Weyl algebra
Abstract definition
Let be a field and be an -vector space with basis
, where is some non-empty
index set
. Let be the tensor algebra of and let
be the ideal in generated by the set
where
is the Kronecker delta symbol. Then the quotient
is the
-th Weyl algebra.
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.
Title | Weyl algebra |
---|---|
Canonical name | WeylAlgebra |
Date of creation | 2013-03-22 15:27:19 |
Last modified on | 2013-03-22 15:27:19 |
Owner | GrafZahl (9234) |
Last modified by | GrafZahl (9234) |
Numerical id | 5 |
Author | GrafZahl (9234) |
Entry type | Definition |
Classification | msc 16S36 |
Classification | msc 16S32 |
Related topic | HeisenbergAlgebra |
Related topic | UniversalEnvelopingAlgebra |