## 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.