Related papers: Hall algebras and quantum symmetric pairs I: found…

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and…

Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the…

Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to…

According to the Hall algebras of quivers with automorphisms under Lusztig's construction, the polynominal forms of several structure coefficients for quantum groups of all finite types are presented in this note. We first provide a…

In 1959, P. Hall introduced the locally finite group $\mathcal{U}$, today known as Hall's universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in $\mathcal{U}$. It can be…

Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two…

Let $\mathfrak{g}\neq \mathfrak{so}_8$ be a simple Lie algebra of type $A,D,E$ with $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak{n}_-\subset \widehat{\mathfrak{g}}$ a nilpotent subalgebra. Given…

We construct multi-brace cotensor Hopf algebras with bosonizations of quantum multi-brace algebras as examples. Quantum quasi-symmetric algebras are then obtained by taking particular initial data; this allows us to realize the whole…

The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown…

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category…

Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets…

This paper concerns the cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the…

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $U_q(\mathfrak{g})$ denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras $B_{c,s}$ of…

In 1998, A.Alekseev and E.Meinrenken construct an explicit $G$-differential space homomorphism $\mathcal{Q}$, called the quantization map, between the Weil algebra $\Weil{\g}= \sym{\co{\g}} \otimes \ext{\co{\g}}$ and $\NWeil{\g}=\U{\g}…

The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum…

A universal quasitriangular $R$--matrix for the non-standard quantum (1+1) Poincar\'e algebra $U_ziso(1,1)$ is deduced by imposing analyticity in the deformation parameter $z$. A family $g_\mu$ of ``quantum graded contractions" of the…

We study superpotential algebras by introducing the notion of quantum-symmetric equivalence defined relatively to two fixed Hopf coactions. This concept relies on the non-vanishing of a bi-Galois object for the two coacting Hopf algebras,…

The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…

We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants…

We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of $\mathbb Z_2$-graded complexes of representations of a quiver with (possible) loops. To overcome…