Related papers: The quantum group $\dot U$ and flag manifolds over…

A purely combinatorial construction of the quantum cohomology ring of the flag manifold $G/B$ is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of two of…

All unitary representations of the quantum ``az+b'' group are found. It turns out that this quantum group is self dual i.e. all unitary representations are 'numbered' by elements of the same group. Moreover, the formula for all unitary…

Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be…

A flag manifold over a semifield K can be partitioned into "half i-circles" which are orbits of a K-action on that flag manifold. Here i is fixed and it corresponds to a simple reflection in the Weyl group. We prove (for certain K) a…

We present a combinatorial proof of the $q$-Pfaff--Saalsch\"utz identity by a composition of explicit bijections, in which $q$-binomial coefficients are interpreted as counting subspaces of $\mathbb{F}_q$-vector spaces. As a corollary, we…

We survey some recent developments on the theory of dual canonical bases for quantum groups and $\imath$quantum groups. The $\imath$quiver algebras were introduced by Wang and the first author, which are used to give two realizations of…

Let $ G $ be a connected reductive algebraic group and its symmetric subgroup $ K $. The variety $ \dblFV = K/Q \times G/P $ is called a double flag variety, where $ Q $ and $ P $ are parabolic subgroups of $ K $ and $ G $ respectively. In…

We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra $H$ associated to the factorization of a finite group into two subgroups. The representations of the quantum double…

We find a quantum group structure in two-dimensional motions of a nonrelativistic electron in a uniform magnetic field and in a periodic potential. The representation basis of the quantum algebra is composed of wavefunctions of the system.…

This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and…

We give a presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, as a quotient of a polynomial ring by an explicit ideal. This is the torus-equivariant version of our previous result, which gives a…

Consider an almost-simple algebraic group G and a choice of complex root of unity q. We study the category of quasi-coherent sheaves $\mathscr{X}_q$ on the half-quantum flag variety, which itself forms a sheaf of tensor categories over the…

Let U^+ be the plus part of the quantized enveloping algebra of a simple Lie algebra and let B^* be the dual canonical basis of U^+. Let b,b' be in B* and suppose that one of the two elements is a q-commuting product of quantum flag minors.…

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the…

We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$, generalising previous work of a number of authors (including the…

We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the {\em double-bosonisation}, associated to every braided group $B$ in the category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$ appears as the…

In this paper, we study Markov dynamics on unitary duals of compact quantum groups. We construct such dynamics from characters of quantum groups. Then we show that the dynamics have generators, and we give an explicit formula of the…

We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent super Cartan datum by constructing an automorphism (called twistor) in the setting of covering quantum groups. The…

In this article, we study the multi-parameter quantum groups defined by generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category $\mathcal O$. This presentation…

We give a survey of some recent results on the fusion semirings of compact quantum groups (computations of and applications to discrete quantum groups) by using the following simplifying terminology: we say that a compact quantum group G is…