Modified Braid Equations for SO_q (3) and noncommutative spaces

Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups $GL_{q}(N), SO_{q}(N)$ and $Sp_{q}(N)$

Modified braid equations satisfied by generalized ${\hat R}$ matrices (for a {\em given} set of group relations obeyed by the elements of ${\sf T}$ matrices ) are constructed for q-deformed quantum groups $GL_q (N), SO_q (N)$ and $Sp_q (N)$ with arbitrary values of $N$. The Baxterization of ${\hat R}$ matrices, treated as an aspect complementary to the {\em modification} of the braid equation, is obtained for all these cases in particularly elegant forms. A new class of braid matrices is discovered for the quantum groups $SO_{q}(N)$ and $Sp_{q}(N)$. The ${\hat R}$ matrices of this class, while…

RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum matrix $T$ as input a general solution of the $RTT$ relations is sought without imposing the Yang - Baxter constraint for $R$ or the braid equation for $\hat{R} = PR$. For three biparametric deformatios, $GL_{(p,q)}(2), GL_{(g,h)}(2)$ and $GL_{(q,h)}(1/1)$, the standard,the nonstandard and the hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K$,such that only for two values of $K$, given explicitly for each case, o…

Aspects of a new class of braid matrices: roots of unity and hyperelliptic $q$ for triangularity, L-algebra,link-invariants, noncommutative spaces

Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$ corresponds to polynomial equations for $q$, the solutions ranging from roots of unity to hyperelliptic functions. The algebras of $L-$ operators are studied. As a crucial feature one obtains $2N$ central, group-like, homogenous quadratic functions of $L_{ij}$ constrained to equality among themselves by the $RLL$ equations. They are studied in detail for $N =3$ and …

修正量子群上的相对辫群对称性及其模

Relative braid group symmetries on modified iquantum groups and their modules

本文研究了修正量子群及其模的相对辫群对称性，提出了三种显式公式并验证了其相容性。

Mapping spaces and automorphism groups of toric noncommutative spaces

We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application we study the `internalized' automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations.

Moduli spaces of noncommutative instantons: gauging away noncommutative parameters

Using the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_θ. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an approp…