2007/03/12 by V. V. Bavula, Bavula, V. V.
Mathematics · Physics and Astronomy · #14L17 #14M20 #14R10 #14R15 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #Rings and Algebras (math.RA) #math.AG #math.RA #msc:14L17 #msc:14M20 #msc:14R10 #msc:14R15
paper · pdf · doi:10.48550/arxiv.math/0703352
73 pages
arxiv created 2007/03/12 · openalex publication_date 2007/03/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Σ -- a sophisticated (and large) part of the group of automorphisms of the Grassmann algebra Łn. It is proved that the Jacobian group Σ is a rational unipotent algebraic group. A (minimal) set of generators for the algebraic group Σ, its dimension and coordinates are found explicitly. In particular, for n≥ 4, dim (§) = (n-1)2n-1 -n2+2 if n is even, (n-1)2n-1 -n2+1 if n is odd. The same is done for the Jacobian ascents - some natural algebraic overgroups of Σ. It is proved that the Jacobian map \s ↦ det ((\der \s (xi))/(\der xj)) 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.