Affine Lie algebras of

Let with its Chevalley basis and let be the suggestively named1 involutive isometries of defined by #m/def/lie

and let . Furthermore we let

We consider the untwisted or twisted affine Lie algebra 2 which have bases

The 1-dimensional subalgebra generates the natural Heisenberg algebras

as Lie subalgebras and we have3

Isomorphism of extended Lie algebras

and are isomorphic (but not as graded Lie algebras) under

There also exist grade-preserving isomorphisms between for .

Via formal series

Taking a formal series approach on , the exact characterization varies with .

For

In we define the formal sums

the commutation relations are more conveniently expressed as

where .

For

In we define the formal sums

the commutation relations are more conveniently expressed as

For

In we define the formal sums

the commutation relations are more conveniently expressed as

and we also have

Representations

#state/develop | #lang/en | #SemBr

Footnotes

  1. For , we can conjugate by Pauli matrices for the same result. ↩

  2. FLM use and ↩

  3. 1988. Vertex operator algebras and the Monster, §3.1, pp. 62–67 ↩