Let be a ring and let be an integer. The -th Weyl algebra over is defined as follows. First we define For we define to be the ring of polynomials in variables with coefficients in and subject to the relations
for all where is the Kronecker delta. We will assume that every element of commutes with all variables and So, for example, is the ring generated by with coefficients in and subject to the relation An element of is in the form . It is not hard to prove that the set of monomials in the form
is an -basis for Also note that If is a domain, then is a domain too. It is straightforward to show that if is a field of characteristic zero, then is a simple noetherian domian.
Linear automorphisms of Now suppose that is field. Define the map on the generators by We would like to see under what condition(s) becomes a -algebra homomorphism. Well, if is a homomorphism, then since we must have
Simplifying the above will give us and since we get We can now reverse the process to show that if then is a homomorphism. So is a homomorphism if and only if But then the map defined by
will also be a homomorphism and Thus is an automorphism of if and only if In terms of matrices, the matrix defines a linear automorphism of if and only if
We can extend the above result to Let a matrix with entries in Let and define the map by Clearly is a -algebra homomorphism if and only if satisfy the same relations that do, i.e.
for all Let be the identity matrix and let be the zero matrix. Let Then, in terms of matrices, becomes
Clearly if satisfies then is invertible and thus will be an automorphism. So is in fact the necessary and sufficient condition for to be an automorphism of
A matrix which satisfies is called symplectic. See that if is a matrix, then is symplectic if and only if