arXiv subjects
Publications and source records attributed to Atibur Rahaman.
In this paper we study a Kahler structure on finite points. In particular, we study the edge Laplacian of a graph twisted by the Kahler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic Dolbeault-Dirac spectral triple and show that the points have a finite diameter with respect to Connes' distance.
We show that there is an almost complex structure on a differential calculus on finite points coming from a bidirected finite graph without multiple edges or loops. We concentrate on a polygon as a concrete case. In particular, a `holomorphic structure on the exterior bundle' built from the polygon is studied. Also a positive Hochschild 2-cocycle on the vertex set of the polygon, albeit a trivial one, is shown to arise naturally from the almost complex structure.
We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a `non-trivial' example of a pair of $2$-graphs that are not quantum isomorphic to each other.
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 construct a braided analogue of the quantum permutation group and show that it is the universal braided compact quantum group acting on a finite space in the category of $\mathbb{Z}/N\mathbb{Z}$-$\textrm{C}^*$-algebras with a twisted monoidal structure. As an application, we prove the existence of braided quantum symmetries of finite, simple, undirected, circulant graphs, explicitly compute it for several examples, and obtain a generalization of a result of Banica in this direction. Finally, in an appendix, we briefly describe the irreducible representations of this braided analogue of the quantum permutation group and their fusion rules.
We construct a family of $q$ deformations of $E(2)$ group for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle group $\mathbb{T}$ viewed as a quasitriangular quantum group with respect to the unitary R-matrix $R(m,n):=(\zeta)^{mn}$ for all $m,n\in\mathbb{Z}$. For real $0<|q|<1$, the deformation coincides with Woronowicz's $E_{q}(2)$ groups. As an application, we study the braided analogue of the contraction procedure between $SU_{q}(2)$ and $E_{q}(2)$ groups in the spirit of Woronowicz's quantum analogue of the classic In\"on\"u-Wigner group contraction. Consequently, we obtain the bosonisation of braided $E_{q}(2)$ groups by contracting $U_{q}(2)$ groups.