Timeline for Weyl algebra acting on a polynomial ring
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 2, 2016 at 16:21 | comment | added | Yang Zhang | Thanks a lot, Allen! I agree the module of interest is $W/\langle (\partial_i), \mathcal D \rangle$. Then the difficulty is that $\langle (\partial_i), \mathcal D \rangle$ is a "mixture" of left and right ideals in $W$. Since for my case, to get $\langle (\partial_i), \mathcal D \rangle$, I need to multiply $(\partial_i)$ by elements in $W$ from left, while multiply $\mathcal D$ by elements in $W$ from right. | |
| Jun 1, 2016 at 22:37 | comment | added | Allen Knutson | You start with "$W$ acts on $R$", i.e. $R = W/\langle (\partial_i)\rangle$. Then your module of interest is $R/\mathcal D\bullet R = W/\langle (\partial_i), \mathcal D\rangle$. | |
| Jun 1, 2016 at 22:29 | comment | added | Allen Knutson | In usual (commutative polynomial) Gröbner basis theory, one has a gradation and therefore filtration on the polynomial ring $R$, and an ideal $I$, so therefore $gr\ R$ acts on $init\ I := gr\ I$. A generating set for the latter ideal lifts to a set in $I$ that automatically generates, and gets called a Gröbner basis. In standard treatments $R$ and $gr\ R$ are identified, which is confusing when one wants to deal with situations like yours where the filtration doesn't come from a gradation (and $gr\ W \not\cong W$). | |
| Jun 1, 2016 at 14:31 | comment | added | Yang Zhang | Keith, yes, I want $\mathcal D$ to be a finite set since for my problem, the context is that $D_i$'s are coming from some tangent vector fields of an affine variety. | |
| Jun 1, 2016 at 14:21 | comment | added | Yang Zhang | Thanks Ketil! I tried Macaulay2: Consider the {\it right}-ideal I generated by $\mathcal D$. (Since Macaulay2 deals with left ideals, I used the antihomomorphism of $W$ to switch between the left/right). Let the $\{g_i\}$ be the Gr\"obner basis of $I$, by the division in $W$, $F=\sum_i g_i \cdot q_i + O$. Act this relation on $R$, so $F=(O\bullet 1) +( {\text polynomials\ in\ } \mathcal D\bullet R)$. $(O\bullet 1)$ seems like ``normal form'' of $F$. However, I am not sure how to choose a good weight vector of $W$, so this division works for some cases but failed for others. | |
| Jun 1, 2016 at 8:22 | comment | added | Ketil Tveiten | Gröbner bases work fine for the Weyl algebra, e.g. Macaulay2 has a good implementation. Also, do you really want $\mathcal{D}$ to be a finite set, or the left ideal in $W$ generated by that finite set? | |
| May 31, 2016 at 23:39 | comment | added | Aaron | There is a filtration on the Weyl algebra such that the associated graded ring is just a polynomial algebra in $2n$ variables, and so I suspect that the non-commutative analogue of Groebner bases works out very nicely. Have you had any luck doing computations when $|\mathcal D|=1$? | |
| May 31, 2016 at 21:46 | history | edited | YCor |
edited tags; edited tags
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| May 31, 2016 at 21:41 | review | First posts | |||
| May 31, 2016 at 22:34 | |||||
| May 31, 2016 at 21:38 | history | asked | Yang Zhang | CC BY-SA 3.0 |