q-commuting polynomials

[tscrim]

These show up in quantum mechanics computations and are a variation of skew commuting polynomials (cf. the exterior algebra) satisfying yx = q^{B_xy} xy for all variables that index fixed a skew symmetric bilinear matrix B.

We provide a simple initial implementation.

CC: @fchapoton

Component: algebra

Keywords: q-commuting polynomials

Author: Travis Scrimshaw

Branch/Commit: 49129ea

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/34412

[tscrim]

Commit: d7f05ce

[tscrim]

Branch: public/algebras/q_commute_polys-34412

[tscrim]

comment:1

Here is a very simple implementation for them that feeds off as much as possible. Likely could be improved (e.g., not relying on FreeMonoid as a backend for the basis), but it works for now.

---

New commits:

d7f05ce

Initial implementation of q-commuting polynomials.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

49129ea

Allowing an arbitrary skew-symmetric matrix as input to determine the multiplication.

[sagetrac-git]

Changed commit from d7f05ce to 49129ea

[tscrim]

Description changed:

--- 
+++ 
@@ -1,3 +1,3 @@
-These show up in quantum mechanics computations and are a variation of skew commuting polynomials (cf. the exterior algebra) satisfying `xy = qyx` for all variables `x != y`.
+These show up in quantum mechanics computations and are a variation of skew commuting polynomials (cf. the exterior algebra) satisfying `yx = q^{B_xy} xy` for all variables that index fixed a skew symmetric bilinear matrix `B`.
 
 We provide a simple initial implementation.

[tscrim]

comment:4

Bot is green.

[fchapoton]

comment:5

ok, ok. I was just wondering for myself whether this fits into the setting of Ore algebras, and hence could be done using the "ore_algebra" package.

[tscrim]

comment:6

Thanks for looking at this. I didn't mean to pressure you into it.

I thought about that too. However, that viewpoint makes one of the variables more special than the others, which is not desirable to me. For example, if we take two variables R[x,y]_q, then to consider this as an Ore algebra, we take it to be the Ore extension R[y][x; phi], where phi(y) = qy. For more variables, we would then iterate this construction, leading us to something like R[x_n][x_{n-1}; phi_n] ... [x_2; phi_3][x_1; phi_2], which would be isomorphic but different for any ordering of the variables. Contrast this with R['x,y'] and R['x']['y'] and R['y']['x'] as polynomial rings within Sage. I suspect this will also be slower too.

[fchapoton]

comment:7

is there any other structure on that algebra ? a coproduct or co-module structure ?

With just x and y, this is sometimes called the quantum plane, right ?

minor remark:

return tuple([self.monomial(g) for g in self._indices.gens()])

could create directly the tuple