数理情報科学セミナー 2015

Finding the infinite dimensional representations of the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ has been very exciting research topic. We consider the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ as a topological algebra over the ring of formal power series in $p$. This talk deals on the one hand with the existence of the dynamical quantum $\mathcal{Z}_k$-algebra structure in the level-$k$ $U_{q,p}(\widehat{\mathfrak{g}})$-module for general untwisted affine Lie algebra $\mathfrak{g}$. We discuss the level-$k$ irreducible highest weight representation of $U_{q,p}(\widehat{\mathfrak{g}})$ in term of the dynamical quantum $\mathcal{Z}_k$-module and the module of level-$k$ elliptic bosons. We show that the irreducible $\mathcal{Z}_k$-module guarantees the irreducibility of level-$k$ $U_{q,p}(\widehat{\mathfrak{g}})$-module. We present the level-1 irreducible highest weight representations of $U_{q,p}(\widehat{\mathfrak{g}})$, which we call the standard representations, for some types of affine Lie algebras $\widehat{\mathfrak{g}}$. On the other hand we discuss the construction of the higher level representation of $U_{q,p}(\widehat{\mathfrak{sl}}_2)$ by taking an elliptic analogue of the Drinfeld coproduct of the level-1 standard representation of $U_{q,p}(\widehat{\mathfrak{sl}}_2)$. We also study an elliptic analogue of the integrable condition of such representation.

Wavelets analysis and compression tools to study fluid and plasma turbulence are reviewed and different applications are presented. We introduce the continuous and the orthogonal wavelet transform and detail several statistical diagnostics based on the wavelet coefficients. We then show how to extract coherent structures out of fully developed turbulent flows using wavelet based denoising. Finally some multiscale numerical simulation schemes using wavelets are described. Several examples for analyzing, compressing and computing turbulent flows are presented.