相关论文: Double-bosonization and Majid's Conjecture, (III):…
Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rank-inductive and type-crossing construction for $U_q(\mathfrak g)$'s is still a remaining open question. In this paper, working with Majid's framework,…
Majid developed in \cite{majid3} his double-bosonization theory to construct $U_q(\mathfrak g)$ and expected to generate inductively not just a line but a tree of quantum groups starting from a node. In this paper, the authors confirm the…
The purpose of the paper is to build up the related theory of weakly quasitriangular dual pairs suitably for non-standard $R$-matrices (irregular), and establish the generalized double-bosonization construction theorem for irregular $R$,…
This paper aims to develop a grafting method to address Majid's conjecture, which enables the construction of a larger target quantum group by grafting two given smaller ones. This method is significant for advancing the understanding of…
Double-bosonisation associates to a braided group in the category of modules of a quantum group, a new quantum group. We announce the semiclassical version of this inductive construction.
Lie bialgebras occur as the principal objects in the infinitesimalisation of the theory of quantum groups - the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to…
An explicit construction of the braided dual of quantum $E(2)$ groups is described over the circle group $\mathbb{T}$ with respect to a specific $R$-matrix $R$. Additionally, the corresponding bosonization is also described.
We obtain new family of quasitriangular Hopf algebras $C^{0|n}_q\lcross \widetilde{U_q(su_n)}\rcross C^{0|n}_q$ via the author's recent double-bosonisation construction for new quantum groups. They are versions of $U_q(su_{n+1})$ with a…
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…
Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of…
We present a general theory of braided quantum groups in the C*-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of…
We apply Majid's transmutation procedure to Hopf algebra maps $H \to \mathbb C[T]$, where $T$ is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of $T$ by subgroups that are…
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 collect here some less well-known results and formulae about the bosonisation construction which turns braided groups into quantum groups. We clarify the relation with biproduct Hopf algebras (the constructions are not the same), the…
Beginning with a skew-symmetric matrix, we define a certain Poisson--Lie group. Its Poisson bracket can be viewed as a cocycle perturbation of the linear (or "Lie-Poisson") Poisson bracket. By analyzing this Poisson structure, we gather…
We develop a general theory of `quantum' diffeomorphism groups based on the universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its various quotients. Explicit formulae are introduced for this construction, as well…
We start with the recently conjectured 3d bosonization dualities and gauge global symmetries to generate an infinite sequence of new dualities. These equate theories with non-Abelian product gauge groups and bifundamental matter. We uncover…
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We…
It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal…
Let M be a manifold with an action of a Lie group G, $\A$ the function algebra on M. The first problem we consider is to construct a $U_h(\g)$ invariant quantization, $\A_h$, of $\A$, where $U_h(\g)$ is a quantum group corresponding to G.…