Alladi Ramakrishnan Hall

Hilbert series of partially commutative Lie algebras

R. Venkatesh

IIT Madras

Let G be a finite simple graph with the vertex set V and the Edge
set E . Partially commutative Lie algebra 𝓅 ​(G)​ associated to G is the Lie
algebra freely generated by the variable Xi​, i∈ V​ subject to the relations
[Xi, Xj]=0 for (i, j)∉ E​. We will realize 𝓅 ​(G) as a subalgebra (positive
part) of Borcherds-Kac-Moody algebra and use the celebrated denominator
identity of Borcherds-Kac-Moody algebras to compute Hilbert series of 𝓅 ​(G)
in terms of independent set polynomial of G​.