Український математичний конгрес - 2009

Volodymyr Bavula (The University of Sheffield, Sheffield, United Kingdom)

The Jacobian map, the Jacobian group and the group of automorphisms of the Grassmann algebra

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (eg, the Grassmann algebras are dual of polynomial algebras as quadratic algebras). We look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the Jacobian set -- the set of all algebra endomorphisms of a polynomial algebra with Jacobian 1 -- 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.