . Monomial bases and PBW bases for these Hall algebras and ıquantum groups are constructed. In the special case of quivers of diagonal type, our construction reduces to a reformulation of Bridgeland's Hall algebra realization of quantum groups. MING LU AND WEIQIANG WANG 3.3. Λ ı -modules with finite projective dimensions 3.4. Projective Λ ı -modules 3.5. Singularity category of Gorenstein algebras 3.6. Singularity category for ıquivers 4. Hall algebras for ıquivers 4.1. Euler forms 4.2. Modified Ringel-Hall algebras for Λ ı 4.3. A Hall basis 4.4. Hall algebras for ıquivers 4.5. Hall algebras for ısubquivers 5. Monomial bases and PBW bases of Hall algebras for ıquivers 5.1. Monomial basis of H(kQ) 5.2. Monomial bases for Hall algebras of ıquivers 5.3. PBW bases for Hall algebras of ıquivers Part 2. ıHall Algebras and ıQuantum Groups 6. Quantum symmetric pairs and ıquantum groups 6.1. Quantum groups 6.2. The ıquantum groups U ı and U ı 6.3. ıQuantum groups of type ADE 6.4. Presentation of U ı 7. Hall algebras for ıquivers and ıquantum groups 7.1. Computations for rank 2 ıquivers, I 7.2. Computations for rank 2 ıquivers, II 7.3. The homomorphism ψ 7.4. ıQuantum groups via ıHall algebras 8. Bridgeland's theorem revisited 8.1. A category equivalence 8.2. Quantum group as an ıquantum group 8.3. Bridgeland's theorem reformulated 9. Generic Hall algebras for Dynkin ıquivers 9.1. Hall polynomials 9.2. Generic Hall algebras, I 9.3. Generic Hall algebras, II Appendix A. Modified Ringel-Hall algebras of 1-Gorenstein algebras by Ming Lu A.1. Hall algebras A.2. Definition of modified Ringel-Hall algebras A.3. 1-Gorenstein algebras A.4. Semi-derived Hall algebras A.5. Tilting invariance References

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Title: Organic electron-rich N-heterocyclic compound as a chemical bridge: building a Brönsted acidic ionic liquid confi ned in MIL-101 nanocagesThis work introduces a facile post-synthetic modifi cation strategy to synthesize a novel functionalized MIL-101 material in which a Brönsted acidic quaternary ammonium salt ionic liquid is confi ned inside well-defi ned nanocages. It shows excellent catalytic performance for acetalization.

Modulating the emission wavelengths of materials has always been a primary focus of fluorescence technology. Nanocrystals (NCs) doped with lanthanide ions with rich energy levels can produce a variety of emissions at different excitation wavelengths. However, the control of multimodal emissions of these ions has remained a challenge. Herein, we present a new composition of Er3+‐based lanthanide NCs with color‐switchable output under irradiation with 980, 808, or 1535 nm light for information security. The variation of excitation wavelengths changes the intensity ratio of visible (Vis)/near‐infrared (NIR‐II) emissions. Taking advantage of the Vis/NIR‐II multimodal emissions of NCs and deep learning, we successfully demonstrated the storage and decoding of visible light information in pork tissue.

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We show that the morphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall algebra of the weighted projective line is injective if g is of finite or affine type. As a byproduct, we use the whole ıHall algebra of the cyclic quiver C n to realise the ıquantum loop algebra of affine gl n . Contents1. Introduction 1 2. Preliminaries 2 3. ıHall algebras and ıquantum loop algebras 11 4. ıHall algebras of cyclic quivers and U ı v ( gl n ) 13 5. Injectivity for finite and affine type cases 21 References 28

A quantum symmetric pair consists of a quantum group 𝐔 and its coideal subalgebra 𝐔 𝚤 𝝇 with parameters 𝝇 (called an 𝚤quantum group). We initiate a Hall algebra approach for the categorification of 𝚤quantum groups. A universal 𝚤quantum group Ũ𝚤 is introduced and 𝐔 𝚤 𝝇 is recovered by a central reduction of Ũ𝚤 . The semiderived Ringel-Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algebras of Gorsky and motivated by Bridgeland's work, are extended to the setting of 1-Gorenstein algebras, as shown in the Appendix by the first author. A new class of 1-Gorenstein algebras (called 𝚤quiver algebras) arising from acyclic quivers with involutions is introduced. The semi-derived Ringel-Hall algebras for the Dynkin 𝚤quiver algebras are shown to be isomorphic to the universal quasi-split 𝚤quantum groups of finite type. Monomial bases and PBW bases for these Hall algebras and 𝚤quantum groups are constructed.