Block Extensions, Local Categories and Basic Morita Equivalences

Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

Longitudinal mapping knot invariant for SU(2)

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots [Formula: see text], their mirror images, and the figure eight knot for the group SU(2).