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Nous montrons que le morphisme de Frobenius quantique construit par Lusztig dans le cadre des algèbres enveloppantes quantiques $U_{\mathcal{B}}$ spécialisées en une racine de l’unité admet un scindage multiplicatif (non unitaire). Nous utilisons pour ce faire une base de la partie torique de la petite algèbre quantique constituée d’idempotents orthogonaux deux à deux et de somme 1 et faisons de même dans le cas «modulaire» pour l’algèbre des distributions d’un groupe algébrique semi-simple.

In these notes we collect some results about finite-dimensional representations of $U_{q}(\mathfrak {gl}(1\mid1))$ and related invariants of framed tangles, which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite-dimensional $U_{q}(\mathfrak {gl}(1\mid1))$ -representations and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a non-zero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_{q}(\mathfrak {gl}(1\mid1))$ .

For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.