Group algebras and algebras of Golod-Shafarevich

Abstract:

In [2], Golod, using results of Golod and Shafarevich [1], has constructed a finitely generated algebra $A = K\left \langle {{y_1}, \ldots ,{y_d}} \right \rangle$, over any field $K$, such that the ideal generated by ${y_1}, \ldots ,{y_d}$ is nil, but ${\dim _K}A = \infty$. Moreover, when char $K = p > 0$, the subgroup $G$ of the group of units of $A$, generated by $1 + {y_1}, \ldots ,1 + {y_d}$, is an infinite $p$-group. The main purpose of the present paper is to show that $K[G]$, the group algebra of $G$ over $K$, is not isomorphic to $A$ for "most" Golod-Shafarevich groups $G$.