Formal Derivatives, Weyl Algebra

Aura Polynomials

Let R be a ring that may or may not be commutative, and let y be an indeterminant. The aura polynomials R[y] call upon a function δ from R into R, such that y*a = a*y+δ(a). Thus the coefficients can always be moved to the left. A "normalized" term in an aura polynomial is something in R times a power of y.

If δ(R) = 0 we have the traditional polynomial ring R[y].

By the distributive law, y*(a+b) = ya+yb. Therefore δ(a+b) = δ(a)+δ(b). In other words, δ is a group homomorphism that respects addition in R.

By associativity, (y*a)*b = y*(a*b). Therefore δ(ab) = δ(a)*b+a*δ(b).

Conversely, assume δ is a function from R into R with these two properties. Build the aura polynomials as described above, and prove the result is a ring. The properties of δ make multiplication associative and distributive, thus producing a ring. I'll leave the details to you.

Noetherian

If R is noetherian, is the aura ring R[y]/δ also noetherian? I believe so, but I don't have a solid proof. Perhaps some variation of hilbert's basis theorem will do the trick.

Weyl Algebra

Let R[x,y] be the ring of polynomials in x and y, with coefficients in R, where x and y do not commute. However, R commutes past x and y. Mod out by the relation yx = xy+1 to get the weyl (biography) algebra over R.

A polynomial is in normal form if each term is a coefficient in R, times a power of x, times a power of y. When two such terms are multiplied, normalize the result by pulling powers of y past powers of x. Thus xy2 times 3xy = 3x2y3 + 6x2y2.

Imagine multiplying y by p(x), where p is a polynomial in R[x]. By the distributive law, we can approach this term by term. Start with yxn and replace yx with xy+1 n times. The result is xny + nxn-1. Therefore yp = py + p′, where p′ is the formal derivative of p with respect to x.

View this as an aura polynomial in R[x][y], where δ implements differentiation. If p and q are two polynomials in R[x], δ(p+q) = δ(p) + δ(q), and δ(pq) = δ(p)q + pδ(q). These are the standard rules for differentiating a sum and product. Since δ has the required properties, the result is a valid ring. The weyl algebra is indeed a ring.

Note that R is a subring that commutes with x and y. If R is commutative, the weyl algebra meets the technical definition of an R algebra.

Ideals and Partial Differentiation

There is a symmetry here. Normalize ynx and get xyn + nyn-1. Thus a polynomial in y is differentiated, when multiplied by x on the right, just as a polynomial in x is differentiated, when multiplied by y on the left.

Let H be an ideal in the weyl algebra. Let t be a term of a polynomial in H. Note that yt-ty = t∂x, and xt-tx = t∂y. The ideal H is closed under partial differentiation.

Simple Rings

Let R be a simple ring of characteristic 0. Consider a nonzero ideal H in the weyl algebra. Let p be a nonzero polynomial in H of least degree. Both p∂x and p∂y produce polynomials of lesser degree. By minimality, these are 0, hence p is a constant. The contraction of H to R yields a nonzero ideal in R, which is all of R. Thus H contains 1, and is the entire algebra. Therefore the weyl algebra, like R, is a simple ring.

Notice the chain of descending left ideals generated by the powers of y. If the simple ring were left artinian it would be the n by n matrices over a division ring. Obviously this is not the case. This is an example of a noetherian simple ring that is not artinian.

If R has characteristic p, multiply by xp on the right. This differentiates, with respect to y, p times. At some point a multiple of p is introduced as a coefficient, and the result is 0. Therefore xp commutes with everything, and generates a proper ideal. The same holds for yp. This is valid even if R is simple, e.g. Zp.

Domain

Let R be a domain, and multiply two polynomials together. Concentrate on the terms of highest degree. After normalization, the result is what you would expect if x and y commute, plus some other terms, i.e. derivatives, of lower degree. The result is nonzero, and the weyl algebra is a domain.

Note that x is not invertible, hence the result is not a division ring.

Higher Order Weyl Algebras

Adjoin a set of indeterminants x1 x2 x3 etc, and a corresponding set of indeterminants y1 y2 y3 etc, to R. Let indeterminants commute past each other, except for yixi, which equals xiyi+1. Multiply by yi on the left, and introduce a term that is the partial derivative with respect to xi. Multiply by xi on the right, and introduce a term that is the partial derivative with respect to yi. The previous theorems generalize to this higher order weyl algebra.