Heisenberg algebra

Natural Heisenerg algebras

Let be a nondegenerate finite-dimensional quadratic space over 1 endowed with an abelian bracket, and and denote untwisted and twisted affine Lie algebras respectively, which are each triangular. Then the commutator ideals2

define Heisenberg algebras, called the - and -natural Heisenberg algebras3 induced by . #m/def/lie For or , we have the commutation relations

for and .4

Modules

Heisenberg modules

Let or and . The -Heisenberg module is then isomorphic as a -graded vector space

setting and denoting we have

for and . In addition is a -graded irreducible module where acts as a degree operator, and if , act trivially.5

Triangular modules

Defining the linear form

Then then the triangular module and Heisenberg module as -modules. Thus given the involutive antiautomorphism

there exists a unique contravariant form

with the properties

for , , and . The first conditions implies that if are homogenous of different degrees.6

Natural Heisenberg module

Let or and let denote the - and -module . Let , , and be a 1-dimensional -module defined by

We define the natural -module to be the tensor product of graded vector spaces7

with the -action given by the Tensor product of Lie algebra representations

In the case or 8 this amounts to a shifted graded module by . If

Conventionally we will take9

Virasoro representation

Letting be an orthonormal basis of , extending the ground field if necessary, we define the following (basis-independent) operators on

where

Then

is a graded representation of the Virasoro algebra .10

Proof

For , , and , it follows from the commutation relations on that

Hence for with and ,

The only remaining case is essentially that and , since the case may be reduced to either zero or another case by the alternating property. In this case, from the expression

it follows

where is some constant which we get from the term and reversing the order of some where necessary.11

We will compute using the application of of to a vacuum vector of , e.g. . We note the following facts: From above there is a unique contravariant form such that and

for , , and . It follows by the definition of that

for and . We also have

Now consider the case . Then

where we have used an Iverson bracket and the fact that for we can commute the positively graded operators to annihilate . Now consider each of the terms

where , so that

We have the following cases

  • for , since we may again commute and annihilate;
  • for whence ;12
  • otherwise

Thus

as required.

In fact, the choice and being trivial uniquely determine the central term in the commutation relations for the Virasoro algebra. Letting , the above choice of gives

for . The -eigenvalue of a homogenous element is termed the weight and denoted so that

and

See also

#state/tidy | #lang/en | #SemBr

Footnotes

  1. ↩

  2. Note that the even subspace of under is trivial, so the decomposition of matches the above. ↩

  3. This is my own terminology, FLM do not give a name for these constructions. ↩

  4. 1988. Vertex operator algebras and the Monster, §1.7, pp. 24–25 ↩

  5. This works because . ↩

  6. 1988. Vertex operator algebras and the Monster, §1.8, pp. 29–30 ↩

  7. 1988. Vertex operator algebras and the Monster, §1.9 pp. 34–35 ↩

  8. Since in either of these cases carries a trivial representation of . In fact, FLM only define this way for and do not use a tensor product construction for . ↩

  9. 1988. Vertex operator algebras and the Monster, §1.9, p. 41 ↩

  10. 1988. Vertex operator algebras and the Monster, §1.9 pp. 35–42. This seems to be the first theorem in the book. ↩

  11. It should already be clear at this point that exhibit a central extension of the Witt algebra, which must be equivalent to the Virasoro algebra. It turns out that the Virasoro algebra, and these operators, are engineered precisely so that gives the right coëfficient. ↩

  12. In this calculation, keep the canonical realization of the Heisenberg commutation relations in mind. ↩