Lie algebras MOC

Affine Lie algebra

Let be a quadratic Lie algebra with a symmetric -invariant bilinear form . The corresponding affine Lie algebra is a certain graded ^central of the Loop algebra .¹ Thence one can construct the corresponding extended affine Lie algebra by adjoining the degree derivation. A generalization is the Twisted affine Lie algebra.

Construction

Let be an algebra over with some bilinear form .² Further let be the ^degreeDerivation on , and construct the vector space

with the bilinear product defined by the conditions

the latter being equivalent to

Then is a Lie algebra, called the affine Lie algebra associated with and , lie iff is a Lie algebra and is a symmetric -invariant bilinear form, and we have the ^central

Proof

First note the bracket on is alternating iff that on is. Let . Then the ^Jacobi on is equivalent to

which holds iff the Jacobi identity holds for along with the identity

for all such that . The latter is equivalent to the bilinear map being symmetric and -invariant, as can be shown by varying .

We extend to a degree derivation of by

so that homogenous subspaces are the eigenspaces of . One obtains the extended affine Lie algebra associated with and by adjoining the degree derivation lie

giving the -gradation³

Properties

• If is an Abelian Lie algebra, the (extended) affine Lie algebra has a triangular decomposition
• See Formal series over an (un)twisted affine Lie algebra

Functoriality

These constructions may be extended to functors from Category of quadratic Lie algebras to [[Category of graded Lie algebras|]], so that if is an isometric Lie algebra homomorphism, one defines by

and similarly

We also have a natural inclusion .

Particular affine Lie algebras and related constructions

• Natural Heisenberg algebras
• Affine Lie algebras of sl_2

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Footnotes

1. This definition, following FLM, is much more general than the traditional one, which restricts to be a semisimple Lie algebra. ↩

2. 1988. Vertex operator algebras and the Monster, §1.6, p. 17ff. ↩

3. We identify with . ↩