Timeline for Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 26, 2023 at 20:27 | comment | added | Leonid Positselski | What does $(-1)^{|x||y|}$ mean when $|x|$, $|y|\in G$ and $G$ is a finite abelian group? | |
| Sep 26, 2023 at 10:03 | comment | added | jack | For a $G$ graded vector space, where $G$ is a finite abelian group, what should be the ideal that we consider ? Of course one considers the elements of the form $xy-(-1)^{|x||y|}$. Here there is no $\mathbb Z_2$-grading and hence no odd elements, what kind of elements would replace $z^2$ ? I would like that it should commute with the base change. | |
| Nov 30, 2009 at 11:06 | comment | added | Leonid Positselski | Yes it does. No, it apparently doesn't: for in characteristic 2, the permutation operators do not depend on the parity, so they cannot distinguish between the symmetric and exterior generators. This may be a common situation for superstructures in small characteristic: e.g., in the definition of a Lie superalgebra one has to pay attention not only to char=2, but also to char=3. The correct definition requires, in addition to the Lie bracket, the data of a "squaring" map q: g_odd -> g_even. When 2 is invertible in your ground ring, this problem does not present itself. | |
| Nov 30, 2009 at 8:34 | comment | added | Marc Nieper-Wißkirchen | For graded vector spaces V = V_even + V_odd this yields the classical symmetric algebra over V_even tensored with the classical exterior algebra over V_odd, doesn't it? However, does this definition make sense in the context of general symmetric monoidal categories? | |
| Nov 29, 2009 at 22:41 | history | answered | Leonid Positselski | CC BY-SA 2.5 |