Manchester Geometry Seminar 2008/2009

Thursday 9 October 2008. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4pm

The Jacobian Map, the Jacobian Group, and the Group of Automorphisms of the Grassmann Algebra

Vladimir Bavula (University of Sheffield)

There are nontrivial dualities and parallels between the polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual to the polynomial algebras as quadratic algebras). We look at the Grassmann algebras from the angle of the Jacobian conjecture for the polynomial algebras (which is the question/conjecture about the Jacobian set -- the set of all algebra endomorphisms of a polynomial algebra with the unit Jacobian -- the Jacobian conjecture claims that the Jacobian set is a group). We study in detail the Jacobian set for the Grassmann algebra, which turns out to be a group -- the Jacobian group -- a sophisticated (and large) part of the group of automorphisms of the Grassmann algebra. For the Jacobian group a (minimal) set of generators, its dimension (as an algebraic group) and coordinates are found explicitly.