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

• Volume: 138, Issue: 1, page 39-117
• ISSN: 0037-9484

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There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic 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 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 agroup). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a group – the Jacobian group$\Sigma$ – a sophisticated (and large) part of the group of automorphisms of the Grassmann algebra ${\Lambda }_{n}$. 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 $n\ge 4$,$dim\left(\Sigma \right)=\left\{\begin{array}{cc}\left(n-1\right){2}^{n-1}-{n}^{2}+2\hfill & \text{if}\phantom{\rule{4.0pt}{0ex}}n\phantom{\rule{4.0pt}{0ex}}\text{is}\phantom{\rule{4.0pt}{0ex}}\text{even},\hfill \\ \left(n-1\right){2}^{n-1}-{n}^{2}+1\hfill & \text{if}\phantom{\rule{4.0pt}{0ex}}n\phantom{\rule{4.0pt}{0ex}}\text{is}\phantom{\rule{4.0pt}{0ex}}\text{odd}.\hfill \end{array}\right$/extract_itex]The same is done for the Jacobian ascents - some natural algebraic overgroups of $\Sigma$. It is proved that the Jacobian map $\sigma ↦det\left(\frac{\partial \sigma \left({x}_{i}\right)}{\partial {x}_{j}}\right)$ is surjective for odd $n$, and isnotfor even $n$ though, in this case, the image of the Jacobian map is an algebraic subvariety of codimension 1 given by a single equation. How to cite top Bavula, Vladimir V.. "The jacobian map, the jacobian group and the group of automorphisms of the Grassmann algebra." Bulletin de la Société Mathématique de France 138.1 (2010): 39-117. <http://eudml.org/doc/272487>. @article{Bavula2010, abstract = {There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic 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  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 agroup). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a group – the Jacobian group\Sigma  – a sophisticated (and large) part of the group of automorphisms of the Grassmann algebra \Lambda _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\ge 4,\[\dim (\Sigma )=\{\left\lbrace \begin\{array\}\{ll\} (n-1)2^\{n-1\} -n^2+2& \text\{if n is even\},\\ (n-1)2^\{n-1\} -n^2+1& \text\{if n is odd\}.\\ \end\{array\}\right.\}$The same is done for the Jacobian ascents - some natural algebraic overgroups of $\Sigma$. It is proved that the Jacobian map $\sigma \mapsto \det (\frac\{\partial \sigma (x_i)\}\{\partial x_j\})$ is surjective for odd $n$, and isnotfor even $n$ though, in this case, the image of the Jacobian map is an algebraic subvariety of codimension 1 given by a single equation.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Grassmann algebra; jacobian group; algebraic group},
language = {eng},
number = {1},
pages = {39-117},
publisher = {Société mathématique de France},
title = {The jacobian map, the jacobian group and the group of automorphisms of the Grassmann algebra},
url = {http://eudml.org/doc/272487},
volume = {138},
year = {2010},
}

TY - JOUR
TI - The jacobian map, the jacobian group and the group of automorphisms of the Grassmann algebra
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 1
SP - 39
EP - 117
AB - There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic 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 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 agroup). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a group – the Jacobian group$\Sigma$ – a sophisticated (and large) part of the group of automorphisms of the Grassmann algebra $\Lambda _n$. 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 $n\ge 4$,$\dim (\Sigma )={\left\lbrace \begin{array}{ll} (n-1)2^{n-1} -n^2+2& \text{if n is even},\\ (n-1)2^{n-1} -n^2+1& \text{if n is odd}.\\ \end{array}\right.}$The same is done for the Jacobian ascents - some natural algebraic overgroups of $\Sigma$. It is proved that the Jacobian map $\sigma \mapsto \det (\frac{\partial \sigma (x_i)}{\partial x_j})$ is surjective for odd $n$, and isnotfor even $n$ though, in this case, the image of the Jacobian map is an algebraic subvariety of codimension 1 given by a single equation.
LA - eng
KW - Grassmann algebra; jacobian group; algebraic group
UR - http://eudml.org/doc/272487
ER -

References

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1. [1] V. V. Bavula – « Derivations and skew derivations of the Grassmann algebras », preprint arXiv:0704.3850, to appear in J. Algebra Appl. Zbl1200.16052MR2597282
2. [2] F. A. Berezin – « Automorphisms of a Grassmann algebra », Mat. Zametki1 (1967), p. 269–276. Zbl0204.03602MR208547
3. [3] N. Bourbaki – Algèbre, chapitres 1–3, Springer, 1998. Zbl0904.00001
4. [4] D. Ž. Djoković – « Derivations and automorphisms of exterior algebras », Canad. J. Math.30 (1978), p. 1336–1344. Zbl0405.15023MR511568

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