The Algebra & Combinatorics Seminar meets on Fridays from 2:30 to
3:30 pm, in Lecture Hall LH1 of the IISc Mathematics Department.
The organizers are Apoorva Khare and R. Venkatesh.


Abstract.
I will give a gentle introduction to the Diamond Lemma. This is a useful
technique to prove that certain "PBWtype" bases exist of algebras given
by generators and relations. In particular, we will see the PBW theorem
for usual Lie algebras.


Abstract.
Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$
is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$
modulo the relations $[u, v]=0$ if there is no edge between the vertices
$u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the
properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$
is naturally graded and the graded dimensions of the Lie algebra
$\mathfrak{g}$ of $G$ have some deep connections with the vertex
colorings of $G$. In this talk, I will explain how to get the generalized
chromatic polynomials of $G$ in terms of graded dimensions of the Lie
algebra of $G$. We will use this connection to give a Lie theoretic proof
of of Stanley's reciprocity theorem of chromatic polynomials.





