Mathematics – Rings and Algebras
Scientific paper
2007-04-29
Mathematics
Rings and Algebras
23 pages
Scientific paper
Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let $\L_n = K\lfloor x_1, ..., x_n\rfloor$ be the Grassmann algebra over a commutative ring $K$ with ${1/2}\in K$, and $\d$ be a skew $K$-derivation of $\L_n$. It is proved that $\d$ is a unique sum $\d = \d^{ev} +\d^{od}$ of an even and odd skew derivation. Explicit formulae are given for $\d^{ev}$ and $\d^{od}$ via the elements $\d (x_1), ..., \d (x_n)$. It is proved that the set of all even skew derivations of $\L_n$ coincides with the set of all the inner skew derivations. Similar results are proved for derivations of $\L_n$. In particular, $\Der_K(\L_n)$ is a faithful but not simple $\Aut_K(\L_n)$-module (where $K$ is reduced and $n\geq 2$). All differential and skew differential ideals of $\L_n$ are found. It is proved that the set of generic normal elements of $\L_n$ that are not units forms a single $\Aut_K(\L_n)$-orbit (namely, $\Aut_K(\L_n)x_1$) if $n$ is even and two orbits (namely, $\Aut_K(\L_n)x_1$ and $\Aut_K(\L_n)(x_1+x_2... x_n)$) if $n$ is odd.
Bavula V. V.
No associations
LandOfFree
Derivations and skew derivations of the Grassmann algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Derivations and skew derivations of the Grassmann algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Derivations and skew derivations of the Grassmann algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-408945