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No abstract uploaded!

No abstract uploaded!

The main facts about Hausdorff and packing measures and dimensions of a Borel set$E$are revisited, using$determining$set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$ , where $\mathcal{B}_E$ is the family of all balls centred on$E$and α is a real parameter. With mild assumptions on φ_{α}, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set$V$in ℝ^{$D$}, these notions are used to introduce the$interior$and$exterior$measures and dimensions of any Borel subset of ∂$V$. We stress that these dimensions depend on the choice of φ_{α}. Two determining functions are considered, φ_{α}($B$)=Vol_{$D$}($B$∩$V$)diam($B$)^{α-$D$}and φ_{α}($B$)=Vol_{$D$}($B$∩$V$)^{α/$D$}, where Vol_{$D$}denotes the$D$-dimensional volume.

In our previous work, we studied positive representations of split real quantum groups U q q~ (g R ) restricted to their Borel part and showed that they are closed under taking tensor products. But the tensor product decomposition was only constructed abstractly using the GNS representation of a C*-algebraic version of the Drinfeld–Jimbo quantum groups. Here, using the recently discovered cluster realization of quantum groups, we write the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.