Hall 123

Finiteness of Group Algebras

Issan Patri

IMSc

Kaplansky conjectured that for any field k and any group G, the group algebra k[G] is directly finite, i.e. for xy=1 iff yx=1 for any x,y in k[G]. We prove this in the complex case for any group. A quick overview of recent results in the positive characteristic case and of other famous conjectures of Kaplansky (idempotent conjecture, zero-divisor, unit conjectures) will also be given.