2008-11-07

Vladimir Bavula, University of Sheffield

Jacobian Algebras

Jacobian algebras are obtained from the Weyl algebras by inverting certain elements (not in the Ore sense). Surprisingly, these algebras have little in common with the Weyl algebras: in fact, they have almost opposite properties. The Jacobian algebras appeared in my study of polynomial automorphisms and the Jacobian Conjecture, which is a conjecture that makes sense only for polynomial algebras in the class of all commutative algebras (it is a theorem). In order to solve the Jacobian Conjecture, it is reasonable to believe that one should develop techniques which make sense only for polynomials. The Jacobian algebras are a step in this direction (they exist only for polynomials and make no sense even for Laurent polynomials).