A more concrete definition
If the field has characteristic zero we have the following more concrete definition. Let be the polynomial ring over in indeterminates labeled by . For any , let denote the partial differential operator with respect to . Then the -th Weyl algebra is the set of all differential operators of the form
where the summation variable is a multi-index with entries, is the degree of , and . The algebra structure is defined by the usual operator multiplication, where the coefficients are identified with the operators of left multiplication with them for conciseness of notation. Since the derivative of a polynomial is again a polynomial, it is clear that is closed under that multiplication.
The equivalence of these definitions can be seen by replacing the generators with left multiplication by the indeterminates , the generators with the partial differential operator , and the tensor product with operator multiplication, and observing that . If, however, the characteristic of is positive, the resulting homomorphism to is not injective, since for example the expressions and commute, while and do not.
|Date of creation||2013-03-22 15:27:19|
|Last modified on||2013-03-22 15:27:19|
|Last modified by||GrafZahl (9234)|