Editing Weyl algebra (section)

=== Degree ===
{{Math theorem
| name = Theorem
| note = 
| math_statement = <math>
A_n
</math> has a basis <math>
\{q^m p^n : m, n \geq 0\}
</math>.{{Sfn|Coutinho|1995|p=9|loc=Proposition 2.1}}
}}

{{Math proof|title=Proof|proof=
 By repeating the commutator relations, any monomial can be equated to a linear sum of these. It remains to check that these are linearly independent. This can be checked in the differential operator representation. For any linear sum <math>
\sum_{m, n} c_{m,n} x^m \partial_x^n
</math> with nonzero coefficients, group it in descending order: <math>
p_N(x) \partial_x^N + p_{N-1}(x) \partial_x^{N-1} + \cdots + p_M(x) \partial_x^M
</math>, where <math>
p_M
</math> is a nonzero polynomial. This operator applied to <math>
x^M
</math> results in <math>
M! p_M(x) \neq 0
</math>.
}}

This allows <math>
A_1
</math> to be a [[graded algebra]], where the degree of <math>
\sum_{m, n} c_{m,n} q^m p^n
</math> is <math>
\max (m + n)
</math> among its nonzero monomials. The degree is similarly defined for <math>
A_n
</math>.

{{Math theorem
| name = Theorem
| math_statement = For <math>A_n</math>:{{sfn | Coutinho | 1995 | pp=14-15}}
 
* <math>
\deg(g + h) \leq \max(\deg(g), \deg(h)) 
</math>
* <math>
\deg([g, h]) \leq \deg(g) + \deg(h) - 2 
</math>
* <math>
\deg(g h) = \deg(g) + \deg(h) 
</math>
}}

{{Math proof|title=Proof|proof=
We prove it for <math>A_1</math>, as the <math>A_n</math> case is similar.

The first relation is by definition. The second relation is by the general Leibniz rule. For the third relation, note that <math>\deg(g h) \leq \deg(g) + \deg(h)</math>, so it is sufficient to check that <math>gh</math> contains at least one nonzero monomial that has degree <math>\deg(g) + \deg(h)</math>. To find such a monomial, pick the one in <math>g</math> with the highest degree. If there are multiple such monomials, pick the one with the highest power in <math>q</math>. Similarly for <math>h</math>. These two monomials, when multiplied together, create a unique monomial among all monomials of <math>gh</math>, and so it remains nonzero.
}}

{{Math theorem
| name = Theorem
| math_statement = <math>A_n</math> is a [[Simple algebra|simple]] [[Domain (ring theory)|domain]].{{sfn | Coutinho | 1995 | p=16}}
}}
That is, it has no [[Ideal (ring theory)|two-sided nontrivial ideals]] and has no [[zero divisor]]s.
{{Math proof|title=Proof|proof=
Because <math>\deg(gh) = \deg(g) + \deg(h)</math>, it has no zero divisors.

Suppose for contradiction that <math>I</math> is a nonzero two-sided ideal of <math>A_1</math>, with <math>I \neq A_1</math>. Pick a nonzero element <math>f \in I</math> with the lowest degree. 

If <math>f</math> contains some nonzero monomial of form <math>xx^m\partial^n = x^{m+1} \partial^n</math>, then 
<math display=block>
[\partial, f] = \partial f - f \partial
</math>
contains a nonzero monomial of form 
<math display=block>
\partial x^{m+1} \partial^n -  x^{m+1} \partial^n \partial = (m+1) x^m \partial^n.
</math>
Thus <math>[\partial, f]</math> is nonzero, and has degree <math>\leq \deg(f)-1</math>. As <math>I</math> is a two-sided ideal, we have <math>[\partial, f] \in I</math>, which contradicts the minimality of <math>\deg(f)</math>.

Similarly, if <math>f</math> contains some nonzero monomial of form <math>x^m\partial^n\partial</math>, then <math>[x, f] = xf - fx</math> is nonzero with lower degree.
}}