Root graded Lie algebras and homology of their coordinates

Yun Gao
(York University)

Central extensions of root graded Lie algebras can be characterized as certain homology of their coordinates. In this talk, I will focus on the simplest case: the elementary matrix Lie algebra $sl_n(R)$, where $R$ is an associative algebra. It turns out that the universal central extension of $sl_n(R)$ is the first Connes cyclic homology group of $R$ if $n>2$. Some variations of cyclic homology such as K\"ahler differentials, Hochschild homology and (skew) dihedral homology will also be introduced.