Heisenberg Lie Algebra Example
The Heisenberg Lie algebra is defined by the commutation relations:
One representation is to define the operators b, in terms of the dummy variables x as:
and b0 = 0.
This can be made into a vertex algebra by the definition:
where :..: denotes normal ordering (i.e. moving all derivatives in x to the right). Thus Y(1,z) = Id.
T is defined by the conditions T.1 = 0 and
Setting
it follows that
and hence:
The vertex operators may also be written as a functional of a multivariable function f as:
if we understand that each term in the expansion of f is normal ordered.
Read more about this topic: Vertex Operator Algebra
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