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

Abstract

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $ $ {\em Jacobian set} -- the set of all algebra endomorphisms of a polynomial algebra with the Jacobian 1 -- the Jacobian conjecture claims that the Jacobian set is a {\em group}). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a {\em group} -- the {\em Jacobian group} $\Sigma$ -- a sophisticated (and large) part of the group of automorphisms of the Grassmann algebra $\L_n$. It is proved that the Jacobian group $\Sigma$ is a rational unipotent algebraic group. A (minimal) set of generators for the algebraic group $\Sigma$, its dimension and coordinates are found explicitly. In particular, for $n\geq 4$, \dim (\S) = (n-1)2^{n-1} -n^2+2 if $n$ is even, (n-1)2^{n-1} -n^2+1 if $n$ is odd. The same is done for the Jacobian ascents - some natural algebraic overgroups of $\Sigma$. It is proved that the Jacobian map $\s \mapsto \det (\frac{\der \s (x_i)}{\der x_j})$ is surjective for odd $n$, and is {\em not} for even $n$ though, in this case, the image of the Jacobian map is an algebraic subvariety of codimension 1 given by a single equation.