Lie algebras MOC

Twisted affine Lie algebra

Let be a quadratic Lie algebra over with a symmetric -invariant bilinear form , and be an involutive isometry of . The corresponding twisted affine Lie algebra and extended twisted affine Lie algebra are generalizations of the corresponding untwisted counterparts. motivate

Construction

Let be a Lie algebra over with an involution , and let be a -invariant bilinear form which is also invariant under in the sense that

for all .¹ Then is -graded into orthogonal² even and odd subspaces

Let be the -graded algebra of Laurent polynomials in indeterminate and be its degree derivation. Constructing

with the same bilinear product defined for the (untwisted) affine Lie algebra gives a Lie algebra. Defining the involution on we extend to the following involution on

The twisted affine Lie algebra associated with , , and is the even subalgebra of under lie

As in the untwisted case, extends to a derivation of

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

Further generalizations

This may be generalized to automorphisms of any finite order.

Properties

1. In case , these constructions yield their untwisted counterparts.

Functoriality

Let denote the category where an object is a Quadratic Lie algebra with an involutive isometric, and a morphism is an isometric homomorphism of Lie algebras such that . Then this constructions forms a functor .

Particular examples

• Affine Lie algebras of sl_2

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.6, p. 19–20 ↩

2. In the sense . ↩